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Generally covariant actions for systems of multiple DO-branes Ling, Henry Ho-Kong 2007

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Generally covariant actions for systems multiple DO-branes by Henry Ho-Kong Ling B.Sc, The University of Calgary, 1999 M.Sc, The University of British Columbia, 2001 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F THE REQUIREMENTS FOR T H E DEGREE OF Doctor of Philosophy in The Faculty of Graduate Studies (Physics)  The University Of British Columbia September 16, 2007 © Henry Ho-Kong Ling 2007  of  11  Abstract This thesis focuses on understanding how general coordinate invariance can be incorporated into effective actions for systems of many DO-branes coupled to bulk supergravity fields. We present progress in two special cases. First, we discuss the implementation of covariance under arbitrary spatial diffeomorphisms. A method of constructing actions with manifest covariance under these diffeomorphisms is developed. While the matrix DO-branes coordinates transform in a complicated manner under spatial diffeomorphisms, we find that it is possible to replace these with matrix-valued fields in space with a simple vector transformation law. Using this vector field, we define a distribution function that serves as a matrix generalization of the delta function, and which describes the location of the DO-branes. The covariant Lagrangians then take the form of an integral over space of a scalar built from the various fields times the matrix distribution function. Next, we approach the problem of implementing covariance under coordinate transformations that mix the space and time directions. As a first step towards understanding this problem, we consider in detail the simpler case of incorporating Poincare invariance into actions for multiple DO-branes in Minkowski space. We find evidence for a non-trivial Lorentz transformation rule for the matrix DO-brane coordinates by using the Poincare algebra as a guiding consistency condition. We determine the necessary conditions that must be satisfied by the leading term of any Poincare invariant action, and find an implicit method of constructing a Poincare invariant completion of any such leading term. The approach is based on using matrix-valued Lorentz covariant fields defined on space-time, built from the matrix DO-brane coordinates.  iii  Contents Abstract  ii  Contents  iii  Acknowledgements  vi  Co-Authorship Statement  vii  1  Introduction 1.1 Orientation 1.2 D-brane actions 1.2.1 Basic characteristics of D-branes 1.2.2 Effective actions for a single D-brane 1.2.3 Effective actions for multiple D-branes 1.3 Directions of research  2  Invariance under spatial coordinate transformations 21 2.1 Overview 21 2.2 Transformation rule for matrix coordinates 23 2.2.1 A first attempt 24 2.2.2 Non-trivial dependence on the metric 26 2.2.3 A consistent transformation law 29 2.3 A matrix-valued vector field 30 2.3.1 Describing a point by a vector field 30 2.3.2 Matrix vector field from a matrix transformation law . 33 2.3.3 Matrix transformation law from a matrix vector field . 35 2.3.4 A differential equation for V 37 2.3.5 Solving the constraints 38 2.4 Generally covariant actions 39 2.4.1 The matrix distribution function 40 1  1 1 5 6 8 13 19  Contents 2.4.2 2.4.3 2.4.4 2.4.5  2.5  3  4  iv  A large class of generally covariant actions Expansion i n powers of X How large is the large class? Relation to the base-point independence approach of de Boer and Schalm 2.4.6 Examples A d d i t i o n a l constraints ' 2.5.1 Emergence of the geodesic equation 2.5.2 T - d u a l i t y 2.5.3 Geodesic distance criterion 2.5.4 Agreement w i t h known results 1  41 41 42 44 46 48 48 48 49 50  Invariance under Poincare transformations 3.1 Overview 3.2 Poincare transformations for multiple DO-branes 3.2.1 Transformation rules for a single brane 3.2.2 T h e Poincare algebra as a consistency condition . . . . 3.2.3 Order-by-order solution 3.2.4 E x p a n d i n g i n the number of commutators 3.2.5 Uniqueness of the transformation rule up to field redefinitions 3.3 Poincare invariant actions for multiple DO-branes . . . . . . . 3.3.1 Constraints for leading order invariant terms 3.3.2 Order-by-order construction of invariant actions . . . . 3.3.3 N o n - A b e l i a n generalization of the relativistic kinetic term 3.4 Covariant objects 3.4.1 A space-time vector field 3.4.2 Generalization to the case of multiple DO-branes . . . . 3.4.3 A covariant m a t r i x distribution 3.5 Manifestly Lorentz invariant DO-brane actions 3.5.1 T h e most general Poincare invariant action 3.5.2 Examples 3.6 Lorentz covariant currents  52 52 54 54 56 58 58  Summary and outlook  79  Bibliography  59 61 62 63 65 68 68 69 70 72 72 74 76  82  Contents  v  A A p p e n d i x to chapter 2 A . l Spatial metric conventions and useful formulae A.2 Solving the constraints for V A . 3 A technical result about commutator expressions  92 92 93 97  1  B  A p p e n d i x to chapter 3 100 B. l A consistent transformation law to sixth order in X 100 B.2 Characterization of Galilean invariant non-Abelian actions . . 102 B.3 The matrix space-time vector-field V*(y) 105 1  vi  Acknowledgements First, I acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC), the University of British Columbia, and the C. D. Howe Memorial Foundation. I owe my supervisor, Mark Van Raamsdonk, many thanks. He has generously shared his ideas, guided my research, encouraged me to follow my own directions, and helped me when I got lost. I have learned much about physics and research from him. I also thank Gordon Semenoff for teaching me through courses as well as by personal example, and for encouraging me throughout my time as a graduate student. I thank Dominic Brecher and Kazuyuki Furuuchi for an enjoyable collaboration, leading to my first ever publication. I also wish to thank my friends and collaborators Greg van Anders and Brian Shieh, for all of the interesting discussions, and especially for our last project together. Thanks to Tami Pereg-Barnea and Karene Chu for supporting me at crucial moments. To Paige Demong, thanks for your many years of friendship. I thank my former teachers, especially J. S. Murphree, David Hobill, Eric Donovan, Alfredo Louro, Debbie Hearn, and R.B. Hicks, as well as all of my friends in the physics department at the University of Calgary. They were an important part of my memorable undergraduate education in physics. I especially thank my family for, well, basically everything. Mom and Dad, Grandma, Elizabeth, Arthur, Amy, Joseph, Brittany, Kristen, Yvonne, Matthew, Cynthia, Ethan, and Deborah. And to my love Wan Yan. Thank you for the constant love, encouragement and support.  Vll  Co-Authorship Statement Chapter 2 of this thesis and the associated appendix is based on the paper "Generally covariant actions for multiple D-branes" with Brecher, Furuuchi, and Van Raamsdonk [1]. There are some differences in exposition, but the results are unchanged. Chapter 3 and the associated appendix is based on the paper "Poincare invariance in multiple D-brane actions" with Brecher, Koerber, and Van Raamsdonk [2]. The main difference between this chapter and the published work is the omission of certain results (based on a method of expanding in the number of commutators of the matrices X\ but working to all orders in X ) obtained entirely by my collaborators. This has also been indicated at the relevant points in the text. Some unpublished material has been included in the appendix to this chapter. A l l collaborators on the two papers indicated above are senior, and the work presented in this thesis has not been submitted for any other degree or diploma. Additional work has been published as part of the PhD program, but has not been presented in this thesis. These are "Little string theory from a double-scaled matrix model" with Mohazab, Shieh, van Anders, and Van Raamsdonk [3], and "Little string theory from double-scaling limits of field theories" with Shieh and van Anders [4]. %  1  Chapter 1 Introduction 1.1  O r i e n t a t i o n  This thesis is about certain aspects of the actions which govern the dynamics of D-branes in string theory. Before stating precisely the problem that we wish to consider, we first describe the setting of this discussion (namely string theory), introduce the relevant characters (the D-branes) and explain the motivation of our study. String theory is based on a seemingly simple idea: the different types of elementary particles that we observe in nature are actually different vibrational states of a single type of object, the string. The ambitious goal is to use this idea to achieve a unified description of matter and all forces, including gravity, within a single self-consistent framework that satisfies the principles of quantum mechanics and relativity [5, 6, 7, 8, 9]. Twenty years ago, an extremely brief summary of the state of knowledge in string theory might have included the following points. There appears to be five mathematically consistent theories describing strings propagating and interacting in Minkowski space. All of these theories include closed strings (strings without ends, forming closed loops). Gravity emerges naturally in each of these theories, because the vibrational states of a closed string include a state with zero mass and spin two. In other words, when the closed string is vibrating in this state, it has the properties of the graviton, the mediator of the gravitational force. Each of the five consistent string theories exhibits a symmetry that relates the bosonic string states to the fermionic string states, known as supersymmetry. Accordingly, these five theories are sometimes called superstring theories. Curiously, the dimension of space-time is not an input in these theories, but is determined by mathematical self-consistency to be ten. To reconcile the appearance of spatial extra dimensions with the fact that we ordinarily experience three spatial dimensions only, a procedure called compactification is invoked. The basic idea is that the six extra dimensions  Chapter  1.  Introduction  2  have not yet been observed because they are curved up into a much smaller size t h a n the distances at which we can currently experimentally probe. Interestingly, certain methods of compactifying one of the five superstring theories produces models that resemble semi-realistic, supersymmetric extensions of the standard model of particle physics, although no models have yet been found which are consistent w i t h a l l features of the world we observe. Our modern day picture of string theory is much richer than this twenty year-old story. T h i s is largely due to the discovery, over a decade ago, that string theory is more than a theory of strings only. In fact, there exists a host of other basic objects that resemble extended objects of various dimensionalities. These objects can be regarded as non-perturbative states of string theory. B y this, we mean that such states are not simply excitations of one or any finite number of strings. In the language of field theory, one might refer to these states as solitons; i n the language of condensed matter physics, one might refer to these as collective excitations. O u t of a l l the stringy solitons, those known as D-branes have played an especially important role i n many of the major developments i n string theory over the past decade. Part of the reason that D-branes have taken such a prominent place is that they have a remarkably simple description: a Dp-brane can be defined as a p-dimensional hypersurface i n space on which the ends of open strings are constrained to move [10]. One should not however think of D-branes as fixed defects i n space. Instead, they are dynamical objects, carrying mass as well as a certain type of conserved charge. In fact, the dynamical behaviour of D-branes is encoded i n the theory of open strings whose ends are attached to the D-brane. We illustrate the fundamental role that D-branes have played i n string theory w i t h a few famous examples. T h e first example is development of the M a t r i x theory description of M-theory. Let us explain briefly. T h e current picture of string theory is that the five seemingly distinct superstring theories are i n fact different limits of one unique underlying theory. Even more surprisingly, there also appears to be a limit of this underlying theory i n which space-time is eleven-dimensional, and which cannot be described as a theory of weakly interacting strings. T h i s is sometimes called the M-theory limit of string theory. A t the time that this picture was first proposed, there was very little knowledge of the nature of M-theory, beyond the statement that it has a classical description i n terms of a field theory known as eleven-dimensional supergravity. Shortly after the discovery of D-branes, it was conjectured that the low energy dynamics of a system of many DO-branes actually provides  Chapter  1.  Introduction  3  a complete description of M-theory [11]. T h i s is now known as the M a t r i x theory conjecture [12]. T h e second example concerns the study of string theory as a quantum theory of gravity. Black holes are one of the types of physical systems for which we expect that a complete understanding of the physics requires a consistent quantum theory o f gravity. We might naturally wonder if string theory is able to provide any insight into the quantum mechanics of black holes. T h e discovery of D-branes made possible an impressive derivation of the Bekenstein-Hawking formula for black hole entropy. T h e derivation is essentially based on the hypothesis (supported by explicit computations) that certain types of black holes can be equivalently described by systems of many D-branes. A derivation of the black hole entropy formula is then obtained by directly counting the relevant states of the system of D-branes, thus providing an understanding of black hole entropy based on conventional statistical mechanical ideas [13, 14]. T h e idea that systems of many D-branes can be equivalently described by some non-trivial space-time geometry was developed much further, beyond the applications to the quantum mechanics of black holes. Ultimately, the study of multiple D-branes along these lines led to the proposal of the gauge/gravity duality [15, 16]. In its most encompassing form, this is the conjecture that every non-Abelian gauge theory is equivalent to some quant u m theory of gravity i n the form of a string theory. T h e development of the gauge/gravity duality has had a great impact on string theory, and has been one of the dominant directions of research i n the field for the past decade. For a recent general review of this subject, see [17]. T h e three examples that we have just considered might be regarded as part of the purely formal development of string theory. We should note that the discovery of D-branes has also greatly influenced the way i n which string theorists have thought about how to connect the theory to the real world. D-branes have been used to devise new approaches to compactification of the extra dimensions. These involve the idea that our four-dimensional world is located at the intersection of systems of multiple coinciding D-branes [18]. T h i s approach has yielded models which resemble supersymmetric grand unified extensions of the standard model of particle physics, w i t h certain features that differ qualitatively from the models obtained from older methods of compactification. A t t e m p t s have also been made to use D-branes to establish contact between string theory and cosmology. For instance, brane inflation is a specific  Chapter  1.  Introduction  4  string theory realization of the inflationary scenario in cosmology [19]. Dbranes are involved in two aspects of the brane inflation scenario. The first has to do with the mechanism of inflation. In this case, the inflation is driven by a scalar field that has a geometric interpretation as the separation between a D3-brane and an anti-D3-brane. The potential of the inflation is the interaction energy due to the mutual attraction of the D3-brane and the anti-D3-brane. The second has to do with the production of Dl-branes at the end of inflation. This point has led to some interest in the possibility of Dl-branes as cosmic topological defects (see for example [20]). Finally we also mention the use of the gauge/gravity duality as a tool for understanding strongly coupled gauge theories. For example, there have been recent efforts towards obtaining a string theoretical description of quantum chromodynamics, the part of the standard model of particle physics that deals with the strong nuclear interactions (see for example [21]). The role that Dbranes play in these investigations is to introduce quarks into the gauge theories. This is a rapidly growing subject, with many related directions of research, such as the use of these methods to understand certain properties of the strongly interacting quark gluon plasma produced by relativistic heavy ion collisions (see for example [22]). Having elaborated on the importance of D-branes, we now discuss the specific aspect of D-brane physics that we are interested in, namely the actions that govern the dynamics of D-branes. As we have mentioned earlier, the dynamical behaviour of a D-brane is encoded by the theory of open strings with endpoints attached to the D-brane. However, at low energies it is useful and sometimes advantageous to describe the D-branes in more conventional terms, using an action. Such actions have been exploited in many applications of D-branes including, for instance, the last two examples of the uses of D-branes that we discussed above. The action that governs the behaviour of a single D-brane is relatively well-understood and has been widely used. On the other hand, while much progress has been made towards understanding effective actions for system of multiple D-branes, it is fair to say that the state of knowledge is rather incomplete. One basic issue that remains unresolved is the problem of implementing local symmetries, particularly general covariance. Let us explain more precisely what is meant by this. As we have noted earlier, the five superstring theories all contain gravity. In the low energy field theory, gravity is represented by the space-time metric. When we write down an action for D-branes, the D-brane degrees of freedom must therefore be coupled to this  Chapter  1.  Introduction  5  space-time metric. In formulating a gravitational theory in terms of metric degrees of freedom, it is important for the action to have the same form for whatever system of space-time coordinates we choose to adopt. This property of coordinate independence is often called general covariance. There are several perspectives on why general covariance is an important property to demand. One is that the behaviour of a physical system should be independent of the coordinates that we choose to describe the system. Another is that general covariance is the gravitational version of gauge invariance, necessary to remove the unphysical degrees of freedom of the space-time metric. In any case, it is important that this symmetry be incorporated into the actions for D-branes in order to ensure that the D-branes are coupled to gravity in a consistent manner. This thesis is devoted to the problem of implementing general covariance in D-brane effective actions, with specific focus on systems of multiple D0branes. One of the main motivations of this work is to explore whether there exists a formalism that keeps the general covariance manifest and in which multiple D-brane actions can be expressed simply. We focus on system of DO-branes because many of the essential difficulties are present in this case already, and a thorough understanding of it should provide insight to the general case of higher dimensional D-branes. In the following section, we will review the basics of the current understanding of D-brane actions, and in the process explain exactly what are the essentially difficulties that we have just alluded to. The final section of this chapter summarizes once more the relevant issues and the precise questions that are being addressed in this thesis.  1.2  D - b r a n ea c t i o n s  We first recall the basics of D-branes and D-brane actions in more detail. The following discussion presupposes basic knowledge of string theory at a technical level. The literature on D-branes is vast and extensive, here we can only touch on a few basic points. For systematic introductions to string theory and D-brane physics, we refer the reader to the textbooks [23, 7, 8], and the review articles [24, 25]. Throughout, we use string units where h = c = 2ira' = 1.  Chapter  1.2.1  1.  6  Introduction  Basic characteristics of D-branes  We consider D-branes in the context of ten-dimensional type IIA and type IIB string theory. We first recall some facts about the corresponding low energy supergravity theories. The effective field theory that describes type IIA closed string theory at low energies is type IIA supergravity. This theory has 32 space-time supersymmetries, with the bosonic field content 9  F  L  U  ,  0,  C$\  C%  (1.1)  These are, respectively, the space-time metric, the Neveu-Schwarz (NS) twoform, the dilaton, the Ramond-Ramond (RR) one-form potential, and the R R three-form potential. The effective field theory that describes type IIB closed string theory at low energies is type IIB supergravity. This theory also has 32 space-time supersymmetries. In the bosonic sector, the basic difference between the IIA theory and the IIB theory resides in the types of allowed R R potentials. The bosonic field content of type IIB supergravity is <j>, C<°\ C<J, C$ .  9 l i v i  pc  (1.2)  The various R R potentials are generalizations of the usual gauge potential in textbook electrodynamics. For example, an object that carries electric R R four-form charge should be extended in three spatial dimensions, and the coupling is given by integrating over the four-dimensional world-volume of the object in ten-dimensional space-time. To describe the coupling to an object carrying magnetic charge, we need to introduce dual potentials. For example, the dual to the one-form potential is a sevenform potential , such that the field strengths are related by Hodge duality dC^ = *dC^. A n object that carries magnetic R R one-form charge has six spatial dimensions, and the coupling is given by integrating over the seven-dimensional world-volume in space-time. The basic carriers of R R charge [26] are the Dirichlet (D)-branes [10]. In the perturbative formulation of string theory, a D-brane is defined simply as a hypersurface in space-time on which the ends of open strings are constrained to move [10, 27]. We discuss this definition in more detail. Take 1  We have neglected the nine-form potential of type IIA string theory and the ten-form potential of type IIB string theory. These are not associated with propagating degrees of freedom, but do play an important role in the theory. See [8] for a more complete discussion. J  Chapter  1.  Introduction  7  the background to be ten dimensional M i n k o w s k i space w i t h coordinates y , fi = 0 , . . . , 9. T h e perturbative string theory is defined by a two-dimensional supersymmetric conformal field theory on the string worldsheet. T h e worldsheet field content includes a set of scalars that represent the embedding coordinates of the string i n space-time. To specify the open string theory, boundary conditions for these fields need to be chosen, which determines the behaviour of the string end points. T h e canonical example of a D-brane is given by imposing N e u m a n n boundary conditions {d Y^ = 0) on the first p + 1 open string embedding coordinates and Dirichlet boundary conditions {Y^ = 0) on the remaining coordinates. T h i s constrains the string end points to move along a flat p +1 dimensional hyperplane i n space-time, a Dp-brane. Quantizing the open string w i t h these boundary conditions, we obtain a set of massless modes, as well as an infinite tower of massive modes w i t h masses proportional to the string scale. O n l y the massless modes are relevant at low energies. T h e massless bosons can be described by a gauge field A (y ) and a set of scalars 4>i{y ). Here the index a takes values from 0 to p and the index % takes values from p + 1 to 9. These fields depend on only the first p + 1 space-time directions, and as a result, they can be thought of as naturally defined on the Dp-brane world-volume. There are also a set of world-volume fermions. Altogether, the massless field content can be obtained from dimensional reduction of the ten-dimensional J\f = 1 t / ( l ) supersymmetric gauge multiplet to p + 1 dimensions. A l t h o u g h we have described the Dp-brane as a fixed hyperplane, it should not be thought of as a rigid non-dynamical object. T h e description we have just given should be understood i n the context of perturbative expansions. T h e hyperplane represents a background configuration of the D-brane about which it dynamically fluctuates. T h e fluctuation modes are the modes of the attached open strings. In particular, the massless scalars ^i(y ) describe the transverse fluctuations i n the shape of the Dp-brane. We w i l l return to this in the discussion of D-brane effective actions. Dirichlet boundary conditions arise naturally by considering the action of T - d u a l i t y i n a theory w i t h open strings [27, 28, 29]. Compactify one direction along which the open strings satisfy N e u m a n n boundary conditions. N o w send the radius of compactification to zero R —* 0. In the T - d u a l description, the compactification radius goes to infinity l/(2irR) —• oo and the open strings satisfy Dirichlet boundary conditions. A s a result, T - d u a l i t y along a direction parallel to a D-brane produces a D-brane w i t h one less dimension. O n the other hand, T - d u a l i t y along a direction perpendicular to a D-brane M  n  a  a  a  a  Chapter  1.  8  Introduction  produces a D-brane w i t h one greater dimension. In this manner, we can start w i t h any Dp-brane and T-dualize until we obtain a space-filling D9-brane, which corresponds to open strings w i t h Neumann boundary conditions i n a l l directions. T h i s is one way to understand why the massless spectrum of a Dp-brane can be obtained from dimensional reduction of a ten-dimensional Af = 1 A b e l i a n gauge multiplet. 2  A Dp-brane naturally couples to the R R p + 1 form potential [10, 30, 31]. We w i l l discuss this further i n the next section. T o match the types of R R potentials appearing i n type I I A / I I B string theory, we have Dp-branes of all even p i n type I I A string theory and Dp-branes of a l l odd p i n type I I B string theory. T h i s is consistent w i t h the action of T - d u a l i t y on D-branes, and the fact that T - d u a l i t y exchanges type I I A and type I I B closed string theory [10].  1.2.2  Effective actions for a single D-brane  The action for a single D-brane has been well-studied, and here we review some selected results. For a more comprehensive review, see for example [32, 33]. For the most part, we ignore the world-volume fermions, and focus attention on the bosonic part of the action. T h i s action can be written as the sum of two terms, known as the Born-Infeld term and the Wess-Zumino term  S = 5BI + Swz-  (1-3)  Let us first state explicitly the manner i n which the D-brane is represented in this context. T h e D-brane world-volume is labelled by the coordinates o , a = 0 , . . .p. T h e embedding of the world-volume i n space-time is x (a) where the index ^ runs from 0 to 9. There is also an A b e l i a n gauge field A (a) defined on the world-volume. T h i s description seems slightly different from the open string theory description of the previous section, but we w i l l discuss the connection later. a  p  a  The Born-Infeld t e r m describes the dynamics of the D-brane and its interactions w i t h the metric, the N S two-form, and the dilaton. In the conventions of [34], it can be expressed i n the form [35] S B ? = -T  j d a p+1  p  e^yf-det  (d x»d x»( a  b  gixl/  -  + F ). ab  (1.4)  Actually, T-dualizing a single D8-brane to a D9-brane in ten-dimensional Minkowski space is not really consistent. In this case, we should really be considering type I string theory, which contains 16 D9-branes, in a constant background gauge field in the 9-direction. 2  Chapter  1.  9  Introduction  Here T is the tension of the Dp-brane, and the gauge field strength is Fab — d A — dbA . The precise value of the Dp-brane tension T depends on the normalization conventions for the supergravity fields, particularly on the conventions for the dilaton </>. Once these are fixed, T can be determined in terms of the string tension and the string coupling. We will not need the explicit expression for the Dp-brane tension for our purposes. A l l bulk fields are evaluated at the x^(o). If we set the bulk fields except for the metric equal to zero and ignore the world-volume gauge field, then the action is simply proportional to the proper p + 1-dimensional volume of the D-brane. For p = 0, we recover the usual relativistic action for a point particle, proportional to the proper length (time) of the worldline. The Wess-Zumino term describes interactions with the R R potentials [37, 38]. p  a  b  a  p  p  3  (1.5) q<p+l  Here, p, is the charge of the Dp-brane. The exponential is defined by the power series, and all products are wedge products of forms. The integral is over the p + 1-dimensional world-volume of the Dp-brane, so the spacetime forms have to be pulled back to the Dp-brane world-volume, using the embedding rc^cr). The sum is over odd or even R R potentials, depending on whether the D-brane is in type IIA or IIB string theory. When this expression is expanded out, it includes the basic coupling p  (1.6) For p = 0, this has the same form as the coupling between an electrically charged point particle and the gauge potential in ordinary electrodynamics. The additional terms involving and F b are required by consistency with T-duality [39, 40, 37] and by anomaly cancellation on the world-volume theory [41, 42]. Physically, these additional terms imply that a single Dpbrane with a non-trivial world-volume field strength F b actually carries R R q-iovm charge for all q < p + 1 (with q restricted to be either odd or even as appropriate to the IIA or IIB theory). The two terms SBI and Swz separately invariant under arbitrary world-volume reparametrizations. This is a gauged symmetry that ensures a  a  a v e  Here we have neglected additional massive type IIA supergravity terms involving the nine-form potential [36]. 3  Chapter  1.  10  Introduction  the world-volume theory has the correct number of physical degrees of freedom. One choice of gauge that clearly exhibits the physical degrees of freedom is known as static gauge. Expand the supergravity fields about a flat background, and consider fluctuations of the Dp-brane around a flat configuration. The space-time coordinates can be chosen such that the flat D-brane world-volume coincides with the hyperplane defined by the vanishing of the last 9 — p space-time coordinates. Consider any Dp-brane configuration with embedding coordinates x^(o), corresponding to a small deviation from flatness. Then as long as the Dp-brane does not develop any folds (so that the first p + 1 embedding functions x (a) define an invertible mapping), we can use a world-volume reparametrization to set x (a) = a . Now the shape of the Dp-brane is completely encoded in the remaining 9 — p embedding coordinates x (o). The background configuration corresponds to x (o) = 0, and any non-vanishing x (o) represents a local transverse displacement of the Dp-brane. The use of static gauge allows us to make contact with the previous discussion of D-branes based on open string theory. If we consider, in static gauge, the leading terms of S B I that are independent of the fluctuations of the bulk fields, we find that the gauge field A (a) and the transverse coordinates x (a) are massless. Comparing this with our discussion of open strings on Dbranes in the previous section, this suggests that the transverse coordinates x appearing here should be identified with the massless scalars & appearing in the quantized open string spectrum. For our purposes, it is important to note the relation between general covariance in space-time and world-volume reparametrization invariance. The original action S = SBI + 5\vz ^ manifestly covariant under arbitrary spacetime coordinate transformations (diffeomorphisms). This is achieved by the use of redundant world-volume fields and the associated world-volume reparametrization invariance. However, if we go to static gauge x^(a) — (a ,x (o)), then general covariance becomes obscured. The action is still manifestly invariant under diffeomorphisms involving only the first p + 1 space-time coordinates, and those involving only the last 9 — p space-time coordinates. However, a coordinate transformation that mixes these sets of coordinates does not preserve the static gauge condition. Such coordinate transformations must be combined with a compensating world-volume reparametrization that restores the static gauge. In the above discussion we have been treating the D-brane action as a low energy effective action in the following sense. When the D-brane aca  a  l  l  l  a  1  l  s  a  l  a  Chapter  1.  11  Introduction  tion is expanded about fiat space in static gauge, then tree level scattering amplitudes obtained from this action should agree with the corresponding open string disk amplitudes [32]. These scattering amplitudes include interactions that involve only open strings, as well as interactions between open and closed strings. From this point of view, the low energy effective action should in principle be derivable by examining string scattering amplitudes [43, 44, 45, 46]. In practice, the action SBI has been derived using more powerful techniques. We mention two general approaches with different points of view. The original derivation [35] of 5 B I used the worldsheet beta function approach. This emphasized the role of the action is a summary of the equations of motion, whose solution defines a possible background (to leading order in the string coupling) of open string perturbation theory [47, 48, 49, 50, 35]. The equations of motion were obtained by computing the one-loop beta function, working to lowest order in the curvature Of the space-time metric g^, the field strength of the NS two-form, the extrinsic curvature of the embedding x , and derivatives of the world-volume gauge field strength F . This is often expressed by the statement that S B I is valid for slowly varying fields. Another approach is based on the calculation of the disk partition function Z with a boundary coupling to a background gauge field (see [33] for a review of this approach). The original emphasis was on the connection of open string theory to non-linear electrodynamics, and the calculations were done for the case of what we would now call a space-filling D-brane. For the case of Dpbranes with p < 9 [51], the boundary couplings include the transverse scalars as well as the gauge field. For constant bulk fields, constant F and constant d x , the disk partition function Z was computed exactly in [51], the result being precisely SBI- These types of partition function calculations bear an immediate resemblence to calculations of disk scattering amplitudes with an arbitrary number of vertex operator insertions. The physical meaning of the disk partition function Z as an effective action that reproduces open string disk amplitudes is discussed in more detail in [52, 53]. Both of these derivations suggest that S B I is a good approximation to the complete effective action for slowly varying fields. For the static gauge action expanded around flat space, this statement can be made more precise. M  ab  ab  l  a  4  Interestingly, these techniques have been adapted to compute effective actions in quantum field theory [54]. 4  Chapter  1.  12  Introduction  Neglecting the coupling to the supergravity fields, we have SBI = -T  P  J  ^/-det (77,5 + d^dtf  + F ).  (1.7)  AB  The square root can be expanded out in powers of the fields. To keep track of the relative contributions of the various terms to low energy scattering processes, it is convenient to adopt a power counting scheme that simply counts the number of derivatives. Although the complete effective action contains terms beyond SBI, according to this power counting scheme the dominant contribution to any low energy scattering process is obtained from interactions in SBI- A S previously mentioned, the disk partition function calculation is exact for constant field strengths F and constant d x\ so any corrections to SBI must involve higher order derivatives of these quantities. The contribution of such terms to any low energy scattering process will be suppressed by powers of small momenta (in string units) relative to the contributions from the action SBI- We should mention that there is a large literature on computing corrections to SBI involving higher derivatives of the world-volume fields. For a review of this topic, see [55]. Finally, we briefly discuss the terms in the action involving the worldvolume fermions. These are obtained by demanding supersymmetry. For a Minkowski space background, ignoring the coupling to supergravity fields, a supersymmetric generalization of SBI was first obtained by [56, 57]. This supersymmetric action can be expressed in a form similar to its bosonic counterpart. The difference is that the D-brane world-volume is embedded into ten-dimensional J\f = 2 superspace instead of ten-dimensional Minkowski space. A n important feature is the presence of a local world-volume fermionic symmetry, known as kappa symmetry. It plays a similar role to world-volume reparametrization symmetry in the purely bosonic case, allowing the action to be written in a manifestly supersymmetric form. By choosing a physical gauge and fixing the kappa symmetry, half of the 32 fermionic embedding coordinates in superspace are removed. In the end, the massless world-volume degrees of freedom agree with those obtained by directly considering the open string spectrum. A B  a  For general bosonic background supergravity fields, the supersymmetric generalizations of SBI and Swz were first obtained by [58, 59, 60]. Besides embedding the D-brane into superspace, one also needs to include superfields on superspace. Component field formulations were developed in [61, 62, 63]. We should note that the supergravity fields considered in this case are taken  Chapter  1.  13  Introduction  to be on-shell (satisfying the supergravity equations of motion). This point of view is somewhat different from our previous discussion, where the action was taken to reproduce open plus closed string scattering amplitudes.  1.2.3  Effective actions for multiple D-branes  For a system of N D-branes, it is natural to write down an effective action consisting of iV separate copies of the single D-brane action. This appears to be valid for processes in which the individual D-branes remain widely separated from each other. For situations where the D-branes come within close contact or even intersect, it is necessary to examine the microscopic open string theory description more closely. We return once again the perturbative string theory viewpoint. Consider perturbing around a ten-dimensional Minkowski space background, but this time with a system of N flat parallel Dp-branes. Before discussing the low energy effective action that describes fluctuations of the system about this background, the relevant low energy degrees of freedom should first be identified. Open strings that begin and end on the same D-brane have the usual set of massless states which appear in the low energy spectrum. In addition to these strings, there are also open strings that stretch between different D-branes. The lightest states of a stretched string have mass proportional to the distance between the D-branes, so whether these string states appear in the low energy spectrum depends on the background configuration. More specifically, for a background in which the parallel D-branes are separated from each other by large distances (in string units), these stretched string modes do not appear in the low energy spectrum. In this case, the low energy field theory is described by N copies of the single Dp-brane action, as we might have expected. However, as first pointed out by Witten [64], when the N D-branes are nearly coincident the lightest stretched strings will appear in the low energy spectrum, and these need to be included in the low energy field theory. In this case, the total number of relevant degrees of freedom is N times that of a single D-brane. Specifically, the low energy degrees of freedom consist of a non-Abelian gauge field A , a set of adjoint scalars fa, and a collection of adjoint fermions defined on the world-volume of the coincident D-branes. Altogether the massless field content is obtained by dimensional reduction of the ten-dimensional J\f = 1 U(N) gauge multiplet to p + 1 dimensions. Now we ask what is the low energy effective action that governs the in2  a  Chapter  1.  14  Introduction  teractioris of these degrees of freedom with each other and with the bulk supergravity degrees of freedom? We review some of the past work on effective actions for multiple D-branes (in no way is this intended to be a complete review of all past contributions). Note that we are dealing with the simplest case of parallel D-branes in ten non-compact dimensions. The discussion focuses on the bosonic part of the action, the world-volume fermions are ignored. We begin with the action for a system of space-filling D9-branes in flat space that was first proposed by Tseytlin [65]. The world-volume coordinates of the D9-branes can be identified with the space-time coordinates, and the only bosonic world-volume field is the non-Abelian gauge field A . The action takes the form a  (1.8) where the field strength is F b = d Ab — dbA + i[A , Ab]. This action is often called the non-Abelian Born-Infeld action. Here, the determinant is taken over the world-volume a, b indices only, and STY denotes the symmetrized trace, where all products of F b appearing in the expansion of the square root are symmetrized before taking the overall trace of matrices. The action for coinciding Dp-branes with p < 9 is obtained by T-duality. Formally, this is achieved by dimensionally reducing the action to p + 1 dimensions, and making the substitution Ai —• fa for i — p + 1,... 9. In analogy with the single D-brane case, we anticipate that the adjoint scalars will play the role of transverse coordinates for the D-branes in some sense, even though they are now N x N Hermitian matrices. So we use the notation X in place of fa. The action is then a  a  a  a  a  1  (1.9)  where the gauge covariant derivative is D X — d X + i[A ,X ], and we have the expression Q = 5 ' + i[X ,X^] with Qij defined as the inverse of Q . In this case, the STr means that all products of F b, D X\ and [X\ X ] are symmetrized before taking the overall trace. Apart from the appearance of the matrix Q , this action does suggest that the X are like the non-trivial components of the D-brane embedding in static gauge. We will return to this point when we consider the coupling to the bulk supergravity fields. l  a  u  tJ  l  l  a  a  l  lj  j  a  lj  1  a  Chapter  1.  15  Introduction  The non-Abelian Born-Infeld action S ^ was obtained by considering the disk partition function with boundary coupling to a background nonAbelian gauge field [65]. In the Abelian (single D-brane) case, if the terms in the sigma model action containing derivatives of the field strength F are ignored, then the partition function can be computed exactly. In the non-Abelian case, even if the terms in the sigma model action containing covariant derivatives of the field strength are ignored, an exact calculation does not appear to be possible, due to the presence of path-ordering in the boundary coupling. However, Tseytlin proposed to define as simply the part of the effective action that contains neither covariant derivatives nor commutators of the gauge field strength. Then according to this definition, ^NBI obtained from the partition function by formally treating all matrices F j, as commuting variables, and the path ordering becomes trivial. The leading non-trivial term in the expansion of the square-root in S^BI is the dimension 4 term that corresponds to the usual Yang-Mills action (here bosonic fields have mass dimension 1). Based on the calculations of string scattering amplitudes [66, 67] it appears that there are no corrections to the symmetrized trace proposal S ^ up to terms of dimension 8. A computation of the two loop beta function of the worldsheet sigma model [68] independently confirms the terms up to dimension 6. However, the non-Abelian Born-Infeld action appears to have a more limited domain of validity as compared to its Abelian (single D-brane) counterpart. Discrepancies between the predictions of S N B I d string theory calculations were first discovered [69] by studying the spectrum of fluctuations about backgrounds with non-trivial field strength F b [70]. Apparently, corrections terms of the form F involving commutators of matrices need to be added in order to resolve the discrepancy [71]. Since these initial investigations, a large literature on computing derivative and commutators corrections to S N B I has developed. A wide variety of approaches to the problem have been explored, and thorough review of the various methods with an extensive list of references is given in [55]. Now we turn our attention to the interactions between D-branes and the bulk supergravity fields. The first attempts to generalize the single D-brane actions SBI and Swz (including the coupling to bulk fields) to the case of ab  c  a  n  D e  a  a n  a  6  5  Since the appearance of this review, several additional papers on computing higher order corrections to the non-Abelian Born-Infeld action have appeared. We list these here for completeness [72, 73, 74]. 5  Chapter  1.  16  Introduction  multiple D-branes were based on the use of T-duality [34, 75, 76]. Essentially two strategies were pursued. Starting from the action for multiple D9-branes, T-duality was used to deduce terms in the action for lower dimensional Dbranes. The other strategy to to start with the action for multiple DO-branes and T-dualize to deduce terms in the action for higher dimensional D-branes. These two procedures give complementary information. In the approach of [34, 75, 76], the Born-Infeld part of the multiple D9brane action is taken to be the minimal generalization of Tseylin's flat space action :  SNBI =  *j  T  d W ( T  STr ( e - ^ - d e t  (g  ab  - B  + F ) j.  ab  ab  (1.10)  Again the world-volume and space-time indices are identified here. Similarly, the Wess-Zumino part of the multiple D9-brane action is (1.11) There is no ordering ambiguity in passing from the Abelian Wess-Zumino term to the non-Abelian one. The sum is over all even R R potentials. A T-duality transformation is then performed to obtain the multiple Dp-brane action in static gauge S = + S^wz- The supergravity fields also have to be transformed under T-duality according to the rules given in [77, 78]. To apply these T-duality rules, the bulk fields must have no spatial dependence in the directions along which the T-duality transformation is being applied. Consequently, the resulting action 5 has no information on how the collection of Dp-branes interacts with fields that have non-trivial dependence in the transverse directions. Also, from our discussion of the non-Abelian Born-Infeld term in fiat space, we might anticipate that the Born-Infeld part of this action cannot be valid beyond terms of order F . Alternatively, one starts with the action for multiple DO-branes coupled to weak supergravity fields [75]. The derivation of this action was based on the observation that the Matrix theory [11] action is given by the SusskindSeiberg-Sen (SSS) limit [79, 80, 81] of the multiple DO-brane action. Using the action for Matrix theory in weak background fields found in [82], the terms of the multiple DO-brane action that survive the SSS limit were reconstructed in [75]. Starting from this multiple DO-brane action, T-duality can be applied to obtain the action for multiple Dp-branes. Again, the bulk fields must have Dp  D p  4  Chapter  1.  17  Introduction  no spatial dependence in the directions along which the T-duality transformation is being applied. Consequently, the resulting action provides no information about the coupling of the Dp-branes to bulk fields with nontrivial dependence on the Dp-brane world-volume directions. Note that this is complementary to the information obtained from T-dualizing the action for multiple D9-branes. Also note that the supergravity fields in this action are restricted to be weak, and only the terms that survive the SSS limit are determined by this method. The action obtained by putting together these various pieces allows us to understand the role of the adjoint scalars X more clearly. Consider the interactions of the scalars X with the various bulk supergravity fields, starting with the dilaton (f>. T-duality from the DO-brane action shows that the dilaton has to be evaluated in the following manner 1  1  OO  <j>{a\ X\o))  1  =n=0£ - A n  . . . c \ > ( r y \ O ) ^ ( a ) . . . X^{o).  (1.12)  1  Tl.  The appearance of these Taylor expansions in the action receives some support from calculations of string scattering amplitudes [83]. This gives further support to the interpretation that (1.13)  x>)EE(ry\xV))  is in some sense the static gauge embedding functions for the multiple Dbrane system. The scalars then appear to be the transverse coordinates of the collection of D-branes, just as in the single D-brane case, but they are now matrix-valued. The form of the interactions with the other supergravity fields is also suggestive. It appears that the coupling to supergravity fields with tensor indices naturally involves non-Abelian pull-backs (this was already suggested in [84, 85]) using the static gauge embedding X (a). More specifically, in [76] the proposed action was written using expressions of the form (1.14) ti  where b.  tin  D X» a  = (6+ ,D X% a  where D is the gauge covariant derivative. a  a  (1.15)  Chapter  1.  18  Introduction  The T-duality approach we have just described also reveals the presence of certain terms in the Wess-Zumino part of the action that involves commutators of X . These commutator terms describe the interactions of multiple Dp-branes with g-form potentials where q > p + 1. String scattering amplitudes have been computed to support the existence of such terms [86]. The appearance of commutators means that these couplings have no counterparts in the single D-brane case (indeed a single Dp-brane can only couple to qform potentials with q < p + 1). These terms lead to interesting physical effects involving a type of non-commutative geometry associated with the matrix coordinates of multiple D-brane systems [32]. 1  We briefly summarize an example of this first presented in [76], illustrating what is now known as the D-brane dielectric effect. Consider a system of N DO-branes in ten-dimensional Minkowski space, but in the presence of a static background R R three-form potential of the form Co\J(y) = 2nSijky - The indices i, j, k take values from 1 to 3, all other components of the three-form potential vanish. Also we have an arbitrary positive constant n. Using the expressions in [34, 75, 76] that we have just discussed, expand the multiple DO-brane action about this background, and consider the leading terms that are independent of the fluctuations in the supergravity fields. k  rDO _  (1.16)  To  The first term comes from the Born-Infeld part of the action, and the second comes from the Wess-Zumino part. Here we have set the world-volume (world-line) gauge field A to zero by gauge transformations, and the dot denotes the derivative with respect to t. Now look for time-independent solutions to the classical equations of motion. For K = 0, any static configuration of commuting matrices X solves the equations of motion. Classically, the energy of such configurations is zero, which is also the minimum energy. Since the matrices commute, they can be simultaneously diagonalized. Such a configuration describes N individual DO-branes located at specific points in space. For a non-trivial background three-form potential the situation is much more interesting. There exists static solutions of the form Q  1  X * = («/xo/T ) J * , 0  (1.17)  Chapter  1.  19  Introduction  where the matrices J form an iV x N representation of the SU(2) alegbra 1  [J\J }=is J . j  ijk  (1.18)  k  Classically, such configurations have negative energy, with the irreducible N x N representation Having the lowest energy out of all these [76]. Since the matrix coordinates X do not commute, they cannot be simultaneously diagonalized, and cannot represent DO-branes located at individual points in space. We can also understand this by observing that the state (1.17) has spherical symmetry whereas it is impossible to arrange a finite collection of points into a spherical symmetric configuration. This configuration of DO-branes is often referred to as a fuzzy sphere [87, 88, 89], or a non-commutative sphere (see also [90] for a discussion in the context of matrix theory). The dielectric effect then generally refers to the expansion of a collection of D-branes into some higher-dimensional non-commutative configuration due to the influence of external R R fields. For a more complete review of the dielectric effect and non-commutative phenomena associated with multiple D-brane systems, see [32]. 1  1.3  Directions of research  This thesis is devoted to the problem of implementing general covariance in effective actions for multiple DO-branes. Let us summarize the relevant issues once more. The transverse positions of N nearly coincident DO-branes are represented by a collection of nine N x N Hermitian matrices X . In a configuration corresponding to commuting matrices, each DO-brane occupies a well-defined point in space. For non-commuting configurations, the system of D-branes takes on the characteristics of a single higher-dimensional object, described by non-commutative geometry. The low energy action that describes the coupling of the DO-branes to supergravity must be generally covariant. To incorporate general covariance, we must therefore determine how the various degrees of freedom of the DObranes transform under general space-time diffeomorphisms. The matrices X pose a distinctive problem, because they are interpreted as spatial coordinates; from the known couplings between X and the bulk supergravity fields, there is a precise sense in which X^(t) = (t,X (t)) are the embedding functions for the system of DO-branes in static gauge. Clearly, the fact that the time and space directions are not treated on the same footing is 1  1  1  l  Chapter  1.  Introduction  20  problematic. Even in the case of a single D-brane, we have seen that the static gauge obscures the covariance of the action with respect to diffeomorphisms that mix the world-volume and transverse directions. In the many D-brane case, the problem is worse because the transverse coordinates X are matrix-valued. Even if we restrict to diffeomorphisms that involve only on the transverse directions, we still have to understand how they act on matrix-valued coordinates. In this thesis, we present progress towards the goal of implementing general covariance in multiple DO-brane actions. In chapter 2, we focus on spatial diffeomorphisms, and explore in detail the problem of defining general coordinate transformations on matrix-valued coordinates. We develop a method of constructing actions for DO-branes coupled to the space-time metric that satisfy general covariance with respect to arbitrary spatial diffeomorphisms. In chapter 3 we turn our attention to diffeomorphisms that mix the space and time directions. While we are ultimately interested in general covariance and the coupling to supergravity fields, we attempt to gain some insight by studying the related problem of Lorentz invariance for multiple DO-branes in Minkowski space. Physically, the meaning of Lorentz invariance is quite different from general covariance. Nevertheless, we are interested formally in coordinate transformations that mix the space and time directions, with Lorentz transformations in Minkowski space being the simplest case to consider first. 1  21  Chapter 2 Invariance under spatial coordinate transformations 2.1  O v e r v i e w  In this chapter, we consider the problem of implementing invariance under arbitrary spatial diffeomorphisms for a system of DO-branes described by the matrix spatial coordinates X . There are two main questions to address. First, how do the matrix coordinates of DO-branes transform under a general change of coordinates? A n d second, how do we write down actions for DO-branes that are invariant under these transformations? In section 2.2, we take up the task of generalizing the transformation law for ordinary coordinates to the case of matrix coordinates. We rely on various physical and formal consistency conditions to find a transformation rule for the matrix coordinates. For example, one condition is that the result must respect the multiplication (composition) law of the diffeomorphism group. Working in powers of the matrix coordinate, we construct a transformation rule to fourth order that satisfies our consistency conditions. Interestingly, the result depends on the metric and its derivatives, consistent with the earlier analysis of [91]. Even at this order, the matrix transformation rule is quite complicated, and while it appears that our approach can be continued to higher orders in principle, the computations quickly become rather cumbersome in practice. In section 2.3, we propose a strategy to deal with the complicated matrix transformation rule. The most familiar tool used to implement general covariance in field theories is the tensor calculus. Accordingly, we find a matrix valued object constructed from X that transforms like an ordinary vector field, so that all the standard tensor methods are at our disposal. This idea is motivated by the work of Van Raamsdonk [92] on the related problem of incorporating local gauge invariance into effective actions for systems of intersecting D-branes. The matrix-valued vector field is obtained as a generalization of the fol1  1  Chapter  2. Invariance  under spatial coordinate  transformations  22  lowing geometric construction. For any point x i n a Riemannian space, there is an associated vector field v defined i n a neighborhood of x such that v (y) is simply the Riemannian version of the displacement vector from y to x . We show that a m a t r i x generalization V of the vector field v exists so long as there exists a consistent m a t r i x transformation rule. More specifically, given a transformation rule for X that satisfies our postulated consistency requirements, there is a straightforward algorithm to directly construct the matrix-valued vector field V\ Furthermore we have found a method of determining V as a function of X and the metric gij without making explicit use of the m a t r i x transformation rule. l  l  l  1  l  %  l  1  1  1  1  In section 2.4, we use this matrix-valued vector field V to construct a large class of generally covariant actions for DO-branes. It is easy to form scalar fields C using V w i t h any tensor fields constructed from the metric. T h e usual procedure i n field theory would be to integrate such an expression over space and time (with a suitable invariant measure) to obtain a generally covariant action. However, i n this case the values of V at different points i n space do not really represent independent degrees of freedom. Specifically, in going from X to V , we have not really introduced any additional degrees of freedom, so that i n fact the value of the vector field V at any single point contains a l l the information of X . To take care of this redundancy i n the object V\ we are led to introduce an additional ingredient: a matrix distribution function S (V) that is inserted into the integral w i t h the effect of localizing the action to the vicinity of the branes (d is the dimension of space). T h e object 6 (V) reduces to a diagonal m a t r i x of ordinary delta functions when V and X are diagonal, and it describes an extended distribution for general non-commuting matrices. T h e final actions take the form 1  1  1  1  1  1  1  d  d  1  x  (2.1) Here, £ is any matrix-valued scalar field constructed from V and tensors built out of the metric. T h i s represents a large class of generally covariant actions. T o illustrate the use of these methods, we demonstrate that essentially any generally covariant action depending only the m a t r i x coordinates and the metric can be expressed i n this form, and give some explicit examples of generally covariant actions. We also discuss the connection between our approach and the earlier "base-point independence" proposal of de Boer and Schalm [91], and show that our methods may be used to implement their proposal. 1  Chapter  2. Invariance  under spatial coordinate  transformations  23  Finally, general covariance is but one of the symmetries that can be used to constrain the form of the low energy effective actions for DO-branes, and in section 2.5 we discuss the compatibility of our results with additional constraints that have been proposed in the literature. Throughout this chapter we only consider the matrix coordinate degrees of freedom X and neglect any other degrees of freedom of the DO-branes, such as the gauge field and fermions. Also we assume a static background metric of the form 1  ds  = -dt? + g {y)dy dy ,  2  i  j  ij  (2-2)  and for the most part all other background fields taken to be trivial (but we do include a brief discussion of the coupling to other background fields near the end of section 2.4). In short, we have restricted ourselves to the problem of finding a realization of the diffeomorphism group on the space of metrics gij and matrices X , and developing methods to write down actions S[gij, X ] that are invariant under the resulting transformations. The dimension of space is denoted by d and is left unspecified. While our discussion is set in this simple context, our methods should apply to more general scenarios in bosonic and supersymmetric string theory. Determining the correct effective action for any given string theory scenario would require a separate analysis for all of the additional issues that are specific to that situation. For example, a specific string theory scenario will have a particular set of low energy degrees of freedom, a particular menu of admissable background fields, and perhaps additional symmetries specific to that scenario. 1  1  2.2  Transformation rule for matrix coordinates  We would like to write effective actions describing multiple DO-branes in a background metric in such a way that general covariance is respected. Naturally, the first question that needs to be addressed is how the various fields transform under general coordinate transformations. Under a general coordinate transformation y = F (y), the transformation rule of the metric is familiar ^ l  l  =  %% 9kl{v)  (2  -  3)  Chapter  2.  Invariance  under spatial coordinate  transformations  24  but the transformation rule for the DO-brane matrix coordinates X (t) needs to be considered in more detail. The situation is clear for a single DO-brane, since we just have ordinary coordinates x (t) for the single DO-brane rather than matrix coordinates, and the transformation rule is simply l  l  ~x\t)  = F\x{t)).  (2.4)  Physically, a single DO-brane is just a point-like object and the transformation rule for its coordinates is the same as that of any point particle. For multiple DO-branes, while there are diagonal matrix configurations that do represent simply a collection of point-like objects with well-defined individual locations, there are also examples of non-commuting matrix configurations that describe extended higher-dimensional objects [90, 93]. As a result, there is little geometric intuition as to how the matrix coordinates should transform, and we rely on a formal approach to the problem.  2.2.1  A first attempt  Here we discuss a first guess at the transformation rule for the DO-brane matrix coordinates. For any given coordinate transformation y —> y = F (y), we would like to know the corresponding transformation of matrices 1  l  l  X  i  = &(F,X).  (2.5)  Some basic points serve as an initial guide. For example, we intepret diagonal matrix configurations as representing a collection of N DO-branes with separate well-defined positions. To be consistent with this interpretation, we should demand that the matrix transformation rule have the property &(F,  diag(xi, 4,...  ^ ) ) = d i a g j V ^ ) , F (x ),... i  2  F(x f),  (2.6)  N  so that the elements of diagonal matrix configurations are transformed individually. Also recall that the matrix coordinates are Hermitian N x N matrices which transform under a U(N) gauge symmetry X —+ UX W. For compatibility with U(N) gauge invariance, the simplest scenario is where the matrix coordinate transformations commute with the gauge transformations 1  U$(F,X)U*  =  §{F,UXU ). ]  l  (2.7)  Chapter  2. Invariance under spatial coordinate transformations  25  A n easy way to satisfy this condition is to write the transformation rule as a series expansion in powers of X . In view of these general considerations, a natural ansatz for the transformation rule is just the matrix version of the Taylor series expansion 1  OO  1  X) = ^ n=0  ... d F*(0)X*  ... X*.  jn  (2.8)  -  n  Despite having other desirable properties, the ansatz (2.8) fails to satisfy one important formal consistency condition: it fails to respect the multiplication (composition) rule for the group of diffeomorphisms. More precisely, $ ( # , $ ( F , X)) ^ $(HoF, X).  (2.9)  This violation of the composition rule can be verified by a direct calculation. The first discrepancies appear at third order $(H,$(F,X))-$(HoF,X)  (2.10)  = -^didjH(F(0))d diF (0)d F (0)[X , i  k  j  k  m  [X\X \] m  +  G(X ). 4  The problem is that the matrices on the left and right hand sides of equation (2.9) are ordered differently, as exhibited by the appearance of commutators in (2.10). To remedy the situation, we may try to modify expression (2.8) by a suitable re-ordering of matrices and/or by adding extra correction terms such that the result satisfies the composition rule. But as we now argue, it appears that this is not the correct approach, and that a more basic change is required. Suppose that we write a general ansatz for the transformation rule as a series expansion in powers of X 1  &(F  t  X) = £ n=0  $)^  jn  (V(0), V  0F(O), ddF(0)...  )x*  . . . X*"  t  (2.11)  '  where the coefficients of the expansion are some unknown functions of F and its derivatives evaluated at the origin. Let us make some seemingly innocent assumptions concerning this transformation rule. First assume each coefficient function admits an expansion in powers of its arguments. Also assume that the transformation rule has the form (2.6) for the case of diagonal matrices. Finally assume that the matrix coordinates transform naturally l  Chapter  2.  Invariance  under spatial coordinate  transformations  26  under linear diffeomorphisms. More specifically, for any linear coordinate transformation F (y) = M) y , we have &(F,X) = M) XK While these assumptions may appear reasonable, they are in fact incompatible with the requirement that the transformation rule satisfies the composition rule. This may be demonstrated by contradiction. For if the composite rule is satisfied, then we may apply it to special cases where one of the coordinate transformations is linear. Together with the regularity assumptions of the coefficients functions, this already implies that the coefficient functions must have the form i  j  constant x  ...  dj F (0). l  n  (2.12)  Then the constant is fixed by demanding the correct limit for diagonal matrices, so we just recover the ansatz (2.8). However, we already know that this ansatz fails to satisfy the composition rule, and so we have a contradiction. The argument can be spelled out in more detail, but this is not really necessary for our purposes. We simply want to make the point that if any realization of the diffeomorphism group <& (F, X) on the space on matrices exists, then it must violate at least one of the assumptions we have outlined above. As a result, this path does not seem promising, and we are motivated to take a different approach. l  2.2.2  Non-trivial dependence on the metric  The obstruction that we have just encountered is not a new result, it was already pointed out by de Boer and Schalm in [91]. These authors also went further and presented a calculation that suggests a solution to the problem. The calculation consists of applying the Noether procedure to the action for multiple DO-branes in weak background fields, and the results suggest that the transformation rule for the matrix coordinates involves the graviton field, although a detailed expression for the transformation rule was not written down. This motivates us to search for a matrix transformation rule that depends not only on F and X\ but also on the metric gij. To guide our seach for such a transformation law $ (F, X, g), we rely on the composition rule as the key consistency condition. Note that the composition rule now takes the form (2.13) 1  l  2. Invariance  Chapter  under spatial coordinate  transformations  27  where the first diffeomorphism F also transforms the metric according to the usual tensor transformation rule, gij(y) = (dy /dy" )(dy /dy ) gki(y) with 1  k  f =  l  l  j  F\y).  Suppose we start with the general expression oo  &(F,X,g)  = ]T $;,.. (F(0)^(0),..  . ..X*".  jn  (2.14)  n=0  Here the coefficients of the series expansion depend on F , g^ and their derivatives (represented by the ellipsis) evaluated at the origin. Before we consider the question of how to determine these coefficients, we first make note of an interesting property that follows if the transformation law satisfies the composition rule, and if a simple shift of the origin y —> y + e acts on the matrices in the obvious way X —> X + e\ The composition rule implies l  l  1  ^(FoH- , 1  l  1  %  $(H, X, g),g) = $ ( F , X, g).  (2.15)  Now take the special case where H corresponds to a simple shift in the origin of the coordinate system, and express this in terms of our series expansions. Then we have the interesting result l  oo  Ei. J  i  J  ?  (o),#)-..)^...^  n=0 oo  = E ®l.. (F(y),9(y),  • • . ) ( * - v)  h  Jn  •••(*- v) jn  (- ) 2  16  n=0  In other words, this is saying that the transformation rule can be expressed as a series expansion in powers of X — y for any point y , and not just the origin y = 0. Also note that on the two sides of the equation, we have the same functions $}j...j , but evaluated on different arguments. Now we return to the problem of how to determine the coefficients in (2.14). Write the coefficient as the sum of our previous ansatz (2.8) and a correction term 1  l  1  l  n  $  L i n = ^ - - - ^ ^ ( 0 ) + A$j .., . 1  1  n  (2.17)  Then a set of equations for the correction terms can be obtained by simply expanding out the general form of the composition rule (2.13) using the series expansions. If we can solve them, we would immediately obtain a  Chapter  2. Invariance under spatial coordinate transformations  28  transformation law that satisfies the composition rule. However, they are not very convenient to work with because a direct substitution of one series into the other on the right hand side of equation (2.13) leads to an infinite number of terms contributing at each order in X\ This happens because the two series have a constant (independent of X ) leading term. On the other hand, if property (2.16) holds, we can substitute the first series into the second series expanded in powers of X — F (0) rather than X . Then at each order in X , only a finite number of terms contribute. This gives us a set of equations for the correction terms of the schematic form 1  1  x  z  1  A^ (HoF(0),g(0),...)  (2.18)  jn  A ^ . . . ( H ( F ( 0 ) ) , ~9(H0)), •••)  = d F -(0)...d F »(0) k  h  k  jn  fcl  + 0 if(F(O)) fc  fcn  ... (F(0), p ( 0 ) , . . . ) + remainder. in  The remainder terms depend on the functions F\ H and the correction terms of order less than n. However, a solution to these equations does not immediately give a transformation law that satisfies the composition rule. The point is that we made use of (2.16) to obtain these equations. The correct statement is that any solution to equations (2.18) that also satisfies (2.16) defines a transformation law that satisfies the composition rule. The statements we have just made are somewhat pedantic, but we are simply trying to say the following. We solve equations (2.18) to find a set of correction terms order by order, and check if the result satisfies the coordinate independence constraint (2.16). If so, we have found a matrix transformation law &(F,X,g) that satisfies the composition rule. For example, we use this method to find the following transformation law to third order 1  &{F,X,g)  =.F*(0) + d F (0)X i  j  +^d d d F (0)X^X X i  j  k  k  j  + ^-d d F(0)X X + j  j  (2.19)  k  k  - ^ ^ ( 0 ) ^ ( 0 ) ^ , [X ,X ]}  l  k  l  1  +  0(X ). 4  In this expression, TJ is the Christoffel symbol constructed out of the metric gij. The third order discrepancy in (2.10) is cancelled precisely by the inhomogeneous term in the transformation law for the Christoffel symbol (see appendix A . l for our conventions and relevant formulae). So far, we have only ensured that the composition rule is satisfied. Now what about the other consistency conditions that we discussed in section fc  Chapter  2. Invariance  under spatial coordinate  transformations  29  2.2.1? Since we construct our transformation rule as a series expansion, the result automatically commutes with U(N) gauge transformations, so we have no conflict with gauge invariance. We also mentioned that the transformation law should transform the elements of a diagonal matrix configuration individually. This condition will be satisfied if we find that the correction terms A $ j\'••jji ' . X . . . X ' " can be written in a form that involves a commutator of matrices. Finally, the transformation law will reduce to the natural result for any change of coordinates consisting of a linear transformation and a shift of the origin, as long as the correction terms involve only second or higher derivatives of the diffeomorphism F . It is clear that the expression (2.19) satisfies all of these additional constraints. In fact, we have checked that there exists a solution to all of our consistency constraints to fourth order in X \ We do not see any obstruction to continuing our procedure to even higher orders, but we do not have a proof that there exists a solution to all orders. The fourth order expression for $ ( F , X, g) is given in appendix A.2, and we see that it is already becoming rather complicated at this order. J 1  7  1  2.2.3  A consistent transformation law  To summarize the story so far, we propose as a working assumption that there exists a transformation law for matrix coordinates X —» X — $ (F,X,g) that satisfies the following consistency conditions: (i) it has a series expansion in X and therefore commutes with gauge transformations (ii) for commuting matrices, <b (F, X, g) reduces to the matrix version (2.8) of the Taylor series, (iii) the transformation law satisfies composition rule, and (iv) for a coordinate transformation of the form F (y) = e -\-M ^, we have the natural result $ (F, X, g) = e + MjX . The explicit expression for the transformation law to third order is given by equation (2.19). The fourth order result is found in appendix A.2. Based on our calculations, it also appears consistent to assume the following additional properties for the transformation law: (v) the metric gij appears in the transformation rule only as part of the Christoffel symbol r*-fc or derivatives of the Christoffel symbol (vi) the transformation rule depends linearly on the function F in the sense that $ ( F + H,X,g) = <&(F, X , a) + $(H,X,g). While these last two conditions are useful because they fix some ambiguity that would otherwise be present, we do not see any compelling reasons to demand that a consistent transformation rule must 1  1  l  l  l  l  1  j  l  1  1  l  Chapter  2. Invariance  under spatial coordinate  30  transformations  satisfy them. W e would be willing to abandon (v) and/or (vi) if these were found to be incompatible w i t h the other four conditions at higher order.  2.3  A matrix-valued vector field  T h e dependence of the m a t r i x transformation rule on the metric is non-trivial already at t h i r d order, and at fourth order the expressions become rather unwieldy. For the purposes of constructing generally covariant actions, it is useful to find an object built out of X that has a simple transformation law. T h e approach taken here is motivated by the work of V a n Raamsdonk [92]. In reference [92], V a n Raamsdonk considered the related problem of incorporating local gauge invariance into effective actions describing configurations of multiple point-like branes w i t h i n a collection of higher-dimensional D-branes. There, the presence of matrix-valued D-brane coordinates led to a complicated expression for the gauge transformation of a certain (bifundamental) field living at the location of the point-like D-branes. T o implement gauge invariance i n the effective action, V a n Raamsdonk introduced a gaugecovariant field (constructed using the bifundamental field) defined on the entire space transverse to the D-brane. T h e value of this gauge-covariant field at any point i n the transverse space contains the same information as the original bifundamental field. However, it has a much simpler behaviour under gauge transformations, and is suitable for writing manifestly gauge-invariant actions. %  In the same spirit, we construct out of X a covariant field defined over space to use as a building block for writing down manifestly invariant actions. W h a t we find is that X can be replaced by a vector field V (y) whose value at any spatial point y contains the same information as X . We start by explaining how the construction works i n the case of single isolated DO-branes w i t h ordinary coordinates. 1  1  l  %  2.3.1  1  Describing a point by a vector field  T h e basic idea is to replace the coordinate description of a point i n space by a field description. Consider a single DO-brane w i t h ordinary coordinates x\ then we construct an associated vector field v i n the following manner. For any field observation point y sufficiently close to the location x of the brane, there is a unique geodesic w i t h unit affine length that connects y to l  l  l  x  Chapter 2. Invariance under spatial coordinate transformations  31  x . In other words, there is a unique solution £(A) of the geodesic equation l  with boundary conditions £*(0) = y\ £ ( l ) = x . We define our vector field so that the value of the field at the observation point y is equal to the initial tangent vector d£ /d\\\=o of this geodesic. Another way to express this statement is to use the exponential map that appears in differential geometry. The exponential map takes a point y and a tangent vector at that point u and maps these to a point obtained by flowing a unit affine distance along the geodesic starting at y with initial tangent vector u . More explicitly, it is given by the analytic expression l  l  l  l  1  l  l  l  1  OO  e x p » = y> + u - J2 - H i - i n f o ) " l  7 1  •• •  (2.21)  n=2 n  where the extended Christoffel symbols T^ - are constructed from the Christoffel symbol and their derivatives (the explicit formulae are given in appendix A . l ) . Then we can define our vector field v by the condition n  l  expi(v(y))  = x\  (2.22)  for all y . When we need to be precise with our notation, we will write v to indicate the dependence of the vector field on the location of the brane x . To be clear, the subscript is not referring to the field observation point. When we wish to indicate the dependence on the field observation point as well, we will write v (y). Pictorially, we can think of v as everywhere pointing towards the location of the DO-brane at x . As the position of the DO-brane changes with time, so does the vector field. To get an explicit expression for v (y), we invert the series (2.21). The first few terms of the series are l  x  l  x  x  %  x  <{y)  =  A* + \v) {y)^^  (r}(y) + 3r}(y)rs(y)) A'A*A'  + i  k  w  m  +...,  (2.23) where we have used the abbreviation A = x — y . In a different coordinate system, we can write down the corresponding expression using the Christoffel symbol in that coordinate system cf and the coordinates of the particle x and the observation point y . 1  %  1  r*  l  u  vl{y)  =  A* + if}G/)£>-A* + j fc  (f}(y) + 3f^. (i/)fS(y)) A'A*A + ..., 1  {  w  m  (2.24)  Chapter  2.  Invariance  under spatial coordinate  transformations  32  with A* = x — y . Intuitively, we expect v (y) to have the correct vector transformation law under general coordinate transformations, because we have described the construction in completely geometric terms. In any case, the vector transformation property can be checked explicitly by using the transformation properties of the Christoffel symbol. x  1  l  x  viiy) = f ^ ) -  ( - ) 2  25  For a single DO-brane, it seems that replacing the coordinate description x with the field description v amounts to a lot of unnecessary complication. The reason we introduce v has to do with the difficulties in generalizing the usual method of writing down generally covariant actions. For example, consider the standard kinetic energy term for a single DO-brane (suppressing an overall prefactor of T ) , l  x  l  x  0  S=^jdtg (x)x (t)x^t).  (2.26)  i  ij  This action is generally covariant because x (t) transforms like a vector, and the metric transforms like a tensor. Note that we have to evaluate the metric at the position of the brane. Even in this simple action, we find two elements that we do not know how to generalize to the case of matrix coordinates. First, we find that the derivative of a matrix coordinate X (t) with respect to time does not transform like a vector. The second point is that we do not know how to define the meaning of gij(X), in other words, we do not know how to evaluate a field quantity like gij(y) at a matrix coordinate. On the other hand, consider the following action constructed from v (y): l  l  x  s=  \I I dt  ddy 5d{x  ~ y^(yK(yH(y)-  The time dependence of v (y) comes the time dependence x (t). This action looks like a typical action for field variables, except that there is an unusual delta function in the integrand. Nevertheless, the action is generally covariant. To see what this action is in the standard coordinate description, we use the series expansion for v (y). Using the fact that v (y)\ = x\ we see immediately that the action (2.27) is identical to (2.26). In general, all of the information about the DO-brane's location is contained in the vector field f (y), so we might expect that any conventional action written using the l  x  l  %  x  x  x  y=x  (2.27)  Chapter  2.  Invariance  under spatial coordinate  transformations  33  coordinates x can be rewritten using the vector field v (y). In the following sections, we will find that it is possible to define matrix generalizations of the vector field v (y) and the delta function 5 {x — y). We will then use these as building blocks for constructing actions for multiple DO-branes satisfying general covariance. 1  l  x  l  d  x  2.3.2  M a t r i x vector field from a matrix transformation law  To construct a matrix generalization of v (y), we might first try an order by order approach, similar to the one we took in constructing the matrix transformation law. We write down the matrix vector field as a series expansion in powers of A = X - y : l  x  i  1  l  oo  V (y) = ^ + zZ Lj,A v  n  x  ••• -  (- )  AJn  2  28  n=2  This expression should reduce to the series (2.23) when the matrices are diagonal. Also, the coefficients V£ - must be chosen to be appropriate functions of the metric and its derivatives at y such that V (y) has the correct vector transformation property n  1  x  Vim  = % &V),  (2-29)  V  when y = F (y) and X = § (F,X,g). Using the expression that we found for the transformation law <& (F,X,g), we can impose condition (2.29) and solve for the coefficients V? „• order by order. We have carried out this calculation to third order in A \ However there is a much better way of proceeding that does not require such tedious calculations. The key idea is to realize that the matrix coordinate transformation rule $ acts as a map that promotes any ordinary function F to a function on matrices. Note that we may regard the x dependent vector field v (y) of the previous section also as a y dependent function of x, fy(x) = v (y). This suggests that we may be able to obtain the matrix generalization oiv (y) by promoting the ordinary function f* to a function on matrices using the map <fr. So we define V (y) = &(f ,X,g). (2.30) %  l  1  l  %  x  l  x  l  x  l  x  x  y  Chapter  2. Invariance under spatial coordinate  transformations  34  Using this definition, we can then prove V^(y) transforms like a vector. To make the proof clear, we should first clarify our use of notation. In the original coordinates we have the vector field v (y). Transforming to another set of coordinates y = F (y), the vector field transforms into v (y). T h e corresponding functions are f (x) = v (y) and f^(x) = v (y) respectively. T h e vector transformation property (2.25) of v (y) implies x  l  l  x  l  y  x  x  x  m=%m  (- ) 2  31  A n d since x = F (x), when we suppress the arguments of the functions, we should write this equation as x  l  dy T h i s is all just notation, but it now allows us to write down our proof concisely. Vi(y)  =  &(fy,Xrg)  =  &(fy,$(F,X,g),g)  =  &(fyoF,X,g)  AW  dy-3  V&v).  (2.33)  In going from the second line to the t h i r d line, we used the composition rule for m a t r i x transformations, and i n going from the fourth line to the fifth line, we used the natural behaviour of the matrix transformations under linear coordinate transformations. T h i s demonstrates that if we are given a consistent m a t r i x transformation law (consistent i n the sense that it satisfies conditions (i) to (iv) of section 2.2.3), then it is always possible to define a matrix-valued object V^iy) that transforms like a vector field (2.29). T h i s definition of V^(y) has some nice properties. It reduces to the expression (2.23) for diagonal matrices. Also, when we are working i n a set  Chapter  2. Invariance under spatial coordinate  transformations  35  of Riemann normal coordinates about the point y , then at the point y , all of the symmetrized extended Christoffel symbols vanish (see appendix A . l ) . Therefore the value of the matrix vector field at this point is simply V (y ) =X*- yl To conclude this section, we present the expression for a matrix vector field obtained by this direct method of construction using the matrix transformation law. We need to simply substitute the appropriate coefficients into (2.19) based on the formula (2.23) for the ordinary vector field v (x). To third order in A = X — y\ we find l  l  Q  x  0  0  l  y  1  1  V (y) = A* + \v) {y)A^A  k  x  k  +\ {r)mWki(y)  + ±r}u(y)A'  + rUy)r? (y)) ^ AJA  A A+ k  l  + °(^)-  1  k  (- ) 2  34  In this method of constructing V (y), essentially all the hard work was already done in the original construction of the matrix transformation rule. x  2.3.3  M a t r i x transformation law from a matrix vector field  At this point, we could try to use our matrix vector field to construct general covariant actions, but we first take a detour to elaborate on the relation between the matrix transformation rule and our matrix vector field. A n interesting observation is that we can reconstruct the expression for the matrix transformation rule from the expression for the matrix vector field. To be specific, take the expression (2.34) for V (y), and the corresponding expression for the transformed quantities V~ (y). Substitute these into the two sides of equation (2.29), and solve the resulting equation for X as a function of X to recover the matrix transformation rule $ (F, X, g). This suggests a new strategy for constructing a matrix transformation rule. In this new point of view, we find the transformed matrix coordinates X from the original matrix coordinates X in the following way. Define an object V (y) by the expression x  x  1  1  l  1  1  x  oo  V (y) = A + Y Vi... (g(y),..  .)A* . . . A *  i  x  jn  (2.35)  n=2  where we freely choose some functions V£ of the metric g^ and its derivatives (represented by the ellipsis) evaluated at y . Construct the correspondJ n  l  Chapter  2.  Invariance  under spatial coordinate  ing object i n a different coordinate system y =  transformations  36  F (y)  1  l  oo  = A* + £  Vi(y)  Vt,„(2(y), ••-)A ... A J 1  J n  .  (2.36)  n=2  Note that the i n the two equations are the same functions, but evaluated on different arguments. T h e n substitute these expressions into the equation J n  niv) and solve for X  1  = % *{y\  (2.37)  v  as a function of X .  T h e result is of the form  1  X> = V(F,X,g)  (2.38) oo  = F(y)  + £  V ... SF(y\  g(y),..  .)A* . . . A * ,  h  3  l  and qu and their derivatives evaluated at  n=l  where ^  A are functions of F 31 •••Jn  J  y\ T h e procedure seems simple enough, and it appears at first sight that we obtain a satisfactory m a t r i x transformation rule by arbitrarily specifying the functions V£ - . B y construction, the coefficients of the series (2.38) satisfy the set of equations (2.18) necessary for the composition rule to be satisfied. However, (2.18) by itself is not sufficient, the expression (2.38) must also satisfy the coordinate independence condition (2.16) to ensure that the composition rule is satisfied. Here is where the problem occurs, since general choices of V* lead to an expression ^ ( F , X, g) w i t h an overall dependence on y . In other words . —¥(F,X,g)^0. (2.39) oy n  J n  %  3  Therefore the coordinate independence condition is violated, and we do not obtain a m a t r i x transformation rule that satisfies the composition rule. T h e good choices of V- ^ are those for which the y dependence drops Out. We already know of one such good choice, given by equation (2.34). l  x  Now let us derive some constraints to help identify the good choices of Vy . To do this, differentiate b o t h sides of equation (2.37) w i t h respect to j n  y . T h e derivative of the left hand side is l  Jyi ( ^ ) )  = w  w  m  +  h  i  ( * u w * > - v*®)  (2.40)  Chapter  2.  Invariance  under spatial coordinate  transformations  37  In the first term on the right hand side, the derivative with respect to y is taken with X held fixed. The term in the large square brackets vanishes if and only if 1  1  •i-X  {  = -£-r*(F,  X, g) = 0.  (2.41)  Combining this with the derivative of the right hand side of (2.37), we find  -  %im ^- ^  +  v  if and only if (2.41) is satisfied. To put this into a nicer form, add the appropriate term containing the Christoffel symbol to both sides of the equation to form covariant derivatives, so that we obtain  = w w  V  j  V  j  <  {  y  l  ( 2  -  4 3 )  if and only if equation (2.41) is satisfied. The content of this equation can be expressed in a more useful form that does not refer at all to the transformed quantities. The point is that once we compute the covariant derivative of V (y) as a series in powers of A using the definition (2.35), we can eliminate these factors of A in favour of V (y) by inverting the series (2.35). The result is 1  x  1  x  ViV&y) =  oo  + £ ^..jMvWxiv)  • • • x (y), v  (2.44)  n  n=2  where the _j (g(y)) are functions of ^ and its derivatives evaluated at y . Now VjVx(y) satisfies (2.43) if and only if these functions (g(y)) are tensors. Therefore, our statement is that equation (2.41) is satisfied, and hence a matrix transformation rule satisfying the composition rule is obtained, if and only if the coefficient functions V? , are chosen such that VjV (y) is of the form (2.44) with tensor coefficients T j - . It is clear that V (y) transforms as a vector field under the resulting matrix transformation rule by construction. n  l  jn  x  t  n  x  2.3.4  A differential equation for V  1  We can formalize what we have just learned into a procedure to determine a matrix vector field without prior knowledge of the transformation law. Before  Chapter  2.  Invariance  under spatial coordinate  transformations  38  describing this procedure, we make a quick note about notation: from now on we drop the subscript X and simply write V for the matrix vector field. Now the prescription is the following. Write down the most general expression of the form 1  oo  X  = y* + V + £  1  V  {  ... V*»,  jl  jn  (2.45)  71=2  that reduces for diagonal matrices to the expression for the exponential map exp^(V). Here - is a function of the metric and its derivatives at y\ This implicitly defines V as a series expansion in powers of A = X — y n  1  1  %  l  oo  V  1  Y, L-in v  = A + 1  AJ1  •• •  A J n  •  (- ) 2  n=2  Use this latter expression to calculate the covariant derivative of V as a series in A . Then eliminate all factors of A by substituting back the starting expression (2.45) to obtain a differential equation of the form 1  1  1  oo  VjV* = -5) + £  'll^Vi>  ... V*  (2.47)  n=2  and demand that the T coefficients be tensors. This is not quite a differential equation that can be directly solved for V\ because the T coefficients are not prescribed. So this is just a constraint on V in the form of a differential equation. Note that the Tj - can be completely expressed in terms of the original coefficients A ^ - are their derivatives. Therefore this differential equation constraint on V actually constitutes a set of recursion relations for A* „• . Finally any solution of these recursion relations will define a matrix vector field. A n d if desired, the matrix transformation rule under which V transforms as a vector can be explicitly obtained. 1  5  n  n  1  1  2.3.5  Solving the constraints  The recursion relations for A ^ - are given in appendix A.2. We have solved these recursion relations to fourth order, subject to two additional constraints that ensure the resulting transformation law satisfies the additional conditions (v) and (vi) specified in section 2.2.3. Here we present the third order n  46  Chapter  2. Invariance  under spatial coordinate  transformations  39  result. We find the general solution A* = V* -  \ v )  k  V  3  V  k  - (Ir* + aR) w  + bR\^  kl  V>V V k  l  + 0(V).  (2.48)  The two arbitrary constants a and b have a simple origin. They correspond to the fact that any given matrix vector field V allows us to construct (to third order) a two parameter family of objects 1  V* + (aR)  kl  + bRi ) jt  V'V V k  (2.49)  l  that also transform as vector fields under the same coordinate transformation law. Indeed, the constants a and b appearing in (2.48) do not show up in the associated transformation law. The choice a — b = 0 is singled out if we require that V should satisfy V (yo) — X — y when one is working in Riemann normal coordinates about the point y . This choice corresponds to expression (2.34). Since it is the one choice that is directly related to the matrix transformation rule via (2.30), it represents a canonical choice for the matrix vector field. 1  l  1  l  0  l  0  2.4  Generally covariant actions  We have discussed the construction of a consistent transformation law for the matrix coordinates of DO-branes. Also, we discussed how these DObrane degrees of freedom can be equivalently described by a matrix-valued vector field V (y). Using this vector field and the metric, we can proceed to construct scalars according to the usual rules of tensor calculus. However, it does not seem correct to simply integrate such scalars to define generally covariant Lagrangians. The reason is that the matrix vector field contains a huge redundancy; its value at any single point contains all the information of the original matrix coordinates. For the case of a single DO-brane, we discussed an appropriate way to handle this issue; we write the Lagrangian as the integral of a scalar times a delta function 8 (x — y) that is localized at the position of the DO-brane. The appearance of the delta function is natural since we are dealing with a point particle, so the action should depend on the value of the metric at the position of the particle. A similar situation arises for multiple DO-branes. Although non-trivial matrix configurations of DO-branes may describe higher dimensional fuzzy objects, nevertheless the action should depend on the value %  d  Chapter  2.  Invariance  under spatial coordinate  transformations  40  of the metric and any other fields in the region occupied by the DO-branes. To implement this idea we will use a matrix generalization of the delta function.  2.4.1  The matrix distribution function  The matrix delta function defined by the formal Fourier transform dk j (2TT) d  6 (X  -y)  d  =  oMx-v)  (2.50)  C  has already been used before to describe the coupling of D-branes to the massless fields of type II string theory [76, 94, 75, 34]. Some properties of this matrix delta function are discussed in [95]. However, this expression is not suitable for our purposes, because it does not transform like an ordinary delta function under general coordinate transformations. Instead, we define the matrix distribution function using our matrix vector field S (V) = J -0^  exp (ihV\y))  d  d  .  (2.51)  To see if this definition is sensible, we check the behaviour under general coordinate transformations: (2TT)  r  dy  J  dk  •dy*  C  dy  d  exp  (2^p  (ikiV\y))  dy  5 (V). d  (2.52)  The matrix distribution function therefore transforms like a density, just like an ordinary delta function. This is very important, since this guarantees that the integral of the distribution function times any scalar will be invariant under general coordinate transformations. For the purpose of carrying out computations, we can express the matrix distribution function in an alternative form. Define W (y) i  = V (y)  (2.53)  + y.  i  i  Then the Fourier expansion for the distribution function becomes r dk J —-  /  d  5 (V) d  =  (2ir) d  E n=0  (-l) nl  \ ex (ikW (y))exp(-ik ) l  l  V  d  n  iy  W»(y)...W^(y)}d ...d 5 (y). d  jn  jn  (2.54)  Chapter  2.  Invariance  under spatial coordinate  transformations  41  This form is useful for computing integrals with the matrix distribution function as a factor in the integrand.  2.4.2  A large class of generally covariant actions  Now we have all the tools to write down a large class of generally covariant actions. Simply construct any scalar field using V (y) and tensors built from th.6 metric r (g(y), v{ ), v( ), R( ), VR(y),...) (2.55) l  £(y) =  y  y  y  and use this to form the Lagrangian L = j d y Tr (S (V)C) d  d  .  (2.56)  Any Lagrangian of this form is manifestly invariant under general coordinate transformations, and therefore gives a generally covariant action S = J dt L. Clearly this represents a large class of actions, since C can be any arbitrary function of the matrix vector field, the metric, the curvatures, and any number of covariant derivatives of curvatures. In fact, we will demonstrate that essentially any generally covariant action (depending on g^ and X only) can be expressed in this form. 1  2.4.3  Expansion in powers of X  1  To make contact with known results, it is desirable to re-express the Lagrangian as a series in powers of X . This can be done by integrating over space using the form of the matrix distribution given in (2.54): 1  OO L  = £  1  3 [ h • • • d^Tc (£(y)W (y). d  • • W^(y))]  h  .  (2.57)  n=U  We have the explicit expression (2.34) for V , and therefore W\ so this gives a straightforward algorithm for obtaining the generally covariant Lagrangian L as a series in X once the scalar C(y) is specified. Some care is required to identify the terms in the sum that contribute at each order in X\ nevertheless the number of terms that contribute at each order in X is finite. 1  x  1  Chapter  2.4.4  2. Invariance under spatial coordinate  transformations  42  H o w large is the large class?  We have discussed a method for generating a large class of generally covariant Lagrangians. In fact, we will now show that any generally covariant Lagrangian that can be expanded in powers of X\ with coefficients that are functions of the metric and its derivatives at the origin, and a single overall trace, can be re-expressed using our matrix vector field and matrix distribution function in the form (2.56). To demonstrate this, first we consider the constraints of general covariance on the leading terms (in powers of X ) of the Lagrangian. Denote the leading terms of the Lagrangian by L n , and suppose that these terms are of n-th order in X . If we perform a change of coordinates by an infinitesimal shift of the origin y — y + s\ then the variation of the these leading terms gives an expression of order (n — 1) in X . The variation of the higher order terms in the total Lagrangian do not give any contributions at order less than n. Now the total Lagrangian is invariant under general coordinate transformations by supposition, so these order (n — 1) terms must in fact vanish. We can express this condition in the form 1  1  l  l  1  dL xi  o  = —L (X  + e) = 0.  o  (2.58)  For example, if any X that appears in the leading terms of the Lagrangian is found inside a commutator with another matrix, or has at least one time derivative acting on it, then this condition will be satisfied. To further constrain the form of the leading terms, consider any diffeomorphism F that leaves the origin unchanged F*(0) = 0. Under such a coordinate transformation, the leading terms of the transformed Lagrangian arise entirely from the leading terms of the original action LQ. Generally, LQ is the sum of terms of the form 1  L  (2.59)  A .. MV),dm,---)T^{X) h  A  where Ai...i„ are functions of the metric and its derivatives evaluated at the origin y — 0, and T - (X) is the trace of a product of the n matrix coordinates with some number of time derivatives, and with a given cyclic ordering of the matrices in the trace. Under a coordinate transformation yi, _p(y) i t h F (0) = 0, any such term becomes l  n  Zn  l  =  w  A ...i»W),^(0) ...)T - -(X). i l  1  >  i  (2.60)  Chapter  2. Invariance under spatial coordinate  transformations  43  — 0, we also have  Since F (0) l  Tii-in/x)  d F {0)...  d F (0)T - (X)  il  =  in  h  jl  + D(X ).  jn  n+1  jn  (2.61)  T h e total Lagrangian is generally covariant, so if we match (2.59) w i t h the leading contribution to (2.60) using (2.61), we find that A ^ . . . ^ must be a tensor. T o summarize, the above considerations tell us that the leading terms of the Lagrangian can be written i n the form Lo = T r (Co), where CQ is constructed from tensors built out of the metric Co =  Fo(g(P), X,  (2.62)  X, R(0), Vi2(0),...),  such that its trace satisfies d iTr(£o) = 0. In appendix A . 3 , we argue that i n fact the following stronger statement holds. We can choose the expression Co appearing i n (2.62) such that it satisfies d iC = 0. In other words, the terms occuring under the trace can be rearranged so that (2.58) holds even without t a k i n g the trace. F r o m these properties of Lo, it follows that there exists an integrated Lagrangian L of the form (2.56) whose leading terms are precisely L - We simply choose the scalar C(y) i n the integrand to be X  X  0  0  c(y)  = MM,  ^(y),  v( ), y  R( ), y  VR( ),...)  (2.63)  y  Notice that the function To appearing i n the integrand is the same function as that appearing i n (2.62) but evaluated on different arguments. To see that this integrated Lagrangian has leading terms identical to L , observe that the condition d iCo = 0 implies that none of the terms i n (2.57) involving the expressions . . . dj £,(y)\ for k ^ 0 can contribute to the leading terms of the action. In other words, the leading contribution to the Lagrangian L is contained i n the expression 0  X  k  y=0  Tr (£(0)) = Tr A n d since V ( 0 ) = X l  1  ( ^ ( O ) , V(0),..., + 0(X ), 2  g(0), VJ2(0), . . . ) ) •  (2.64)  the leading terms of L are given precisely  by L . W i t h this result, we can argue that any generally covariant Lagrangian that can be expanded in powers of X w i t h coefficients that are functions of the metric and its derivatives at the origin, and which contains a single overall trace, can be re-expressed i n the integrated form (2.56). T h e argument is Q  %  Chapter  2. Invariance  under spatial coordinate  transformations  44  the following. Given any such Lagrangian L with leading terms of order n, we can construct an integrated Lagrangian L i that has same leading terms as L using the prescription given above. Then the difference between these two Lagrangians L — L i is another generally covariant Lagrangian, but with leading terms of order n +1. Now we can find another integrated Lagrangian L that has the same leading terms as L — L i by the same procedure as before. Then the difference L — L — L is again an invariant action, now with leading terms of order n + 2. We can keep iterating this procedure, and in this manner express the original Lagrangian as an infinite sum L = ^2 L . Incidentally this also shows us how to characterize the most general covariant Lagrangian up to terms of order m in X . We simply write down the most general scalar C(y) constructed from V(y), and tensors built from the metric, up to m-th order in V, that satisfies dyiC = 0. This is substituted into (2.56) to obtain the corresponding Lagrangian. 2  2  x  k  k  1  2.4.5  Relation to the base-point independence approach of de Boer and Schalm  We have just described a method of writing down generally covariant Lagrangians by integrating over an expression involving a scalar field C(y) and a matrix distribution function. But if the scalar field C(y) happens to be independent of y , dC(y)/dy = 0, then we would have l  l  C(F- (y)) 1  = C(y).  (2.65)  In this case, there is no need to integrate over space. We can simply choose an arbitrary reference point y and the trace of the scalar field evaluated at that point Tr (£(z/o)) defines a generally covariant Lagrangian; every choice of reference point defines the same Lagrangian because of (2.65). This suggests the following alternative method of writing down generally covariant actions. Start with any scalar expression £o(y) at some given order in V(y), then check to see if di£o(y) = 0. If not, then find some higher order scalar expression C\(y) such that the leading terms of its derivative diC\{y) cancel the nonvanishing expression d{Co(y). Then continue the process to higher orders and obtain a series Yln^n that is independent of y\ Consequently, the trace of this series evaluated at any reference point yl defines a generally covariant Lagrangian. This alternative procedure is basically the method proposed by de Boer l  0  Chapter  2.  Invariance  under spatial coordinate  transformations  45  and Schalm [91]. These authors developed a formalism for writing down generally covariant DO-brane actions directly in terms of X , based on the use of Riemann normal coordinates about some base-point along with the requirement of "base-point independence". What we just outlined in the previous paragraph is essentially a coordinate-independent description of their proposal. This can be seen more clearly by observing that our canonical choice of V (y) satisfies V (y ) = X — y in Riemann normal coordinates about y . For convenience, we use the term "base-point independent Lagrangian" to refer to the types of Lagrangians constructed by the method of de Boer and Schalm. From the discussion in [91], these base-point independent Lagrangians seem to be natural from the string sigma model perspective (see also [96]). The connection between our integrated Lagrangians and these base-point independent Lagrangians is the following. We claim that any integrated Lagrangian can be rewritten in the form of a base-point independent Lagrangian. To see this, consider any generally covariant Lagrangian defined by the integral expression (2.56). The expansion of this Lagrangian in powers of X gives us L(X,X,g(0),dg(0),...). (2.66) 1  l  l  %  l  0  Q  l  Q  1  Now choose an arbitrary point y and transform to a new system of coordinates y = F (y) such that y is mapped to the origin of the new coordinate system, F (y ) = 0. Furthermore, we can choose this new system of coordinates to be a set of Riemann normal coordinates about the origin y = 0. Now general covariance of the Lagrangian means that we must have l  0  1  l  l  0  l  0  l  L(X X,p(O) 5p(O),..0 = L ( X , X , p ( O ) , ^ ( O ) , . . . ) . >  . (2-67)  >  In the Riemann normal coordinates, derivatives of the metric at the origin can be written entirely in terms of tensors built from the metric and its derivatives evaluated at the origin. Taking the canonical choice of the matrix vector field, we have X = V (0). Making these substitutions, the Lagrangian becomes 1  l  L(X, 1, g(0), dg{0), • • •) = L'{V{0), V(0) g(0), t  £(<>), V £ ( 0 ) , . . . ) ,  (2.68)  where L' is constructed completely from tensors at the origin and is therefore a scalar. Finally transform back to the original coordinate system using F~ so that 0 is mapped to y and x  l  Q  L'(V(0),  V(0),~g(0), R(0), •..) - L'(V{y ), 0  V(y ),g(y ), 0  0  R(y ) 0  • • .)•  (2-69)  Chapter  2. Invariance under spatial coordinate transformations  46  This demonstrates that the Lagrangian can be written as L(X, X, g(0),dg(0),...)  = L'(V(y ), 0  V(y ),g(y ), 0  0  R(y )...), Q  (2.70)  and since the choice of y was arbitrary, the scalar expression on the right hand side of the above equation must be independent of y . Therefore, we have expressed our original integrated Lagrangian as a base-point independent Lagrangian. This result shows us that we can apply our methods to implement the base-point independence method of de Boer and Schalm. More explicitly, consider the scalar field C (y) taken to be the starting point of the basepoint independence method. Instead of trying to find order by order a series of higher order corrections £\, £ 2 , • • • needed to achieve base-point independence, we can simply form 0  l  0  0  (2.71) where CJ is an arbitrary scalar with leading terms that are higher order than those of Co- As we have just shown, this integrated Lagrangian can be rewritten as a base-point independent Lagrangian. If £ satisfies dyiCo = 0, then the leading terms of the resulting base-point independent Lagrangian are precisely Tr(Co). Therefore, the expression (2.71) is a closed form solution to the base-point independence constraints. The arbitrary terms found in [91] should then correspond to the arbitrary terms C. 0  2A . 6  Examples  Here we present some examples of Lagrangians that are invariant under general coordinate transformations. We consider a fixed background geometry satisfying the equations of motion Rij = 0. The DO-brane coordinates X and the space-time coordinates (t,y ) have scaling dimension —1. We restrict to terms with minimum possible scaling dimension. Subject to these restrictions, the most general expression for the kinetic energy L^ (terms with exactly two time derivatives) to fourth order is given 1  %  in  by  Chapter  2. Invariance under spatial coordinate  transformations  47  Here, we have suppressed the overall factor of To i n the action. There are two arbitrary constants a and a . After performing the spatial integral, the result can be written as a base-point independent Lagrangian x  2  Lidn =  (2-73)  Tr(^gijV'Vi  +  R  ..  k l  a i  ) v*y'V'V  f c  - Q + a, + 2 a ) V ^ V ^ W 2  +0(V ) r5  T h i s Lagrangian can be expanded i n powers of X if desired, by using expression (2.34) relating V and X . One observation is that second order leading term completely determines the t h i r d order terms (in X ) of the Lagrangian, because these come from the uniquely determined order X term i n series expansion of V . The most general expression for the (minus) potential energy L ti to sixth order is given by 1  1  1  1  2  1  po  £ °ti =  (2-74)  P  /d y  VW, k  T r [5 {V)^ g [y^  d  d  gij  kl  J  V }+ l  +9rjRkimn(h[V\ V )[V^ V ]{V , V ] + b [{v\ V% ^ [ [ V ™ , V% V"'])}) . k  n  l  m  2  A g a i n we have suppressed an overall factor of T . There are two arbitrary constants b\ and b - A s a base-point independent Lagrangian, this becomes 0  2  -kpoti  =  (2.75)  i  j  k  n  l  m  2  j  + (bi + j^j  -  + (&! - 8 6  {(8b )V V V V V V  +9i Rkimn  + 8b + 2  VV V V VV i  k  :i  vV V VVV  n  i  k  n  j  l  m  l  Tn  2  +  - (2b + ^ 2  vV V V V V'+ J  f c  n  m  VV VV V V i  k  i  n  m  -  l  + (fci + 8b - -^j v K * V W V V ' J <  n  , B  2  +0(V )). 7  Chapter  2.  Invariance  under spatial coordinate  transformations  48  Expanding in powers of X we find that, similar to the case for the kinetic energy, the fifth order terms of the potential energy are completely determined by the fourth order leading term. 1  2.5  Additional constraints  General covariance is one symmetry that can be used to constrain the form of DO-brane effective actions. In general situations and in particular scenarios there are further conditions that help constrain the form of the action. In this chapter, we consider some of the additonal constraints that have been proposed in the literature, and demonstrate the compatibility with our results.  2.5.1  Emergence of the geodesic equation  For a single DO-brane, if we take the kinetic energy term (l/2)gij(x)x x ' to be the complete Lagrangian, then the equation of motion that follows is precisely the geodesic equation. Equivalently, if we express the Lagrangian in Riemann normal coordinates, then the resulting equation of motion is such that x (t) = a t is a solution for any constant vector a . One might wish to demand that the analogous result holds for multiple DO-branes, i.e. that the kinetic energy Lagrangian for multiple DO-branes leads to a matrix version of the geodesic equation. This is one of conditions imposed by de Boer and Schalm [91] in their construction of the DO-brane action. We can implement this constraint in a straightforward manner. Write the Lagrangian L given by (2.73) in Riemann normal coordinates about the origin, and then demand that the equations of motion resulting from the Lagrangian be solved by X (t) = A t for any constant vector of matrices A . A direct calculation shows that this condition is satisfied if the constants a\ and a in (2.73) are chosen to satisfy a\ + 4a = —5/12. t  l  l  3  1  k i n  x  2  2.5.2  x  1  2  T-duality  Any low energy D-brane action that is derived from string theory should be consistent with T-duality. In the simple context of our discussion where the total DO-brane Lagrangian is given by the sum L = Z/ + L t i of the kin  p o  Chapter  2. Invariance under spatial coordinate transformations  49  kinetic and potential terms given in (2.73) and (2.75), we can consider Tduality in the time isometry direction. The result gives us a relation between the kinetic and potential energy terms. Specifically, T-duality implies that formally making the substitution X*-> [X ,**]  (2.76)  0  should give us (up to an overall minus sign) the D-instanton action. On the other hand, we expect the D-instanton action to be identical in form (again up to an overall minus sign) to the potential terms in the DO-brane action, but with Latin indices i = 1,... ,d promoted to Greek indices [i = 0,1... d, and with the Euclidean metric ds — g^dy^dy" = (dy°) +g jdy dy '. ' Demanding that these two procedures give the same answer and implies that the constants oi, a^, &i, and 62 must satisfy a\ = b\ and a = —462. 2  2  t  :  i  2  2.5.3  Geodesic distance criterion  In references [97, 98], Douglas proposed a number of general conditions on actions describing multiple DO-branes in curved backgrounds, one of these being the constraint known as "the geodesic distance criterion". Consider expanding the action around a background configuration X = D + Y , where D = diag(x\ . . . x ) corresponds to branes with well-defined positions at x , a = 1... N, and Y represents the fluctuations about this background. The geodesic distance criterion simply states that the physical fluctuations corresponding to off-diagonal matrix elements Y^p should have masses proportional to the geodesic distance between the corresponding branes at x and x p. For the Lagrangian L = L + L t i , we can constrain the arbitrary constants appearing in (2.73) and (2.75) by using the geodesic distance criterion. This is done by computing the mass matrix for the off-diagonal fluctuations Y£p. For this purpose, it is convenient to work in Riemann normal coordinates about Xp. Then we find 1  1  l  l  l  N  l  l  a  l  a  %  k i n  (Ml )ij p  po  = d pd p (gki(xp)g ixp) k  l  a  a  i3  -d pd pd^pd^{2a k  l  a  a  - g (xp)g i(xp)j ik  (2.77)  j  + %b )g (x )R {xp) j S  2  2  kl  0  imjn  + ...,  where d p = x — Xp. Note that there is no summation over the repeated a and j3 indices in this expression. Since we are working in Riemann normal l  l  a  a  Chapter  2. Invariance under spatial coordinate  transformations  50  coordinates about Xp, the geodesic distance between x and x p is given simply l  l  a  by \Jgij(x/3)d pd p- Therefore the geodesic distance criterion holds (to this order) if and only if we choose a = —4b . Interestingly, this is necessarily true if the T-duality constraint is satisfied. z  3  Q  a  2  2.5.4  2  Agreement with known results  The effective action for DO-branes in type IIA string theory in ten noncompact space-time dimensions and non-trivial background fields has been extensively studied, and many terms of the effective action are known. We can check if our results are compatible with these. For example, the leading terms of the action for DO-branes in a weak background fields have been determined by Taylor and Van Raamsdonk [75]. In the conventions of this chapter, the terms in the Lagrangian describing the coupling to a weak transverse metric gij — rjij + hij are (to linear order in hij), i /d y 9  My)Str(  + {X\X }[X ,X }) k  3  k  S (X 9  - y)).  (2.78)  Here Str denotes the symmetrized trace prescription. For precise agreement with this action, the constants appearing in the kinetic and potential terms (2.73) and (2.75) must be chosen to be a = - 7 / 3 6 , a = - 1 / 1 8 , &i = - 7 / 3 6 and b = 1/72. Therefore, agreement with the linearized results completely fixes the arbitrary constants appearing in (2.73) and (2.75). We see that given the linear couplings in h^, general covariance constrains the non-linear terms in hij. We can also consider applying our methods to the coupling of DO-branes to other background fields. For example, the Lagrangian obtained in [75] includes a linear coupling to the time component of a weak background Ramond-Ramond one-form potential x  2  2  Jd yC (y)Tr(5 (X-y)). 9  (2.79)  9  Q  Then general covariance tells us this term should be generalized to Jd y 9  C (y)Tr(6 (V)(l 9  0  + £'))>  (2.80)  where CJ is some scalar field involving commutators of V of higher order in the background fields. Another example is the generally covariant generalization of the term that gives rise to the D-brane dielectric effect. The 1  Chapter  2.  Invariance under spatial coordinate  transformations  51  linearized coupling is in this case (2.81) The generally covariant term that reproduces this to lowest order is (2.82) Again, £ ' is some scalar field commutator expression of higher order in the background fields. Note that it is the Ramond-Ramond potential itself that appears in the integrand, instead of its derivative. The derivative on the potential that appears in (2.81) comes from the term in (2.57) with a single derivative acting on C.  52  Chapter 3 Invariance under Poincare transformations 3.1  Overview  The restriction to spatial diffeomorphisms in chapter 2 was made specifically to avoid transformations that mix the time direction with spatial directions described by matrices. As we have discussed in chapter 1, such transformations present an additional complication, since they look complicated even in the Abelian (single D-brane) case if we restrict to the static gauge. Before attempting to analyze the full group of space-time diffeomorphisms, it is natural to begin with the simplest case for which the additional complication arises, namely Lorentz transformations for a system of DO-branes in flat space. This is the focus of the present chapter. We begin in section 3.2 with an order-by-order analysis of the transformation law. We determine the infinitesimal Poincare transformations for a single particle in static gauge, and show that the simplest generalization of these to the matrix case does not respect the Poincare algebra. We find that it is possible, to sixth order in X , to add commutator terms to the boost transformation rule such that the Poincare algebra is restored. The success of this procedure is non-trivial and provides evidence that a consistent transformation rule exists to all orders. Assuming this, we show that the boost transformation law is unique up to field redefinitions which do not affect the other Poincare transformations. In section 3.3, we begin our analysis of the invariant actions, now working order-by-order in X in the static gauge. Using the transformation rule from section 3.2, we find that it is possible to add terms order-by-order to the leading Tr (X ) kinetic term and to the simplest potential term Tr [X\ X ] to obtain (independent) Lorentz invariant results (we work up to 0(X )). We 1  1  1  2  7  2  6  Oi course, this should be guaranteed if string theory is consistent and Lorentz invariant in flat space. 1  Chapter  3. Invariance  under Poincare  transformations  53  determine a necessary condition that must be satisfied by the leading term of any Poincare invariant structure, generalizing the necessary conditions of time-reversal and Galilean invariance (including parity) i n the A b e l i a n case. Finally, we show that for any choice of field there is at most one invariant action depending on X and not X or higher derivatives of X . Unfortunately, we find that any invariant generalization of the A b e l i a n kinetic term may be written i n this way for some appropriate choice of field, so it is not clear whether a canonical non-Abelian generalization of the usual relativistic kinetic term exists. 1  1  1  In section 3 . 4 , we look for a more natural way to write Lorentz invariant actions. A s i n our previous studies, we look for matrix-valued covariant objects defined as fields over space-time from which we can b u i l d manifestly invariant actions as integrals over space-time. We find (at least up to fifth order i n X ) that there exists a covariant m a t r i x vector field V^(y) built from X but transforming simply under a Lorentz transformation = S} y as 1  1  1  v  v  V»(y)  = A",r(i/).  In the A b e l i a n case, V is the derivative of the proper distance to the trajectory along a geodesic which intersects the trajectory orthogonally. In addition, discuss the existence of a covariant matrix distribution function ®(y) which reduces i n the A b e l i a n case to 1  0 ( y ) = jdr  J-d x»d x» T  5 \x {r) d+  T  v  -  y»).  In section 3 . 5 , we show that all Poincare invariant actions may be written using these two covariant objects as S = j  d y d+1  Tr  (C(V(y))Q{y)),  where £ is a scalar built from V. T h e independent Poincare invariant structures may be characterized by their leading terms, which may either be / dt T r {X ) or may be written as the integral of a Lagrangian L(X, X, X,...) w i t h an even number of Xs and time derivatives satisfying 2  6\L(X + e, X,...)  = df,L(X,  X + /?,...) = 0,  i.e. a term w i t h all Xs and Xs appearing i n complete commutators. T h i s is precisely the necessary condition we found i n section 3 , so we conclude  Chapter  3. Invariance under Poincare  transformations  54  that the one-to-one correspondence between Poincare invariant structures and Galilean (and time-reversal) invariant leading terms familiar from the Abelian case extends to the non-Abelian case also. In section 3.6, we discuss the couplings to space-time supergravity fields. We note that Lorentz symmetry also implies higher order corrections to these terms, and in particular to the various conserved space-time currents associated with the branes (e.g. the stress-energy tensor or Dp-brane currents). Throughout this chapter, we focus only on the DO-brane matrix coordinate degrees of freedom. In particular, we neglect the fermionic degrees of freedom, and assume that the gauge field A has been set to zero by gauge transformations. 0  3.2  Poincare transformations for multiple DO-branes  Consider our system of N DO-branes in Minkowski space. The low-energy description of these branes coming from open string theory is the static gauge description: there is one N x N Hermitian matrix for each direction in space X (t) given as a function of inertial time, and these are the (matrix-valued) spatial coordinates for the system of DO-branes in this frame. In this section, we would like to understand how these degrees of freedom behave under a Poincare transformation. l  3.2.1  Transformation rules for a single brane  First, recall the Poincare transformation rules for a single DO-brane. The setting is (d + l)-dimensional Minkowski space with inertial coordinates yi = (t,y ). In static gauge we specify the spatial coordinates of the DObrane as a function of time x (t). Clearly, this description is inconvenient for the purposes of discussing Lorentz invariance. For example, consider changing to another system of inertial coordinates by using a general Poincare transformation. f = A V + a" (3-1) 1  %  l  and let x (t) be the static gauge coordinates of the DO-brane in this coordinate system. The relation between the two static gauge descriptions is non-trivial. l  Chapter  3. Invariance under Poincare  transformations  55  The Poincare transformation (3.1) allows us to write down the relation A V + A V W + «°. = A y + A y ' ( t ) + a\  i = x\t)  j  (3.2)  Solve the first equation to obtain f as a function of i, and substitute the result into the second equation to obtain *<(*) =  A'orHi) + a * ^  ( r ^ ) ) + «*> 1  (3-3)  where we have defined f(t) = A° t + A ° j O ; ( £ ) + a . Clearly, this relation is rather complicated if the Poincare transformation involves a non-trivial Lorentz boost. For a single DO-brane, we know of a description that makes Lorentz invariance transparent. Introduce an arbitrary worldline parameter o and describe the brane by a set of embedding functions x (a) whose Poincare transformations are simply l  0  0  M  # * ( a ) = A V ( G T ) + a"-  (3-4)  To ensure that the system has the correct number of physical degrees of freedom, we must also demand the (gauge) invariance of the action under an arbitrary reparametrization of the worldline o-^o'  = h(a).  (3.5)  If we fix the reparametrization invariance by choosing x°(a) = a, then we have recover the static gauge description. The complicated transformation rule for static gauge coordinates is due to the composition of the original linear Poincare transformation with a compensating reparametrization that restores the static gauge. It would be nice if the transformation rules (3.4) could be extended in some simple way to the case of multiple DO-branes, e.g. by introducing matrices for all space-time (rather than just spatial) directions. Unfortunately, there is no obvious way to do this, and such a description would seem to require an analogue of the worldline reparametrization symmetry capable of eliminating an entire matrix worth of degrees of freedom. Therefore, to make progress in understanding the transformation law for the matrix-valued coordinates of a collection of DO-branes, we attempt to directly generalize the static gauge transformation rule (3.3). So far, we  Chapter  3. Invariance under Poincare  56  transformations  have discussed the Poincare transformations as passive coordinate transformations. From this point on, we take the mathematically equivalent point of view that treats the Poincare transformations as active transformations on the state of the system. This is just for notational convenience, so that we avoid writing many tildes in our formulae. Furthermore, to simplify matters as much as possible, we specialize to the case of infinitesimal transformations. Then the static gauge transformation rules for translations, time translations, rotations, and boosts take the form Sax ' = a 1  1  5 ox  %  =  ~a°x  Sx  l  =  ux  Sx  i  =  a  w  0  l  ij  j  ftt-fitfj?  (3.6)  It is the non-linearity in the transformation rule for boosts that makes a generalization to the non-Abelian case quite nontrivial.  3.2.2  T h e Poincare algebra as a consistency condition  The Poincare transformation rules for the matrix coordinates of multiple DO-branes should be some generalization of (3.6). Since they must reduce to (3.6) in the case where all matrices are diagonal, it must be that all corrections involve commutators of matrices. A further constraint comes from demanding that the Poincare algebra is still satisfied by the non-Abelian transformations. The rotation, translation, and time translation rules in (3.6) are linear in X and generalize unambiguously to the non-Abelian case without modification. We will assume that these receive no commutator corrections, since it is consistent with the algebra of rotations and translations (and certainly very natural) to do so. For the boost transformation law in (3.6), an ordering issue arises since there are various non-Abelian generalizations of the quadratic term. The Poincare algebra demands that the correct generalization must satisfy 1  (fipSp ~ SpSp)^ — $uii=ppi-pifrX (S Sp - SpSg)X = 5 o=p. X (5 oSp - SpS^X = S^pX* {SpS -5Jp)X = dpi^jflX* 1  %  l  s  l  a  1  a  i  w  a  (3.7) (3.8) (3.9) (3.10)  Chapter  3. Invariance under Poincare  57  transformations  In fact, it is easy to show that (3.7) is not satisfied for any of the possible orderings of the quadratic term without adding corrections to the transformation law at higher orders i n X * . We w i l l therefore write the putative Poincare transformation rules for the non-Abelian case as 5X  l  s  6 oX  i  a  =  a}  (3.11)  =  -a ^  (3.12)  0  S^X'  = <Jip  8X  = fit - pySymi&X')  i  p  (3.13) + (3 T j  (3.14)  ij  where S y m indicates the symmetrized ordering (Sym(,4.£?) = \(AB + BA)) and T ' stands for some series of terms that vanish for diagonal X . We now ask whether it is possible to choose H e r m i t i a n T built from X and its derivatives such that the constraints (3.7,3.8,3.9,3.10) are satisfied. First, the constraint (3.10) is satisfied automatically as long as T is a tensor under the rotation group. A n y tensor built from the vector X and its derivatives satisfies this constraint. Next, the constraint (3.9) is satisfied as long as T contains no explicit time dependence. The constraint (3.8) implies that 13  1 2  lJ  %3  1  13  d T (X ij  £  + s) = 0,  i.e. all undifferentiated Xs must appear i n commutators. Finally, the constraint (3.7) implies that pyS-p&t - (3 6 S j  ij  3  = f3 p X i  j  j  - i3 p X , l  j  j  (3.15)  where S  ij  = - S y m ^ X ^ + r ^  T h i s turns out to be quite nontrivial, and we resort to an order-by order approach to check whether a solution exists. Note that we are not assuming that the quadratic term is symmetric, since T ' may contain quadratic commutator terms. 2  i  y  Chapter  3.2.3  3. Invariance  under Poincare  58  transformations  Order-by-order solution  It is straightforward to check that (3.15) holds at leading orders with T = 0, but breaks down at order X unless we add commutator corrections T at order X . We find that these must satisfy %3  3  13  4  <W)i  = \P  [X ,X ]}  k  k  J  +  [X',*']]  - [X\ [X ,X ]]) k  ,  i  (3.16)  where 5p indicates the variation keeping only the order X° term in the boost transformation law (3.11). Appropriate corrections are possible at this order, for example 3  T% =  (3.17)  - S y m (~X [X\ k  8  ^  [X , X }} + X [X , k  j  k  k  [XfX ]} 1  - X [X , k  j  [X , X }]) . k  1  Note that the equations (3.16) determining T at this order are overconstrained in the sense that solutions exist only for special choices of the righthand side. Thus, the existence of a solution can be taken as a first piece of evidence that the Poincare transformation rules admit an extension to the non-Abelian case. This expression is not unique, but we will see shortly that all possible solutions are related by a class of field redefinitions. We might now proceed ad nauseam checking at each order in X that a choice of T exists such that the constraint (3.15) is satisfied to the appropriate order. This was done explicitly up to sixth order in X\ and the result is given in appendix B.1. There is however, a more refined approach developed in [2] that we now describe. 13  1  13  3.2.4  Expanding in the number of commutators  As shown in [2], it turns out to be possible to obtain partial all-order results in powers of X, when expanding in the number of commutators. Suppose we have a term consisting of a product of matrices at some order. Then we can always symmetrize this product and compensate by adding appropriate terms with commutators. These extra terms can in turn be symmetrized, where the commutators are considered as a unit under the symmetrization, The operation Sym is assumed to treat commutator expressions as a unit in the symmetrization. 3  Chapter  3. Invariance under Poincare  59  transformations  by adding terms with more commutators and so on. In the end, one obtains a sum of symmetrized products. Because of the overall symmetrization the number of commutators in a term has a definite meaning. Only the first term does not contain commutators and remains in the Abelian limit. The other terms are non-Abelian corrections with a fixed number of commutators. If the non-Abelian corrections are small it would be sensible to calculate only up to a certain number of commutators. Using this approach, it has been checked [2] that a solution to the constraint (3.15) and therefore a consistent boost transformation rule exists to all orders in X at second order in commutators. The existence of a full solution even to second order in commutators is extremely nontrivial and suggests strongly that a consistent boost transformation law exists to all orders. We have not pursued this method of expanding in the number of commutators, for readers interested in the results of this approach, see reference [2]. At certain points in the remainder of this chapter, there are calculations that have been done only by this method; we will indicate this when we come to these points in our discussion.  3.2.5  Uniqueness of the transformation rule up to field redefinitions  It is easy to see that we cannot expect a completely unique solution to the constraints outlined so far for the non-Abelian generalization of the boost transformation law. For, consider a new variable X^X'  + FiX),  (3.18)  where F is a polynomial in X (possibly infinite) defined so that X and X agree in the Abelian case and X has the same transformation rule as X under rotations, translations, and time translations. These will be true as long as • F (X) is a vector built from X and its derivatives that vanishes for diagonal X; l  • F (X) l  has no explicit time dependence;  • F (X) is translation invariant (has all undifferentiated X s appearing in commutators). l  Chapter  3.  Invariance  under Poincare  transformations  60  The transformation rule for X under boosts (obtained by transforming the right side of (3.18) and rewriting all occurrences of X in terms of X by inverting (3.18)) will generally be different from that of X, with the lowest order change in T given by l]  = ti%F* .  P AT j  ij  (3.19)  As above, 5pX — fit denotes the order X° term in the boost transformation law. The new transformation rule will necessarily be consistent with the Poincare algebra, as this follows directly from consistency of the transformation rule for X. Since the other Poincare transformations remain the same, the boost transformation rule for X represents a new solution to the constraint (3.15). On the other hand, it can be shown that all non-uniqueness in the transformation law may be associated with such field redefinitions. For suppose that there exist two different transformation laws 5pX and SpX for which the constraints (3.15) and all other constraints of that subsection are satisfied. Then the leading order difference A T between T and T ° must satisfy i  lJ  lJ  l  0  fi 5° A Ti 2  It follows that A T  Pl  - (3i5° A ri  0  p2  is of the form  i3  0  = Sym ( x  AT  ij  0  = 0.  0  (3.20)  4  f c l  •••X  km  D  i { j k l  -  k m )  ) ,  (3.21)  where D 0' i- ™) is an arbitrary tensor that cannot contain X and X outside of commutators. But (3.19) shows that this is the same leading order difference that arises in making a field redefinition (3.18) with l  fc  fc  pi  =  _L_  rn + 1  •• • x  s m(x X j  Y  kl  v  k m  D  i { j k l  -  k m )  )  '  .  (3.22)  If SpX and 5pX differed at order X , then 5pX and 5pX may differ only at higher order. We may then repeat our procedure, making a further field redefinition to remove the leading discrepancy at this order, and so forth, so that after an infinite number of steps we find some new variable X^ such that SpXoo is the same as 5pX. Note that the F in (3.22) satisfies all the constraints of the previous paragraph, since the discussion before equation (3.15) implies that A T should satisfy these same constraints. n  lJ  0  Here, the round brackets denote symmetrization.  Chapter  3. Invariance  under Poincare  61  transformations  Thus, any two consistent generalizations of the Poincare transformations to the non-Abelian case are related by a field redefinition that is trivial in the Abelian case and preserves the transformation rules for rotations, translations, and time translations.  3.3  P o i n c a r ei n v a r i a n t a c t i o n sf o r m u l t i p l e DO-branes  In this section, we begin to investigate the constraints imposed by Poincare invariance on the form of the effective action. We will assume henceforth that a consistent boost transformation rule exists (generalizing (3.17) to all orders). In particular, we assume that (as in (3.17)) it is possible to write such a transformation law without introducing any dimensionful coefficients, such that all terms will have one less time derivative than the number of Xs. In our discussions below, we consider explicitly only single-trace actions, which arise in string theory at the leading order in g , but we expect that most of the results generalize readily to the case of multi-trace actions. 5  s  Abelian case As a warm up, consider the case of a single DO-brane, for which the leading term in the effective action is simply the non-relativistic kinetic term 6  (3.23) This action is Galilean invariant but not Lorentz invariant. If we demand invariance under the boost transformation in (3.6), we must add a higher order term ( l / 8 ) i to the action so that the variation of (3.23) under the second term in (3.6) is cancelled by the variation of this term under the first term in (3.6). The variation of the new term under the second term in (3.6) must be cancelled by the variation of yet a higher order term, and so forth. 4  In the context of string theory one might wonder if the correct transformation law involves higher order terms with explicit powers of a'. However, our results from the previous section suggest that there does exist a valid transformation law without any cv' dependence and that any a' dependent transformation law should be equivalent to this by a field redefinition (that would necessarily involve explicit factors of a'). The overall factor of the mass has been suppressed. 5  6  Chapter  3. Invariance  under Poincare  62  transformations  Of course, we know it is possible to carry this out to all orders, with one possible Lorentz invariant completion being the relativistic kinetic term S = - J dty/T^i  = - J ds .  2  (3.24)  This result is not unique, since there are higher order Lorentz invariant structures we could add with arbitrary coefficients. The first of these is r,  r,  ((PxUcPxA  -/ H^^) / d  =  f  x  Ix^f  2  \  ((T^)i (r^jt)' +  d t  ,  n  n r  .  ( 3 2 5 )  where s is proper time, and generally, we will have one Lorentz invariant structure for each Galilean (and time-reversal) invariant leading term. On the other hand, (3.24) is the unique Lorentz invariant action depending on x and no higher derivatives of x. We would now like to see how these statements generalize to the nonAbelian case. 3.3.1  C o n s t r a i n t s for l e a d i n g o r d e r i n v a r i a n t t e r m s  Ideally, we would like to be able to write down the most general Poincare invariant action depending on the matrix coordinates X and its derivatives. Such an action would be a general linear combination of all possible independent Poincare invariant structures with arbitrary coefficients. As a first step, we will determine a set of necessary conditions that the leading term (with the fewest Xs) in any such structure must satisfy. Apart from the boost transformation law, the remaining Poincare transformations in (3.11) do not mix terms with different numbers of Xs, so the leading term must be a rotational scalar, have no explicit time dependence, and be invariant under a shift in X by a multiple of the unit matrix, 1  1  d S (X e  0  + e) = 0.  (3.26)  We must also have invariance of the leading term under parity and timereversal transformations, and this requires an even number of Xs and an even number of time derivatives respectively. Finally, the leading term must be invariant under the O(X ) term in the boost transformation rule, since the variation of the full set of terms under the full transformation law will contain no other terms of this order. Thus, we must also have 0  d S {X p  0  + pt) = 0.  (3.27)  Chapter  3. Invariance under Poincare  63  transformations  Note that these are the same conditions as in the Abelian case, and are simply the statement that the leading term must be invariant under the Galilean group (including parity) plus time reversal transformations. It is obvious that any term for which all X s and Xs appear in commutators satisfies (3.26) and (3.27), since in this case, even the variation of the Lagrangian is zero. More generally, we may have terms for which the variation of the Lagrangian in (3.26) or (3.27) is a total derivative. One example is the non-Abelian generalization of (3.23), S = J dt T r ( X ) .  (3.28)  2  0  In appendix B.2, we show that this is the only example which cannot be rewritten by partial integration as a term for which all Xs and Xs appear in commutators. Thus, the lowest-order term of any Poincare invariant action is either (3.28), or can be written as the integral of a scalar Lagrangian with no explicit time-dependence such that all X s and X s appear in commutators. In section 3.5.1, we will argue that these necessary conditions on the leading term are actually sufficient to guarantee the existence of a Poincare invariant completion. For now, in order to gain some confidence in this statement, we will construct the completions order-by-order in a couple of examples using the order-by-order results from section 3.2 for the transformation law.  3.3.2  Order-by-order construction of invariant actions  First, we consider the simplest possible Galilean invariant potential term, S = J dt Tr Q [ X \ X ' ] [ X \ X ' ] ) , J  (3.29)  J  present in the low-energy effective action for DO-branes in weakly coupled string theory. In this case, the first required corrections are at 0 ( X ) and take the form 6  = J dt STr (^[X ,X ][X ,X }+ i  j  i  +-[X ,X ][X ,X ]X X i  2  k  j  k  i  (3.30)  j  j  - l[X ,X ][X ,X>]X X ) i  8  i  i  k  k  +0(X ), 8  /  Chapter  3. Invariance  under Poincare  64  transformations  independent of the choice for the 0(X ) and higher order terms in the transformation law. It turns out that these correction terms reproduce known terms in the DO-brane effective action obtained [65] by T-dualizing the simplest (symmetrized) non-Abelian generalization of the Born-Infeld action for D9-branes. Indeed, it may be checked that the correction terms in (3.30) are precisely the 0(X X ) terms in 4  4  2  S = - Jdt S T r d e t ( Q 0 ( l - X^XJ), /  (3.31)  i  v  where Q = S + i[X\X ] and Qij is the inverse of Q . On the other hand, the required correction to (3.30) at order X includes terms with X in commutators which are not reproduced by (3.31). As a second example, we consider the simplest possible leading term, the non-relativistic kinetic term (3.28). Using the boost transformation rule (3.11,3.17) up to order X , we find that adding the symmetrized version of terms in the Abelian relativistic kinetic term suffices up to 0(X ) to make the action invariant, but this breaks down at 0(X ). Fortunately, it is possible to add terms involving commutators at this order to restore Poincare invariance. For example, using (3.11,3.17) for the boost transformation rule, we find that the variation of ij  ij  j  tj  8  4  5  6  S = -jdt  S t r ( l - \x  2  - \{X ) 2  (x x [x ,x ][x ,x ] i  k  i  i  -3X X [X ,X ][X ,X>] i  k  i  i  k  k  j  2  - ^(X ) 2  (3.32)  3  - ZX X X [X , i  i  k  {xfx }}  k  1  + 2X X [X ,X ][X ,X }) j  k  k  i  j  i  ) +  0(X ). 8  is zero up to 0(X ) terms that would presumably be cancelled by the leading order variation of 0(X ) corrections to the action. The corrections here are not among the known terms appearing in (3.31). We will see in the next subsection that these commutator correction terms can actually be eliminated by a field redefinition. The expressions in this section are certainly not unique, since we can always add with arbitrary coefficients any of the higher order invariant structures discussed in the previous subsection. However, the absence of any obstruction to our order-by order construction at the first non-trivial order can be taken as evidence that a full Poincare invariant completion exists. In section 3.5.1, we will provide stronger, evidence and suggest a way to write manifestly invariant actions in terms of new covariant objects. 7  8  Chapter  3.3.3  3. Invariance under Poincare  65  transformations  Non-Abelian generalization of the relativistic kinetic term  In discussing the Abelian case, we noted that among all invariant actions, there is a special choice, the relativistic kinetic term (3.24), which depends only on x and not on any higher derivatives. To close this section, we would now like to see to what extent this generalizes to the non-Abelian case. To start, we show that any Poincare invariant structure depending only on X must (apart from additive and multiplicative constants) begin with the term (3.28). For, assume the Lagrangian for some other invariant action S(X) had a different leading term L of order X . According to the constraints of the previous subsection, L must have all X s in commutators so that the condition (3.27) holds, and will necessarily have n > 4. The leading contribution to the variation of this term comes from the second term of the boost transformation in (3.11), and using the cyclicity of the trace, we can write b\L = Tr ( S y m ^ X ' X * + fi X X^C^X)). 7  n  n  n  3  n  If the full action is invariant, this variation must combine with the variation of a higher order term under the first term in (3.11) to give a total derivative 5}L  n  + 5°L  = j Tv  n+2  (fiX Q (X)). 3  t  n  Note that we cannot have terms where 0 is contracted with a derivative of X on the right side since this would produce fiX terms which are not present on the left side. Comparing all terms containing a second derivative of X , we have 3  Tr ( S y m ^ ' X O Q ^ X ) ) = Tr (fiX j Q (X)). 3  t  n  (3.33)  Now, on the left side, the fiX always appears adjacent to the X in the trace. On the right side, Q is of order X - , so there will certainly be terms for which the X is not adjacent to fiX . Thus, (3.33) is impossible, and our assumption that there exists a Poincare invariant action depending only on X whose leading term is not (3.28) must be false. 3  n  4  3  This section is not essential to the development in the remainder of the chapter. The reader only interested in the result may skip to the final summary paragraph on a first reading. 7  Chapter  3.  Invariance  under Poincare  66  transformations  It follows immediately that given the non-Abelian transformation rules, there can be at most one independent invariant action depending only on X. If there were more than one, then at least one linear combination would have a leading term other than (3.28), and we have seen that this is impossible. The present result is not quite as strong as it may sound. Since we have assumed a specific transformation law, what we have actually shown is that for any given choice of field, there is at most one action depending only on X. On the other hand, there could be other independent actions which after appropriate field redefinitions depend only on X . In the absence of some canonical choice for the field there would be no sense in which one of these actions would be preferred over another and therefore no canonical generalization of (3.24) to the non-Abelian case. Actually, we will now see that any Poincare invariant generalization of (3.24) to the non-Abelian case can be brought to a form which depends only on X, using a suitable field redefinition. In fact, for any invariant kinetic action, there is a choice of field for which the action takes the form  S = - Jdt Str (Vl-* 2 )  •  (3-34)  For, consider the most general Poincare invariant action of the form S = J dt Tr(^X  + •••).  2  (3.35)  We assume that all higher-order terms have the same number of Xs as time derivatives, since the variation of any other terms will not mix with the variation of these terms under Poincare-transformations. Now, consider the lowest order terms with second or higher derivatives of X, or with X appearing in a commutator. These terms must be translation invariant, so may be written with all undifferentiated Xs appearing in commutators. The terms involving higher derivatives may clearly be written as J dt T V ( X F ( A ' ) ) i  i  )  (3.36)  for some F, where we can use integration by parts to put any terms with three or more derivatives on X in this form. Terms with no higher derivatives but some X appearing in a commutator will be functions of X alone, so integrating by parts to remove the derivative from some X appearing in a commutator will leave a set of terms all of which have a single X. Rearranging  Chapter  3. Invariance  under Poincare  transformations  67  commutators in some terms, we may again bring this set of terms to the form (3.36). In both cases, the resulting F will still have all undifferentiated F s appearing in commutators. Also, since the total number of time derivatives and Xs was assumed to be equal, F will contain at least one undifferentiated X, which must therefore appear in a commutator, so F vanishes in the Abelian case. Thus, F satisfies all of the conditions listed below (3.18) for an allowed field redefinition 8  X  i  X  1  +  F\X).  Under such a field redefinition, the leading modification to the action will come from the change of the leading term in (3.35) and give (after integrating by parts) S ^ S -  Jdt  Tr^F^X))  + higher orders.  which eliminates the lowest order terms in S with either higher derivatives or X appearing in a commutator. By repeated field redefinitions, we can achieve this at any order, ending up with an action that contains no higher derivative terms and no commutators (i.e. a completely symmetrized function of X). A l l terms in such an action survive in the Abelian case, for which the unique Poincare invariant function of x is (3.24), so our resulting action must be precisely (3.34). A t this point, we have fixed the choice of field completely, since any further field redefinitions will introduce additional terms into the action. To summarize the results of this section, we have shown first that for a given definition of the field, there is at most one Poincare invariant action depending on X and no higher derivatives. On the other hand, we have shown any invariant generalization of (3.24) may be written in this way by an appropriate field redefinition, and there will be a unique choice of field for which this action takes the form (3.34). Thus, among the many invariant non-Abelian generalizations we expect for the relativistic kinetic term with a particular choice of transformation law, there is no obvious way to make a canonical choice. Terms involving only X which do not contain any commutators may also be brought to the form (3.36), but in this case, F will not be translation invariant. 8  Chapter  3.4  3. Invariance under Poincare  68  transformations  Covariant objects  The naive order-by-order approach to writing down Poincare invariant actions discussed in the previous section is cumbersome to say the least. Following the approach of chapter 2, we search for a set of covariant objects, which transform simply under Poincare transformations, to serve as the basic building blocks for constructing manifestly invariant actions  3.4.1  A space-time vector field  In this section, we introduce a space-time vector field that carries all the information contained in the static gauge embedding coordinates. Although the approach is motivated by the success of the method in chapter 2, we should note that this space-time vector field discussed here is distinct from the spatial vector field of chapter 2. First consider the single DO-brane case. Let the static gauge embedding coordinates of the DO-brane be x^(t) = (t,x (t)). We define an associated space-time vector field in the following manner. A t any space-time point sufficiently near the DO-brane world-line, there exists a geodesic that passes through y and intersects the DO-brane world-line orthogonally (with respect to the Lorentzian metric). Call the point of intersection x^(t ) — (t , x (t )). Then define the space-time vector v^(y) to be the displacement vector from the point y^ to x^ity). By repeating this construction at every point we obtain a vector field, which should carry all the information about the motion of the DO-brane in space-time. l  M  l  y  y  y  In other words, v»{y) = afty  - y",  (3.37)  where the time t is implicitly determined by the condition y  - 0.  (3.38)  For an accelerating DO-brane, planes orthogonal to the brane will generally intersect each other at points sufficiently far away, so it is clear that v {y) is not globally well-defined. However, this is enough to ensure that there is a well-defined expansion for v (y) in powers of the static gauge coordinates fX  9  fi  For a uniformly accelerating trajectory, v will cease to be well-defined beyond the Rindler horizon. 9  Chapter  3. Invariance under Poincare  69  transformations  x (t) and its derivatives, and it is this expansion that we will use primarily in what follows. Since our definition of v^iy) was coordinate-independent, this must transform as a four-vector under Lorentz transformations, l  In particular, for an infinitesimal boost we have  3.4.2  Sv°(t,y)  =  P-v(t^-0-Wv°(t,y)-P-yd v°(t,yj,  5v(t,y)  = 0v°(t,y^-tP-Vv(t,y)-P-yd v(t,y).  t  (3.39)  t  Generalization to the case of multiple DO-branes  We would now like to see whether v generalizes to the non-Abelian case. That is, we would like to construct a set of matrix-valued functions V^(y) defined as a formal expansion in terms of X {t) which transform as a space-time vector field and which reduce to d i a g ( ^ ( y ) , . . . , v^ (y)) when the matrices X (t) are diagonal. To ensure the latter condition, we may write 11  l  N  l  W (y) = V* (y) + AV>(y), t  (3.40)  m  where V ^ (y) is the expression obtained by replacing all occurrences of x in the expansion of v' (y) with X and using the completely symmetrized product of matrices and AV (y) is an expression that must involve commutators. To see whether the construction is possible, we write the most general expansion of the form (3.40) up to some order in X, and demand that the covariant transformation rules (3.39) are satisfied to this order using the order-by-order results for the transformation rule obtained in section 3.2. Happily, we find that at least up to order X , it is possible to choose AV^(y) so that the covariant transformation rules hold. A t fifth order in X\ our explicit expression for V (y) in terms of X is extremely complicated. A l though it may not be of practical use, we present the result in appendix B.3 for completeness. We do not have a proof that an appropriate V^(y) can be constructed to all orders. If it can, it is easy to see that many such objects exist, since we may always construct others from the original one e.g. = V + d V [V^, d V ]. There may be some canonical choice for V^, as we found for the spatial matrix l  s  m  x  1  p,  5  fJ,  1  M  v  p  w  p  Chapter  3. Invariance under Poincare  70  transformations  vector-field in chapter 2, but we do not know the additional constraints that would select this. On the other hand, we will see that any choice for V (assuming one exists) will allow us to construct the most general Poincare invariant actions. 10  3.4.3  4  A covariant matrix distribution  Assuming that the covariant object V (y) exists in the non-Abelian case, it is now trivial to construct scalar fields C(y) simply by taking any product involving and its derivatives such that all indices are contracted with yf . To obtain an invariant action, we should integrate over space-time, but we still need some analogue of the 8 (V) term in chapter 2 that would localize the action to the well-defined positions of the individual branes in the case of diagonal X . We have not been able to construct such a distribution directly from the covariant object V . However, in [2] it was shown that it is possible to construct an object with the appropriate transformation properties directly, at least up to two commutator terms to all orders in X . Our goal is to construct from X (t) a matrix valued field Q(y) such that actions of the form ,i  v  d  1  M  1  l  S = J d y d+1  Tr(C(y)Q(y)),  (3.41)  will be invariant if £ is a scalar built from V^. Here, Q(y) should transform as a density and should contain the matrix generalization of a delta function reducing the integral over d+ 1-dimensional space-time to an integral over the one-dimensional world-line. In other words, it is the matrix generalization of the distribution 9(y) for the single brane case, which takes the form 9(y) = J dr^-drx^drx, We will call it the covariant matrix  5 \x (r) d+  v  - y») .  (3.42)  distribution.  Under Lorentz transformation a density should transform as 0(Ay) = 6(y),  (3.43)  One constraint that we might impose is that V should satisfy d^V^ =d V . This holds in the Abelian case, since V = — ^d^V . In the non-Abelian case, given any definition of V * we can take = - j ^ f V " ^ ) which ensures that is covariant and that d^V is symmetric. 10  v  tl  2  M  v  Chapter  3. Invariance under Poincare  71  transformations  or specifically under an infinitesimal boost 5 e(t, y) = -tp • ve(t, y)-p-y 0  d S(t,y).  (3.44)  tf"  (3.45)  t  Defining the moments of the distribution as @(n-in)( ) t  =  J  d  d  $) ^... y  y  ,  we find that the constraints (3.44) become Q(il-in)  S  =  p(hQi2...i )  nt  _ p^Qdh-in) dt  n  (3  46  )  To zeroth order in the commutators a solution to this constraint is given by eg^")  = Sym (Vl  -X  X  2  ... X ^  (h  .  in  (3.47)  In fact, this is the only solution (modulo an overall constant and rescaling of X) built solely out of X and X. In terms of the density we have at leading order e m(t,y) sy  = Sym  (Vl-*2  S (X(t) d  - y)) ,  (3.48)  where ^ - ^ / ( ^  ,  ' '  * -  (  v  ,  ' '  so that it indeed contains the required d-dimensional delta-functions in the case where X is diagonal. We must now ask whether it is possible to add correction terms to (3.47) such that the constraints (3.46) are satisfied with the non-Abelian transformation rules (3.11). A lengthy calculation in [2] using a transformation rule valid to second order in commutators and all orders in X gives the necessary corrections terms up to second order in commutators and to all orders in X . For the rest of our discussion, the explicit form of these correction terms will not be necessary so we refer the reader to reference [2] for the details. From now on, we will assume that covariant V (y) and 0(y) exist to all orders, and proceed to discuss the Poincare invariant actions. 1  1  fl  Chapter  3.5  3. Invariance under Poincare  transformations  72  Manifestly Lorentz invariant DO-brane actions  Given the vector field V (y) and the covariant matrix distribution Q(y), it is now manifest that any action fi  J d y d+1  Tr (C(y)Q(y))  (3.49)  will be invariant as long as £ is a scalar field built from V and its derivatives. To obtain an explicit expansion of this action in powers of X and its time derivatives, we may use the expansion = E ^ p ^ ' ^ W  Q^V)  di,... d 6 (y),  (3.50)  d  lp  of 0 in terms of its moments. Then the action takes the form S =  fdtf:Tr(^d ...d C(V)\ e ^(t)). =o \P-  (3.51)  i  h  J  ip  yi=0  J  p  Since 0 ( - * p ) = 0(X ), the leading term in the action will come from the set of all terms for which n + order(d^ • • • d C) is a minimum. The expression (3.49) clearly gives rise to a large class of invariant actions. We now show that essentially any invariant action can be written in this form. Fortunately, this can be done without using the detailed form of the expression for 0 . P  l l  in  3.5.1  The most general Poincare invariant action  In section 3.3.1, we showed that the leading term S of any Poincare invariant action could be written using a rotational scalar Lagrangian built from an even number of Xs and an even number of time-derivatives, such that So = / dt T r ( X ) or all Xs and X s appear in commutators. We will now show that any term satisfying these conditions has a Poincare invariant completion that may be written in the form (3.49), and that these completions form a basis for the full set of Poincare invariant actions. First, if So = J dt Tr ( X ) , we can write a Poincare invariant completion as - j d y Tr(0(y)). 0  2  2  d+1  Chapter  3. Invariance under Poincare  73  transformations  Otherwise, the leading order Lagrangian L may be written as a sum of terms for which all X s and Xs appear in commutators. Now L = Tr (£) is a rotational scalar, and by parity and time reversal invariance, must have an even number of Xs and an even number of Xs. Consequently, the index on each matrix (X )^ will pair with the index on some other matrix (X )^ where m and n are the number of time derivatives on the first and second matrix respectively. We now define a matrix object C(y) built out of V by making the following replacements in £ , depending on whether m and n are both even, both odd, or of opposite parity. If m and n have the same parity, we replace 0  0  1  1  11  (X*)  (2fc)  M  • • • (X ) ° {  —(-d ) V^  ( 2  2  (v')(2W)...(vi)(2W)  • ••  k  {-d )% 2  (3.52)  -\(-&) d,y ---(-d ) d V> k  _ ^  v  i  l  i  v  Since the total number of time derivatives is even, there must be an even number of pairs where m and n have opposite parity. We may then group these arbitrarily into pairs of paired X s , and make the replacement  (X0  (2fc)  • • • (X^+V...  (X ) j  (3.53)  • • • (X Y V  i2p)  J  __>  2q+  -(d ) V^...d (-d ) V ---{-d ) V ---d (-d YV 2  k  2  il  l  a  2  p  v  2  u  a  After these replacements, we are left with an object C that transforms as a scalar field, so the action / d y d+1  Tr(C(y)Q(y))  (3.54)  will be Poincare invariant. Furthermore, it is easy to check that in the replacements (3.52) and (3.54), the contributions on the right side which are of lowest order in X have ^-independent terms which are precisely the terms on the left. It is important here that all expressions and d V^ appear in commutators (since we assume all X s and X s do), so that possible lower order terms from the leading y in V vanish. As a result, the leading order term in £(y = 0) is exactly £ , and all of the y-dependent terms in C(y) u  x  1  In particular, any terms involving an odd number of e tensors will not be invariant under parity/reflections, while terms involving an even number may be rewritten using Ss. There will be additional structures involving es which are invariant under the part of the Poincare group continuously connected to the identity but violate either parity or time translation invariance; we will not discuss them further here. 11  Chapter  3. Invariance under Poincare  transformations  74  will only lead to higher order terms in the action, so the action (3.54) will have leading term J dt L . This completes the proof (assuming the existence of covariant objects £ and 0 ) that all Galilean and time-reversal invariant leading terms have Poincare invariant completions that can be written in the form (3.49). To show that the terms just constructed form a basis for all Poincare invariant actions, let us suppose this were not true. Then consider some action S linearly independent from the set Si we have just constructed. Then among all actions S — J2 iSi there must a subset whose leading terms have maximum order. Choose an action S in this subset, and suppose that S has leading term So at order X . By the results in section 2, this term must be Galilean and time-reversal invariant, and we have just seen that So has some Poincare invariant completion 5" that can be written in the form (3.49). But then S — S' is of the form S — XI CiSi and has a leading term of higher order than S , contradicting our assumption. To summarize, we have now shown that every Poincare invariant action has a Galilean and time-reversal invariant leading term, and any such term has a Poincare invariant completion that may be written in the, form (3.49). Finally, the set of such terms form a basis for all possible Poincare invariant actions. 0  c  max  p  max  max  max  3.5.2  Examples  To close this section, we discuss as examples the Poincare invariant completions of the simplest kinetic and potential terms. First, by the results of this section, the most general Poincare invariant completion of the kinetic term (3.28), allowing only terms with as many time derivatives as Xs (i.e. terms that can mix with the leading term under a Lorentz transformation) i s 12  Jd yTv(e(y)[-l d+l  +  C (V(y))}y 4  where £ 4 is an arbitrary scalar built from Vs and an equal number of derivatives, which may without loss of generality be taken to be a term with at least Note that any choice for £ 4 may be absorbed into a redefinition of Q(y). The arbitrariness in 0 corresponds to the freedom to make such redefinitions. 12  Chapter  3. Invariance under Poincare  transformations  75  two commutators of order V or higher. While the result is by no means unique, it is highly constrained relative to the set of all possible translation and rotation invariant actions. As a precise example of the degree to which the action has been constrained, consider all terms with up to two commutators. In this case, there are only a finite number of independent terms in £ 4 that can contribute. To see this, note that the leading term of any such expression may be written schematically as 4  13  STr([X,X][X,X]X---X)  ,  where the total number of Xs is 4 + 2n for some n , and the total number of time derivatives must be equal to this. For Galilean invariance, all Xs outside commutators must have at least two time derivatives, so there must be at least 4 n time derivatives. Then 4n < 2n + 4,  so we have n < 2. It is then easy to write down all possible leading terms containing two commutators; up to total derivatives we find 8, 17, and 2 terms respectively for n equal to 0, 1, and 2. Thus, the most general Poincare invariant completion of the kinetic term (3.28) contains 27 arbitrary coefficients up to terms involving more than two commutators. On the other hand, the number of independent translation and rotation invariant terms with equal numbers of Xs and time derivatives and up to two commutators is infinite, so we see that the additional requirement of boost invariance is indeed a severe constraint on the action. As a second example, we consider the Poincare invariant completion of the potential term (3.29). Allowing only terms that can mix with the leading term under a Lorentz transformation, the most general invariant completion r  i s  C J d y d+1  Tr (e(y)([V„V }[V^V} l/  +  C (V(y))). 6  where CQ is the general linear combination of all scalars built from n Vs and n — 4 derivatives. Without loss of generality, n may be taken to be at least 6, and all terms in C§ may be taken to have at least 3 commutators. Thus, This follows since the leading term in any higher order invariant action will be at least of order X and by the construction of the previous subsection, we may construct such an action using an C with terms of order V and higher. 13  4  4  Chapter  3. Invariance  under Poincare  transformations  76  the full set of two-commutator terms in the Poincare invariant completion of (3.29) are uniquely determined to be  c Jd  Tr  d+l y  (e (y)([v; ,v; j[v; ^v; '])). t  sym  ymM  ym  yrn  ym  where 0 and V are the symmetrized parts of 0 and V. Note that based on rotation and translation invariance alone, the full set of allowed two-commutator correction terms to potential (3.29) is s y m  sym  STr(b [X\X }{X\  X ]X  j  j  n  2n  + c [X\ X ][X\  X ]X X X ),  j  k  n  j  k  2n  n  so in this case, the additional constraint of boost invariance fixes the infinite series of coefficients b and c„ completely. To close this section, we note that our structures V* and 0 provide an alternate way to write invariant actions even in the Abelian case. For example, using our prescription, the Galilean invariant term \x has Lorentz invariant completion n  1  2  Jd y d+1  d v»d\9{y). 2  Using the Abelian expression (3.42) for 6 and those in appendix B.3 for v^, this reduces precisely to the right side of (3.25).  3.6  Lorentz covariant currents  We have seen that the requirement of Poincare invariance places severe constraints on the form of the effective action. In this section, we note that similar constraints arise in the expressions for the conserved space-time currents associated with the branes. We use the example of the DO-brane current for DO-branes in uncompactified type IIA string theory, which couples to the Ramond-Ramond one-form field of type IIA supergravity. Identical considerations apply to the other currents, which include the stress-energy tensor, the higher brane currents, and the string current (which couples to the NS-NS two-form). The DO-brane current J^(y) appears in the effective action coupled to the Ramond-Ramond one-form CJP as S =  pjd yC^(y)J^y). w  (3.55)  Chapter  3. Invariance under Poincare  77  transformations  Since is a Lorentz vector, J (y) must be some expression built from X (t) transforming as a vector under Lorentz transformations. A t low energies/small velocities, the leading order expression for J^(y) — (p(y),J (y)) (ignoring fermions) is a simple generalization of the Abelian expression [75], p  l  l  p(t,y)  =  / Tr (S (X(t)-y))  \  d  r dk = J-jLL. 9  Tr ( e  i f c l  ^),  (3.56)  r dh 9  J%y) = Tr (X*(t) 5 (X(t) - y)) = J It is easy to check that current conservation, d  c V  ^  Tr ( X V  f e J  = 0,  ^). (3.57)  is satisfied with these definitions. However, we will now see that J does not transform as a vector under Lorentz transformations (without additional correction terms). A Lorentz vector field J should transform under Lorentz transformations as M  M  >(Ay)  =  A" J"(y).  (3.58)  v  This implies that under an infinitesimal boost we have P-J{t,y)-0-Vp(t,y)-P-yd {t,y),  5 p{t,y)  =  6 J(t,y)  = Pp(t,fl-tP-VJ{i,fl-p-ydtJ(t,y).  p  0  tP  (3.59)  It is convenient to define multipole moments of the current components as in (3.45). In terms of these, the constraints of Lorentz covariance read •••<») + n £/? V (  fypfr-'-)  =  p.  fyjfr-*»>  =  ^p( "-^)+nt/?  JIM  i l  ( i l  " -  2  in)  &  JP ' ' {Jiv in)  t  J' "^ -/^^J^ -" " . 2  )  i l  i  )  (3.60)  Using the non-Abelian transformation rules (3.11), we may now check whether these relations are satisfied for the moments that follow from the leading expressions (3.56) for the currents, namely p  ^  ^sym'  = i n )  =  STv(X^---X^), STrCX'X* ---**"). 1  (3.61)  Chapter  3. Invariance under Poincare  transformations  78  It is easy to check that all the constraints (3.60) are satisfied with the expressions (3.61) in the Abelian case or for diagonal matrices, but are not satisfied in general. Thus, the full Lorentz covariant DO-brane currents must include additional higher order terms involving matrix commutators, and these correction terms should be heavily constrained by (3.60). In [2], it was verified that up to two commutator terms and to all orders in X there do exist corrections to the currents such that (3.60) are satisfied. The calculation is quite involved, and we refer the reader to the reference for the details. 1  79  Chapter 4 Summary and outlook In this thesis, we have presented some progress towards understanding general covariance in multiple DO-brane actions. For the restricted problem of general covariance with respect to spatial diffeomorphisms discussed in chapter 2, we have developed a covariant formalism that emphasizes the idea of using fields defined on space-time in favour of world-volume degrees of freedom. Specifically, we discovered that if we could find a matrix-valued field V (y) (constructed out of the matrices X ) that satisfies certain well-defined constraints, then it is possible to explicitly determine the transformation rule for the matrix coordinates X of the DO-branes under arbitrary spatial diffeomorphisms. Furthermore, this V can be used to write down essentially any generally covariant action for X coupled to a spatial metric in a form that makes the covariance obvious. While we were not able to prove the existence of V (y), we were able to construct it as an expansion in powers of X — y to fourth order. This approach can be extended to include more general coordinate transformations. For example, an arbitrary space-time coordinate transformation can be obtained from the composition of a time-dependent spatial coordinate transformation that leaves the time coordinate unchanged l  1  1  1  1  i  1  t = t,  $ = ^{1^),  l  (4.1)  together with a position dependent transformation of the time coordinate, leaving the spatial coordinates unchanged i=F°(t yi) 1  y = y\ {  >  (4.2)  Pictorially, we can imagine a coordinate system defined through a foliation of space-time by a set of hypersurfaces (defining what is meant by different points at the same time), together with a congruence of curves that cut across these hypersurfaces (defining what is meant by the same point in space at different times). The first set of coordinate transformations leaves the hypersurfaces fixed, but changes the congruence of curves. The second set of  Chapter  4.  Summary and  outlook  80  coordinate transformations changes the hypersurfaces that foliate space-time, but leaves the congruence of curves fixed. In unpublished work with David Gosset and Mark Van Raamsdonk, we have investigated the extension of these methods to the case of time-dependent, spatial coordinate transformation (4.1). In this context, the coupling of multiple DO-branes to a general metric (not necessarily restricted to be non-trivial in the spatial directions only) was considered. It appears that the methods presented in chapter 2 can be adapted successfully to this case as well. It would be interesting to make contact with the early efforts of [97, 98, 99] towards constructing actions for D-branes in curved space (especially Kahler manifolds) as possible starting points of defining Matrix theory in non-trivial backgrounds (see also [100]). For this purpose, it may be useful to consider a version of our formalism specific to holomorphic coordinates developed in [101]. It would also be interesting to explore whether the class of covariant actions presented in chapter 2 predicts any interesting generic phenomena for D-branes in curved space, such as the gravitational version of the dielectric effect [102, 103, 104, 105]. As a first step towards understanding general diffeomorphisms that mix the space and time coordinates (including the those which change the time coordinate in a non-trivial manner such as (4.2)), we have considered the problem of incorporating Poincare invariance in multiple DO-brane actions. Our results, presented in chapter 3, are more modest in this case. We attempted to make as much progress as possible while remaining in the static gauge description. There is some evidence that Lorentz transformations can be defined on the matrix-valued static gauge embedding coordinates, although the explicit expressions appear complicated. Again, we attempted to use the strategy of replacing world-volume degrees of freedom with space-time fields. We find evidence for the existence of a number of matrix-valued space-time fields which transform covariantly. These can be used to construct manifestly Poincare-invariant actions. Unfortunately, we have not found a single object out of which all of these covariant objects can be expressed, and it appears necessary to construct each one of the covariant objects separately through an order-by-order procedure. The results we have found indicate a deeper structure which we have not been able to fully uncover. Perhaps one way forward is to take seriously the idea of introducing a matrix X° associated with the time direction as well. The essential problem would then be to find an appropriate gauged symmetry (a generalization of world-volume reparametrization invariance) that is capable of eliminating the  Chapter  4.  Summary and  81  outlook  X° degrees of freedom. Recently, there has been progress along these lines in [106, 107]. These authors do not consider explicitly the matrix degrees of freedom. Instead, they utilize a set of fermionic degrees of freedom on the D-brane world-volume, which upon quantization become matrices. A l l manipulations presented in [106, 107] are done on the world-volume fermions, with the matrices replacing them only at the final stages of the calculation. Finally, if such a description involving matrix coordinates for all directions can be found, then it may be possible to adapt the methods of chapter 2 to achieve complete general covariance. In this context, one would have a description with a matrix-valued space-time vector field V (y ) replacing the space-time embedding coordinates X^. The description for D-branes of other dimensions would appear to be the same, except that the space-time vector field satisfies different constraints depending on the dimension of the D-branes. tx  u  82  Bibliography [1] Dominic Brecher, Kazuyuki Furuuchi, Henry Ling, and Mark Van Raamsdonk. Generally covariant actions for multiple D-branes. JHEP, 06:020, 2004. hep-th/0403289. [2] Dominic Brecher, Paul Koerber, Henry Ling, and Mark Van Raamsdonk. Poincare invariance in multiple D-brane actions. JHEP, 01:151, 2006. hep-th/0509026. [3] Henry Ling, A l i Reza Mohazab, Hsien-Hang Shieh, Greg van Anders, and Mark Van Raamsdonk. Little string theory from a double-scaled matrix model. JHEP, 10:018, 2006. hep-th/0606014. [4] Henry Ling, Hsien-Hang Shieh, and Greg van Anders. Little string theory from double-scaling limits of field theories. JHEP, 02:031, 2007. hep-th/0611019. [5] Michael B . Green, J. H . Schwarz, and Edward Witten. Superstring Theory. Vol. 1: Introduction. Cambridge, Uk: Univ. Pr. (1987) 469 p. (Cambridge Monographs On Mathematical Physics). [6] Michael B . Green, J . H . Schwarz, and Edward Witten. Superstring Theory. V o l . 2: Loop Amplitudes, Anomalies and Phenomenology. Cambridge, Uk: Univ. Pr. (1987) 596 p. (Cambridge Monographs On Mathematical Physics). [7] J. Polchinski. String theory. V o l . 1: A n introduction to the bosonic string. Cambridge, U K : Univ. Pr. (1998) 402 p. [8] J. Polchinski. String theory. V o l . 2: Superstring theory and beyond. Cambridge, U K : Univ. Pr. (1998) 531 p. [9] B . Zwiebach. A first course in string theory. Cambridge, U K : Univ. Pr. (2004) 558 p.  83  Bibliography  [10] Joseph Polchinski. Dirichlet-Branes and Ramond-Ramond Charges. Phys. Rev. Lett, 75:4724-4727, 1995. hep-th/9510017. [11] Tom Banks, W . Fischler, S. H . Shenker, and Leonard Susskind. M theory as a matrix model: A conjecture. Phys. Rev., D55:5112-5128, 1997. hep-th/9610043. [12] Washington Taylor. M(atrix) theory: Matrix quantum mechanics as a fundamental theory. Rev. Mod. Phys., 73:419-462, 2001. hepth/0101126. [13] Andrew Strominger and Cumrun Vafa. Microscopic Origin of the Bekenstein-Hawking Entropy. Phys. Lett, B379:99-104, 1996. hepth/9601029. [14] Curtis G . Callan Jr. and Juan M . Maldacena. D-brane Approach to Black Hole Quantum Mechanics. Nucl. Phys., B472:591-610, 1996. hep-th/9602043. [15] Juan M . Maldacena. The large N limit of superconformal field theories and supergravity.  Adv.  Theor. Math.  Phys., 2:231-252, 1998.  hep-  th/9711200. [16] Ofer Aharony, Steven S. Gubser, Juan M . Maldacena, Hirosi Ooguri, and Yaron Oz. Large N field theories, string theory and gravity. Phys. Rept, 323:183-386, 2000. hep-th/9905111. [17] Gary T. Horowitz and Joseph Polchinski. Gauge/gravity duality. 2006. gr-qc/0602037. [18] Ralph Blumenhagen, Mirjam Cvetic, Paul Langacker, and Gary Shiu. Toward realistic intersecting D-brane models. Ann. Rev. Nucl. Part. Sci., 55:71-139, 2005. hep-th/0502005. [19] S. H . Henry Tye. Brane inflation: String theory viewed from the cosmos. 2006. hep-th/0610221. [20] Edmund J. Copeland, Robert C. Myers, and Joseph Polchinski. Cosmic F- and D-strings. JHEP, 06:013, 2004. hep-th/0312067.  84  Bibliography  [21] Martin Kruczenski, David Mateos, Robert C. Myers, and David J. Winters. Towards a holographic dual of large-n(c) qcd. JHEP, 05:041, 2004. [22] David Mateos, Robert C. Myers, and Rowan M . Thomson. Holographic viscosity of fundamental matter. Phys. Rev. Lett., 98:101601, 2007. [23] C. V . Johnson. D-branes. Cambridge, USA: Univ. Pr. (2003) 548 p. [24] Joseph Polchinski. Lectures on D-branes. 1996. hep-th/9611050. [25] Washington Taylor. Lectures on D-branes, M(atrices). 1997. hep-th/9801182.  gauge theory  and  [26] Gary T. Horowitz and Andrew Strominger. Black strings and p-branes. Nucl. Phys., B360:197-209, 1991.  [27] Jin Dai, R. G . Leigh, and Joseph Polchinski. New Connections Between String Theories. Mod. Phys. Lett, A4:2073-2083, 1989. [28] Michael B . Green. Space-time duality and Dirichlet string theory. Phys. Lett, B266:325-336, 1991. [29] Petr Horava. Background Duality of Open String Models. Phys. Lett., B23L251, 1989. [30] M . Bianchi, G . Pradisi, and A . Sagnotti. Toroidal compactification and symmetry breaking in open string theories. Nucl. Phys., B376:365-386, 1992. [31] Joseph Polchinski, Shyamoli Chaudhuri, and Clifford V . Johnson. Notes on D-Branes. 1996. hep-th/9602052. [32] Robert C. Myers. Nonabelian phenomena on D-branes. Class. Quant, Grav., 20:S347-S372, 2003. hep-th/0303072. [33] A . A . Tseytlin. Born-Infeld action, supersymmetry and string theory. 1999. hep-th/9908105. [34] Washington Taylor and Mark Van Raamsdonk. Multiple dp-branes in weak background fields. Nucl. Phys., B573:703-734, 2000. hepth/9910052.  85  Bibliography  [35] R. G. Leigh. Dirac-Born-Infeld Action from Dirichlet Sigma Model. Mod. Phys. Lett, A4:2767, 1989.  [36] Michael B. Green, Christopher M. Hull, and Paul K. Townsend. Dbrane Wess-Zumino Actions, T-Duality and the Cosmological Constant. Phys. Lett, B382:65-72, 1996. hep-th/9604119. [37] Michael R. Douglas. Branes within branes. 1995. hep-th/9512077. [38] Miao Li. Boundary States of D-Branes and Dy-Strings. Nucl. B460:351-361, 1996. hep-th/9510161.  Phys.,  [39] Enrique Alvarez, J. L. F. Barbon, and J. Borlaf. T-duality for open strings. Nucl. Phys., B479:218-242, 1996. hep-th/9603089. [40] E. Bergshoeff and M. De Roo. D-branes and T-duality. Phys. Lett., B380:265-272, 1996. hep-th/9603123. [41] Yeuk-Kwan E. Cheung and Zheng Yin. Anomalies, branes, and currents. Nucl. Phys., B517:69-91, 1998. hep-th/9710206. [42] Michael B. Green, Jeffrey A. Harvey, and Gregory W. Moore. I-brane inflow and anomalous couplings on D-branes. Class. Quant. Grav., 14:47-52, 1997. hep-th/9605033. [43] Mohammad R. Garousi and Robert C. Myers. Superstring scattering from D-Branes. Nucl. Phys., B475:193-224, 1996. hep-th/9603194. [44] S. S. Gubser, A. Hashimoto, Igor R. Klebanov, and Juan M. Maldacena. Gravitational lensing by p-branes. Nucl. Phys., B472:231-248, 1996. hep-th/9601057. [45] Akikazu Hashimoto and Igor R. Klebanov. Scattering of strings from Dbranes. Nucl. Phys. Proc. Suppl, 55B:118-133, 1997. hep-th/9611214.  [46] Igor R. Klebanov and Larus Thorlacius. The Size of p-Branes. Lett, B37L51-56, 1996. hep-th/9510200.  Phys.  [47] Ahmed Abouelsaood, Curtis G. Callan Jr., C. R. Nappi, and S. A. Yost. Open Strings in Background Gauge Fields. Nucl. Phys., B280:599, 1987.  86  Bibliography  [48] Curtis G. Callan Jr., C. Lovelace, C. R. Nappi, and S. A . Yost. String Loop Corrections to beta Functions. Nucl. Phys., B288:525, 1987. [49] Curtis G. Callan Jr., E . J. Martinec, M . J. Perry, and D. Friedan. Strings in Background Fields. Nucl. Phys., B262:593, 1985. [50] Harald Dorn and H . J. Otto. Open Bosonic Strings In General Background Fields. Z. Phys., C32:599, 1986. [51] A . A . Tseytlin. Self-duality of Born-Infeld action and Dirichlet 3-brane of type IIB superstring theory. Nucl. Phys., B469:51-67, 1996. hepth/9602064. [52] O. D. Andreev and A . A . Tseytlin. Partition Function Representation for the Open Superstring Effective Action: Cancellation of Mobius Infinities and Derivative Corrections to Born-Infeld Lagrangian. Nucl. Phys., B31L205, 1988. [53] A . A . Tseytlin. Renormalization of Mobius Infinities and Partition Function Representation for String Theory Effective Action. Phys. Lett, B202:81, 1988. [54] Christian Schubert. Perturbative quantum field theory in the stringinspired formalism. Phys. Rept, 355:73-234, 2001. hep-th/0101036. [55] Paul Koerber. Abelian and non-Abelian D-brane effective actions. Fortsch. Phys., 52:871-960, 2004. hep-th/0405227. [56] Mina Aganagic, Costin Popescu, and John H. Schwarz. D-brane actions with local kappa symmetry. Phys. Lett., B393:311-315, 1997. hepth/9610249. [57] Mina Aganagic, Costin Popescu, and John H . Schwarz. Gaugeinvariant and gauge-fixed D-brane actions. Nucl. Phys., B495:99-126, 1997. hep-th/9612080. [58] E. Bergshoeff and P. K . Townsend. Super D-branes. B490:145-162, 1997. hep-th/9611173.  Nucl.  Phys.,  Bibliography  87  [59] Martin Cederwall, Alexander von Gussich, Bengt E. W . Nilsson, Per Sundell, and Anders Westerberg. The Dirichlet super-p-branes in tendimensional type IIA and IIB supergravity. Nucl. Phys., B490:179-201, 1997. hep-th/9611159. [60] Martin Cederwall, Alexander von Gussich, Bengt E. W . Nilsson, and Anders Westerberg. The Dirichlet super-three-brane in tendimensional type IIB supergravity. Nucl. Phys., B490:163-178, 1997. hep-th/9610148. [61] Donald Marolf, Luca Martucci, and Pedro J. Silva. Fermions, T-duality and effective actions for D-branes in bosonic backgrounds. JHEP, 04:051, 2003. hep-th/0303209. [62] Donald Marolf, Luca Martucci, and Pedro J . Silva. Actions and fermionic symmetries for D-branes in bosonic backgrounds. JHEP, 07:019, 2003. hep-th/0306066. [63] Luca Martucci, Jan Rosseel, Dieter Van den Bleeken, and Antoine Van Proeyen. Dirac actions for D-branes on backgrounds with fluxes. Class. Quant. Grav., 22:2745-2764, 2005. hep-th/0504041. [64] Edward Witten. Bound states of strings and p-branes. Nucl. Phys., B460:335-350, 1996. hep-th/9510135. [65] A . A . Tseytlin. On non-abelian generalisation of the Born-Infeld action in string theory. Nucl. Phys., B50L41-52, 1997. hep-th/9701125. [66] David J. Gross and Edward Witten. Superstring Modifications of Einstein's Equations. Nucl. Phys., B277:l, 1986. [67] A. A . Tseytlin. Vector Field Effective Action in the Open Superstring Theory. Nucl. Phys., B276:391, 1986; Erratum-ibid. B29L876, 1987. [68] D. Brecher and M . J. Perry. Bound states of D-branes and the nonAbelian Born-Infeld action. Nucl. Phys., B527:121-141, 1998. hepth/9801127. [69] Akikazu Hashimoto and Washington Taylor. Fluctuation spectra of tilted and intersecting D-branes from the Born-Infeld action. Nucl. Phys., B503:193-219, 1997. hep-th/9703217.  88  Bibliography  [70] Frederik Denef, Alexander Sevrin, and Jan Troost. Non-Abelian BornInfeld versus string theory. Nucl. Phys., B581:135-155, 2000. hepth/0002180. [71] Pascal Bain. th/9909154.  On the non-Abelian Born-Infeld action.  1999.  hep-  [72] Luiz Antonio Barreiro and Ricardo Medina. 5-field terms in the open superstring effective action. JEEP, 03:055, 2005. hep-th/0503182. [73] D. Oprisa and S. Stieberger. Six gluon open superstring disk amplitude, multiple hypergeometric series and Euler-Zagier sums. 2005. hepth/0509042. [74] Pedro Bordalo, Lorenzo Cornalba, and Ricardo Schiappa. Towards quantum dielectric branes: Curvature corrections in abelian beta function and nonabelian Born-Infeld action. Nucl. Phys., B710:189-254, 2005. hep-th/0409017. [75] Washington Taylor and Mark Van Raamsdonk. Multiple dO-branes in weakly curved backgrounds. Nucl. Phys., B558:63-95, 1999. hepth/9904095. [76] Robert C. Myers. th/9910053.  Dielectric-branes.  JEEP,  12:022, 1999.  hep-  [77] Enrique Alvarez, Luis Alvarez-Gaume, and Yolanda Lozano. A n introduction to T duality in string theory. Nucl. Phys. Proc. Suppi, 41:1-20, 1995. hep-th/9410237. [78] Eric Bergshoeff, Christopher M . Hull, and Tomas Ortin. Duality in the type II superstring effective action. Nucl. Phys., B45L547-578, 1995. hep-th/9504081. [79] Nathan Seiberg. Why is the matrix model correct? Phys. Rev. 79:3577-3580, 1997. hep-th/9710009.  Lett,  [80] Ashoke Sen. DO branes on T(n) and matrix theory. Adv. Theor. Math. Phys., 2:51-59, 1998. hep-th/9709220. [81] Joseph Polchinski. M-theory and the light cone. Prog. Theor. Phys. Suppl, 134:158-170, 1999. hep-th/9903165.  Bibliography  89  [82] Washington Taylor and Mark Van Raamsdonk. Supergravity currents and linearized interactions for matrix theory configurations with fermionic backgrounds. JHEP, 04:013, 1999. hep-th/9812239. [83] Mohammad R. Garousi and Robert C. Myers. World-volume interactions on D-branes. Nucl. Phys., B542.-73-88, 1999. hep-th/9809100. [84] Harald Dorn. Nonabelian gauge field dynamics on matrix D-branes. Nucl. Phys., B494:105-118, 1997. hep-th/9612120. [85] C. M . Hull. Matrix theory, U-duality and toroidal compactifications of M-theory. JHEP, 10:011, 1998. hep-th/9711179. [86] Mohammad R. Garousi and Robert C. Myers. World-volume potentials on D-branes. JHEP, 11:032, 2000. hep-th/0010122. [87] J. Hoppe. Quantum Theory of a Massless Relativistic Surface and a Two-Dimensional Bound State Problem. Elem. Part. Res. J. (Kyoto), 80:145, 1989. M I T Ph.D. thesis, 1982 (available at http://www.aei-potsdam.mpg.de/~hoppe). [88] B . de W i t , J . Hoppe, and H . Nicolai. On the quantum mechanics of supermembranes. Nucl. Phys., B305:545, 1988. [89] J. Madore. A n Introduction to Noncommutative Differential Geometry and its Physical Applications, 2nd Edition. Cambridge, Uk: Univ. Pr. (1999) 378 p. (London Mathematical Society Lecture Note Series, No. 257). [90] Daniel Kabat and Washington Taylor. Spherical membranes in matrix theory. Adv. Theor. Math. Phys., 2:181-206, 1998. hep-th/9711078. [91] Jan De Boer and Koenraad Schalm. General covariance of the nonAbelian DBI-action. JHEP, 02:041, 2003. hep-th/0108161. [92] Mark Van Raamsdonk. Blending local symmetries with matrix nonlocality in D-brane effective actions. JHEP, 09:026, 2003. hepth/0305145. [93] Soo-Jong Rey. Gravitating M(atrix) Q-balls. 1997. hep-th/9711081.  90  Bibliography  [94] Yuji Okawa and Hirosi Ooguri. How noncommutative gauge theories couple to gravity. Nucl. Phys., B599:55-82, 2001. hep-th/0012218. [95] Koji Hashimoto. The shape of non-Abelian D-branes. JHEP, 2004. hep-th/0401043.  04:004,  [96] S. F . Hassan. N = l worldsheet boundary couplings and covariance of non- Abelian worldvolume theory. 2003. hep-th/0308201. [97] Michael R. Douglas. D-branes and matrix theory in curved space. Nucl. Phys. Proc. Suppl, 68:381-393, 1998. hep-th/9707228.  [98] Michael R. Douglas. D-branes in curved space. Phys., 1:198, 1998. hep-th/9703056.  Adv. Theor. Math.  [99] Michael R. Douglas, Akishi Kato, and Hirosi Ooguri. D-brane actions on Kaehler manifolds. Adv. Theor. Math. Phys., 1:237, 1998. hepth/9708012. [100] Keshav Dasgupta, Mohammad M . Sheikh-Jabbari, and Mark Van Raamsdonk. Matrix perturbation theory for M-theory on a P P wave. JHEP, 05:056, 2002. hep-th/0205185. [101] Kazuyuki Furuuchi. Holomorphically covariant matrix models. 2004. hep-th/0406165. [102] Jan de Boer, Eric Gimon, Koenraad Schalm, and Jeroen Wijnhout. Evidence for a gravitational Myers effect. Annals Phys., 313:402-424, 2004. hep-th/0212250. [103] Yoshifumi Hyakutake. Gravitational dielectric effect and Myers effect. Phys. Rev., D71:046007, 2005. hep-th/0401026. [104] Vatche Sahakian. On DO brane polarization by tidal forces. 04:038, 2001. hep-th/0102200.  JHEP,  [105] Vatche Sahakian. Transcribing spacetime data into matrices. 06:037, 2001. hep-th/0010237.  JHEP,  [106] P. S. Howe, U . Lindstrom, and L . Wulff. Superstrings with boundary fermions. JHEP, 08:041, 2005. hep-th/0505067.  Bibliography  91  [107] P. S. Howe, U. Lindstrom, and L. Wulff. On the covariance of the Dirac-Born-Infeld-Myers action. JHEP, 02:070, 2007. hep-th/0607156.  92  Appendix A Appendix to chapter 2 A.l  Spatial metric conventions and useful formulae  The metrics appearing in chapter 3 are all spatial metrics, with positive signature. The Christoffel symbol is defined by 1% = \g" (-dwjk + d  (A.l)  + d ).  j9kl  k9jl  Under a general coordinate transformation y = F (y), the Christoffel symbol transforms as l  l  « ~, dldy^dT , , _ dy dy df jk\V) dy dyi d y dyi dy dy dy ' The solution to the geodesic equation m  (  =  l  1  k m n { y >  with £ (0) = y and d£, /d\(Q) l  l  n  2  rl  0 0  C(\) =  y  m  n  1  ' '  = u\ is given by  l  i  k  + \u -Y: i  \ -rl., (y)u ... n  (AA)  h  n  n=2  U  -  Here, the extended Christoffel symbols T^  j n  can be denned recursively  r},..;„ = v ( j l r $ 2 . . , n ) ,  (A.5)  where V j is the usual covariant derivative defined by the Christoffel symbol, but acting only on lower indices in the above equation. The round brackets denote complete symmetrization of the indices. A Riemann normal coordinate system about the point y is a coordinate system in which Tjj j (yo) = 0, so that geodesies which pass through the point y are given by £* = y + Xu\ Finally, our conventions for the Riemann tensor are %  0  l  n  0  l  0  R)H = dkT) t  d r + r^r™ t  jk  - r^r™.  (A.e)  Appendix  A.2  A. Appendix  93  to chapter 2  Solving the constraints for V  1  In this section we wish to show in greater detail how the procedure outlined in section 2.3.4 can be carried out. We start with the expression (2.35) V\y) = A* + VJ A A j  + Vj A A A ...,  k  j  k  k  (A.7)  l  kl  where once again A = X — y . To reiterate, our objective is to find a suitable set of coefficients VI „• such that the covariant derivative of V (y) can be expanded as a series in V (y) with tensor coefficients. The series for V (y) can be inverted so that we have 1  x  1  l  l  z  A * = V + A) V V i  j  + A) V V V ....  k  j  k  k  .  l  kl  (A.8)  The lowest order expansion coefficients for the two series are related as follows jk  v  Vj  kl  Vjklm  -  ^jk  = -A)  + (A*- A£j + A J A " )  kl  a  ~ )klm  =  a  + ( ja klm  A  A  fc  + am jkl)  A  A  (A-9)  A<  +(A}fc A" + A * A £ , + A ^ A " ) — a  m  am  -(A) At A» a  b  m  (A A^ A ) b  f c  + A } A L A L + A A<] A  ab  b  lm  a  am  b  kl  +  k  A i  lrn  m  A £ A }  f  c  ) .  We compute the covariant derivative V j V = djV + Tj V using equation (A.7), and then eliminate the factors of A by substitution of (A.8), so that we obtain an expansion in powers of V . 1  1  k  k  1  1  VjV* = -6} + T V V k  + T V VV  l  k  jkl  Explicitly, the coefficients T£ pi  rpi  jk  1  T]  kl  Tjklm  -  =  J n  l  ....  m  jklm  are  T / i  jk  jk  l  V  d V \-V^-Vy4i3  (A.10)  (A.11)  k  — djVklm ~ Vjklm + ( j^ka ~ Vjka) ?m 8  A  Here the underline notation means that we take all terms that can be obtained by moving the first lower index around, but keeping the relative order of the other lower indices fixed, and add them together. For example  Appendix  A. Appendix  94  to chapter 2  Vjkim = Vjkim + kjim + kijm + k\my % subsituting the Vs in terms of A s v  V  V  we can express the Ts in terms of the A s completely. Now we require that the T coefficients be tensors, and since there are no three index tensors that can be built only from the metric, this means Tj vanishes. This allows us to simplify the expressions, and the result is fc  =  Tjfcjm  Aj  — VjA  l  f c i m  klm  — A Tj ka  (A. 12)  — A^ Tj .  lm  m  kl  Here the covariant derivatives on the A s act on the lower indices only. We can write an ansatz for a solution to these equations as a sum of two terms  A  ) l J 2 - j n  =  ~ ~ \  r  ) l h - 3 n  +  C  (A.13)  j l J 2 - 3 n '  where the first term reproduces the definition of the ordinary exponential map, and the second term produces a commutator expression that vanishes when the V matrices are diagonal. Since the T tensors appearing on the left-hand side of equations (A. 12) are unconstrained, these equations do not determine the A s uniquely. We can narrow down the possibilities by demanding that the A s produce a transformation law that satisfy all of the additional constraints outlined in section 2.2.3. Specifically, we can ensure that the resulting transformation law satisfies properties (v) and (vi) in section 2.2.3 by making the ansatz  ,• = E H i 3l'"3n  Z—/  i kT-  Jl-.-JmK  + 1  ,• +T  3m+l---3n  h  i , 31-"3n  1  (A. 14) ^  '  m=2  where T are any tensors constructed out of the Christoffel symbol and its derivatives, without the explicit use of the metric. If it turns out that these extra conditions are too restrictive so as to prevent the existence of any solutions, then we may have to relax them. So far, we have not encountered any problems up to fourth order. Note that the last tensorial T term on the right hand side does not contribute to the transformation rule. For example, at third order these correspond to the arbitrary constants a and b in equation 2.48. We may disallow such pure tensor terms to further reduce the possibilities.  Appendix  A. Appendix  95  to chapter 2  The transformation law and covariant vector at fourth order Up to fourth order in V, we find that the most general possible solution of the explicit constraints we have just outlined (with all pure tensor terms vanishing) is A  i  yi _ ±r Vi jk  =  - ^F V>V V V  - ~T) V V V 3  k  V  k  l  k  kl  + ^ n klrn(y [y ^ i  R  m  (A.15)  m  , V]] + ^ ( V ) )  {  j  Here, the expression C (X)  l  jklm  +  0(V ). 5  indicates any complete commutator expression,  ljkl  -^—C (X de  + e) = 0.  ijkl  m  y  1  There are six independent terms of this type, C (X)  = a[X\X }[X ,X ]  ijkl  j  +d[Xf  k  + c[X\ [X , [X ,X*]]]  + b[X\X ][X ,X ]  l  k  j  [X\ [X , X }}} + e[X , [X\ [Xf X }}} + f[X , k  1  k  fc  j  l  1  [Xf [X\ X }}}.  k  1  With this definition, the covariant derivative of V is a tensor given by V  . y i = _ {-(l { 5  R  ii  \  +  R  l)j  ^  V  (A  .16)  Note that the arbitrary coefficients in (A.15) do not affect (A. 16) until the next order. From the relation between X and V, we find the following transformation law for X up to fourth order in X (the function F , the metric and all of its derivatives are evaluated at y = 0 ) : l  ¥(F,  X, g)  =  F + dFX l  i  j  j  -^d d F m  + \d d F X X i  j  k  +^-d d d F  1  l  n  m  j  k  l  XXX X j  l  m  k  -X X X X  -  XXX X )  l  j  T {X X X Xi k  i  j  XXX X  l  m  l  m  m  k  -  k  l  ^d d d d F X^X X X  -  k  j  k  +X X X X j  klm  i  j  +  k  m  n  ^-d d diF X X X  +  Tf (X X X X  i  k  +  k  T™,[Xf [X ,X ]}  k  j  j  k  j  l  l  k  j  k  X XXX k  l  j  m  m  m  m  k  l  m  Appendix  A.  Appendix  to chapter 2  -X X X X k  j  l  +^-d d F  T T (X X X X  +^-d d F  Y T {X X X X  m  i  n  n  p  i  j  p  jk  n  n  j  p  kp  l  k  m  lm  j  l  m  +  XX XX )  -  XXX X )  -2X X X X l  m  j  -^dnF*  m  R (X [X^ , n  m  klm  k  k  [X \X ]) l  3  l  l  m  m  k  XX XX j  k  l  m  X XX X k  l  m  j  2X X X X )  -  j  k  j  +  k  +X X X X l  j  +  k  m  96  k  j  + V (X)) klm  l  m  +  O(X').  A l l of the arbitrary coefficients in V appear in the transformation law. This is expected, since the arbitrary coefficients that do not affect the transformation law are associated with pure tensor terms in (A. 14), and we have not allowed such terms to enter into our solution. Useful expressions for expanding actions about a point In writing out the expansion of our integrated actions in powers of X\ it is convenient to have, for reference, the following expressions for the Taylor series coefficients K--i  n  = ^dtr  • • diAK + y%=o  in the expansion of V + y in powers of y. Using (A. 15) find y  w} = -r;. x -i(rj ^^ fc  fc  fc  As discussed in section (2.4.5), we can go to Riemann normal coordinaes and replace X by V in any expanded action to obtain a base-point independent action depending only on V and tensors built from the metric. So it is also  Appendix  A. Appendix  97  to chapter 2  useful to have expressions for Ws in Riemann normal coordinates. Going to this system, we have for example, -3-"(fci)j  4  ~Q  \lm)(j;k)  R  +  so that we obtain W = X + 0(X ) l  l  4  wj =  ™*U*' -1 v,j%x<^> - iv Hf  w;  = 0(VVRX).  k  klm  A.3  r a  c m f)  y  A:('x™)  lHfc)  + 0(X  3  )  (A. 18)  A technical result about commutator expressions  In this appendix, we argue that for any expression £ — A^.^X • • •X with more than three Xs such that T r ( £ ) satisfies condition (2.58), that is 11  d Tr(£) xi  = 0,  ln  (A.19)  there is another expression £ ' , equivalent under the trace (Tr(£') = Tr(£)) satisfying d iC = 0. X  For simplicity, we discuss terms without X s , but our argument can easily be extended to include them since they trivially satisfy d iX = 0. Note first that any symmetry properties of the coefficients A can be transferred directly to the product of X s by replacing the ordering shown with an average over all orderings that give the same result when contracted with A. Writing the resulting sum over permutations as X  Q h - i u  =  ^ a j M D ...XM«).  Appendix  A. Appendix  98  to chapter 2  It now follows that (A. 19) will be satisfied if and only if a «Tr(C? - ») = 0, il  (A.20)  i  x  since any cancellations that resulted from symmetries of A will still occur here. Now, by rearranging terms in the trace, we can write all terms such that X appears first, 11  Tr(O - ) h  = TrpC^Oi -*").  in  (A.21)  2  Then (A.20) implies that 0 - y'dxiTriO* -*")  = y Tx{0\ - )  1  h  2  + X^^T^X' ^^-"*").  in  1  m  In order that the right side should vanish, the expressions involving y vanish independently for each m, therefore  lm  Tt(Oi) = 0  dxid  =  must  0.  The second condition implies that 0\ is a sum of expressions built from products of nested commutators. Since the trace of 0\ must also vanish, we should have 1  a  where Aa and B each must take the form of either a single matrix X or an expression built completely from products of nested commutators. Since we assumed that O " contained at least four X s , at least one of A and B must contain more than a single X for each a (we may assume, without loss of generality that it is B ). Then, inserting this expression into (A.21) and rearranging the commutator, we find a  n  mln  a  a  a  Tr(O) = T r ( 0 ' ) , where Q' = Y:[X ,Aa]B . h  a  (A.22)  a  Equivalently, 0\ must be the dimensional reduction to 0+0 dimensions of some gauge covariant expression. 1  Appendix  A. Appendix  to chapter 2  99  Further, it is clear that d iO' = 0 since O' is a sum of complete commutator expressions. Finally, if we define X  £' = ^,., (o'r- , in  n  then Tr(£') = T r ( £ ) and d iCJ = 0 as desired. Note that for terms at order X , both A and B must be single X s , and as a result our assertion fails. X  3  100  Appendix B Appendix to chapter 3 B.1  A consistent transformation law to sixth order in X 1  We obtain a consistent transformation law for the matrix coordinates under a Lorentz boost by solving equation (3.15). The transformation law has the form 6X  = (3H- p Sym{X Xi)  i  j  0  If we define the matrix C i ikj  [ ,  X ]} + [X , [X ,X }}  k  =  j  X  k  j  1  then a consistent choice for the matrix T T  j  = ^Sym(x C k  -x x k  (x [x ,  l  l  -X X k  +  ikj  l  k  l  -(l/12)X  f c  l  k  k  k  k  (B.2)  j  k  to sixth order in X is  u  1  ikj  + (1/3) x  [x\x \]  l  j  l  {[X , [C ,X ]] +[X ,  - [X , [X , X%  + [X ,X ][X ,X }  i  1  l  + x [x ,  j  (\[X ,X ][X ,X ] i  (B.1)  XXX C  [x ,x ]}  l  ftT'i.  to be  lk  C  +  l  klj  k  + [C ,  1  klj  l  +  j  [X ,X }} l  [X ,  [X ,X }}^j  k  l  [X ,X }[X ,X ]) i  + [X ,  1  l  1  j  l  k  [C ,X ]\ ilj  k  [C \ X }} + [X\ [C , X }} + [C , [X , X ]} il  1  ilj  k  ilj  1  k  +3[X ,[C ,X }] l  ilj  k  +[x\ [[X , [X\X%X ]]  + [[x , [x ,x%  ix\x }}  +[X , [{x\ [X ,X%X ]]  + [[x\ [X\X%  [X ,X ]}  +[X\  [[X , [X , X%X }}  + [[X , [X , X% [X\ X }}  +[X ,  [[X\ [X\X%X ]}  + \[X\ [X\X%  +[X\  {[X , [X , X%X }}  + [[X , [X , X*]], [X\ X }}  +[X ,  [[X\ [X , X%X }}  + {[X\ [X , X% [X , X }}  k  k  k  k  1  l  k  1  1  1  1  k  1  1  1  1  k  k  k  l  1  k  1  1  [X ,X ]} k  1  1  1  1  1  k  1  j  Appendix  B. Appendix to chapter 3  +[X ,[[X ,[X ,X l  i  k  j  },x }} + 1 [[xW.x^ux",* ]] l  +[X ,[[X\[X ,X k  l  j  +[X ,[[X ,[X ,X k  i  l  j  ],*']] +  k  j  \[X\[X\X%[X ,X ]] k  1  },x }}[[X ,[XfX%[X ,X ]} },x }}[{X ,[XfX%[X ,X }} 1  -[X ,[[X ,[X ,X l  101  i  k  -[X ,[[X ,[X ,X k  l  j  i  l  1  1  l  -[X ,[[X ,[X ,X l  k  j  i  IX }}  k  1  1  -[X ,[[X ,[X',X\ k  l  [[X ,[X',X%[X ,X ]] l  IX }}1  -[X ,[[X ,[X ,X i  l  j  k  k  1  [[X\[XfX }l[X\X }] k  1  },x }}l[X ,[XfX%[X ,X }} -^^[xWxfxix }}\[X\[XfX%[X\X }} -[X\[[X ,[X ,X },x }}[[X ,[XfX%[X ,X }} -[X ,[[X ,[X ,X },x }}l[X ,[X ,X%[XfX }} +[X ,[[X ,[X ,X IX }} + [{X\[X +[X ,[[X ,[X ,X },x }} +,X%[X ,X }} +[X ,[[X ,[X ,X },x }} +[X ,[[X*,[X ,X ix }} +[[Xf[X ,X%[X ,X }} 1  1  l  1  l  k  j  l  k  1  i  1  j  j  k  l  i  k  j  l  i  l  j  k  i  k  k  1  1  l  k  k  1  l  1  1  l  l  k  1  l  k  1  1  l  l  i  + [[Xf[X ,X%[X ,X ]}  +[X ,[[X',[X ,X  i  IX }}  +[X ,[[X ,[X ,X  i  ],*']] +[{xf{x ,x ux\x \} ],*']] +[[Xf[X ,X%[X ,X }} ],*']] +[[X ,X^[[X\X },X }}  k  l  i  j  l  1  j  l  l  k  -\[X ,X ],[[X X j  k  1  i  1  l  l  IX }}-  k  +[[X ,X%[[XfX l  +[[Xi,X%[[X X 8  l  1  k  1  l  IX ]}-  k  1  1  +[{X\X%[[X\X  +0(X ).  k  1  [[X ,X%[[XfX },X }}  l  l  1  1  -[[X ,X ],[[X ,X k  l  1  l  l  -[[XfX%[[X ,X j  k  k  l  i  l  l  [[X\X%[[X ,XJ},X }} },x }} + l[X ,X%[[X ,X>],X }} ),x }}[[X ,X },[[XfX%X ]}  +{[X ,X^[[X ,X l  1  k  l  +[[X ,X ],[[X ,X' i  k  l  k  +[X ,[[X ,[X ,X k  l  IX }} 1  [[XfX%[[X\X%X }} 1  +  \[X\X%[[X\X%X ]} 1  },x ]} + l  %X }} + [[X\X ],[[Xi,X%X ]] k  1  l  (B.3)  Appendix  B.2  B. Appendix  102  to chapter 3  Characterization of Galilean invariant non-Abelian actions  Here we prove that the minimal conditions (3.26, 3.27) for the leading term of a Poincare invariant action imply that the leading term of the Lagrangian is either Tr ( X ) or can be written in a way such that all Xs and Xs appear inside commutators. We first show that any action satisfying 2  d S{X  + e) = 0  £  (B.4)  can be written as the integral of a Lagrangian with d L(X + e) = 0 i.e. such that all X s in L appear in commutators. For suppose that an action S satisfying (B.4) is the integral of a Lagrangian L. Then employing the symmetrized expansion discussed in section 2, we may write €  1  L = L+J2 m=l  ^ STr(L ,.. T  m  ( i  X ...X*-), i l  i m )  (B.5)  -  where the various terms in L and L(t ...i„) do not contain any free X s . Here, the various individual commutators or differentiated X s appearing in a given term of L are to be symmetrized with the remaining X s . Now, for S S to vanish under a translation 5X — e, 5L must be a total derivative 1  e  5L e  (B.6)  = J^-UK  Generally, U may be written l  ^  = E^STV(^ .., 1  m )  X-...X-),  We axe using the fact that any product of matrices may be written as a sum of completely symmetrized products, where the individual terms in a product must be individual matrices or complete commutators of the form 1-  [x£ \[x£>\[...,[x^\x£r ]...}}} 1  where (n) represents the nth time derivative.  )  Appendix  B. Appendix  to chapter 3  103  )=  (-)  so (B.6) becomes 7^T)j  E  S T r  ( (u..^) L  e i 1  ^  •••  2  X i m  B  7  E ^ S T r C ^ , . ^ , ^ . • • X - ) + ^ - l ^ S - r r C ^ . . ^ ^ ... In this equation, consider the terms with the largest number of free Xs. To be precise, we can substitute X —> X + f3 and compare the terms with the largest power of p. Doing this, we find it necessary that ^(il—in)  =  ^12-in  where C is a commutator, C ~ [(X )^\ STr([(^) '),^j - ' ]X (  1  i  l  i l  • --X ) in  +^"il-ln>  Aj "" "]. Since  j  1  1  = n STr^  " * " ^ , (X )®]  1 -  1  •  j  ••X "), 1  the term in (B.5) involving C can rewritten such that it has only n — 1 free Xs and therefore can be absorbed into a redefinition of L ...i _ . W i t h this redefinition, we now have 2  ix  kn-in)  = U t ,  n  n  x  .  (B.8)  In particular, U must be completely symmetric on all its indices, so we can h-i = (m-3n)Comparing the terms in (B.7) with (n — 1) free Xs, we find  W l i t e  U  U  n  where Cj ...j _ is a commutator. As before, by rearranging terms in the trace, we may eliminate C in favour of a redefinition of £ ( i . . . i _ ) - Thus, 1  n  1  1  n  2  and it must be that t/j ... _ is symmetric in all of its indices. Continuing in this way, we find that by rearranging commutators, it is possible to ensure that all Us are completely symmetric tensors and l 1 2  in  1  (ii-ik)  L  = Uh-ik) U  XJ  + (h-ik) U  (-) B  9  This assertion would be incorrect if A were [X*,X ], but this is impossible, since the i and j indices would have to contract with the indices on two other Xs which are symmetrized. 2  j  Appendix  for all k>l.  B. Appendix  104  to chapter 3  Substituting (B.8) and (B.9) into (B.5), it follows that  £ = 4 Z dt  ^ S T r ( L 7 . . , J ^ • • • X*-) - Tr (il  „n\  (tyX*) + L,  n)  where we have integrated by parts to get the second term. Since L and U do not contain any free X s by assumption, we conclude the action S can be written as the integral of a Lagrangian density with all X s in commutators, d L(X + e) = 0 (up to total derivative terms). Starting from a Lagrangian L that has been written so that it contains no free X s , let us now suppose the action is invariant under 8X = fit. Then we must have l  e  5 L=j P &. fi  t  (B.10)  i  for some We can organize the symmetrized expansions of L and $ in terms of the number of free X s (not appearing in commutators), to write L  =  L+f:—STr(L ... X^...X -), i  {ll  lm)  (B.ll) Here, by assumption, all X s and X s in L and L ^ . . . ^ ) appear in commutators. In ^(ij...^), all X s must appear in commutators by assumption. Also, unless we have $ oc X\ any free X in ®\ .„ ) would remain undifferentiated in at least some terms on the right side of (B.10), and this is not allowed since the left side contains no free X s . Thus, either <3>oc X or all X s and X s in l  ix  im  l  ^(ii-im) P P a  e a r  m  1  commutators.  Inserting the expansions ( B . l l ) in (B.10), we have E  ^ ^ y S T r ^ , . . ^ ) ^ ^ ... X ^ ) = E ^ S T r ( $ ; . . , X - ... X - ) + ^_l i ]  m )  (B.12)  i y [  STr(4 .., X- ... X - ) . i  m )  This equation is exactly analogous to (B.7) above, and the rest of the proof proceeds in parallel to that above. This time, we end up with the statement 3  3  The only change is to the comment in the previous footnote, which should now deal  Appendix  B. Appendix  105  to chapter 3  that = Tt E ^ S T r ( $ . . , X ^ • • • X<") -  L  ( n  The special case that  oc X  Tr ( ^ X * ) + L  n )  corresponds to  1  L  = Tr  0  (X ). 2  Otherwise, all Xs and X s in and L appear in commutators, so after integrating by parts to remove the first term here, we have succeeded in writing the action as the integral of a Lagrangian density for which all X s and X s appear in commutators.  B.3  The matrix space-time vector-field V {y) y,  For reference, we write down the expansion of v^(y) to fifth order in x. The expansion for the time component is v°(y)  = -v° + v° - v° + t , ° - v° + ..., 2  3  (B.13)  5  where =  yx  v% =  x-x  i>3  (y • x)(x • x) + (x • x)(y • x) + (x • x)(y • x)  =  +  (y-x){y-x)  +(y • x) {y 2  v®  =  (x-x)(x-x)  • x) + (l/2)(y • x)(y • x)  (B.14)  2  + (x-x)(x-x)  +  2(y-x)(y-x)(x-x)  +2(y • x)(y • x)(x • x) + (y • x) (x  • x) + (y • x)(y • x)(x • x)  2  + ( l / 2 ) ( x • x)(y • x)  + (3/2)(a; • x)(y • x)  2  2  + (y, xf{y  with three special cases, C oc [X ,Xi], C oc [X ,Xi], and C oc [X ,X ], apparently not true that rearranging the commutators l  %  Tr  (C  I L  ...  j  for which it is  X '.--X -) I  I B  l  • x)  I  leads to an expression with no free Xs and less free X s . But again, none of these cases are realized. The first two are not possible since the commutator is an antisymmetric rotational tensor whose indices must contract with the indices of symmetrized X s , while the third is not possible since it would necessarily have an odd number of time derivatives and violate time-reversal invariance.  Appendix  B.  Appendix  106  to chapter 3  . +(3/2)0/ • x)(y • x){y • xf + (l/6)(z/ • x)(y • x ) t»5  =  (x • if(y  3  • x) + 2(x • x)(x • x)(y • x) + (x • xf(y  • i)  +2(y • x)(x • x)(x • i) + 2(y • x)(x • x)(x • x) + (l/2)(y • x)(x • x)  2  +3(rr • x)(y • x)(x • x) + (x • x)(y • x)(x • x) + 3(y • x) (x • x)(y • x) 2  +3(y • x) {x  • x){y • x) + (y • xf(x  2  • x) + (3/2)(y • x)(x • x)(y • x)  2  +(3/2)(y • x)(x • x)(y • x) + (9/2)(x • x){y • x)(y • x) 2  +{3/2){x • x)(y • x)(y • x)  + 3(y • x){y • x)(y • x){x • x)  2  + (2/3)(x -x)(y  xf + (l/2)(x • x)(y • xf + (l/6)(x • x)(y • xf  +(l/2)(y • x)(y • xf(x +(2/3)(y  2  • x)(y  • x) + (y • x)\y  • x)(y • xf  + (l/2)(y  • x) + 3(y • x)(y • xf(y  • xf(y  • xf  • if  +(l/24)(yx)(yxf.  Note that all xs and ys appearing on the right hand side of the above equations are the spatial x and y . Also, all factors of x and its derivatives are evaluated at y°. The expansion for the spatial components is l  1  v (y) = -y l  l  + v\-v  i  + vi-vi  i 2  + vi + ...,  (B.15)  where v\  =  x  v\  =  x {y • x)  vl  =  x {x-x)+x {y-x){y-x)  v\  =  x (y  i  l  i  +x {y  if  i  =  • x)(y • x) + x (x  l  +x \y • xf(y i ( i • i)(x l  • x) + (l/2)x (y +(1/6)  • i) + i (x  • x)(y • xf  l  x* {y  l  • x) + i (y  +(l/2)x (x  • x)(y • if  +(3/2)i (y  • x)(y • x){y • if  i  i  -\-2x\y • i)(y  • xf(x  l  • x)(y • x)(x • i)  • i) + i\y  • x)(y • x)(x • i) + i \y l  • xf{y  • i)  + ( 1 / 6 ) ^ • x)(y • if  • x)(x • i) + x (y l  • xf(i  • if(y  • xf  +{l/2)x (y  • if(x  • i) + (l/2)x\y  i  • x)(x • x)  l  + (3/2)^(0; • x)(y • if  +{3/2)x {y l  + x (y  if  • x)(x • i) + 2i (y  l  +2x (y • x)(y • i)(x  • x)(y • x)  l  i  (yx)  (B.16)  i  • x){x • x) + x (x  l  l  v\  + (l/2)x (yxf  i  + {l/2)x\y  • i) + x (y l  • x)(y • if • if(y  • if(x  • x)  + {1/2)3*{x • if  • x) + (l/2A)x\y  •  if.  Again, all xs and ys appearing on the right hand side of the above equations are the spatial x and y , and all xs and derivatives are evaluated at y°. The 1  l  Appendix  B. Appendix  107  to chapter 3  matrix generalization V^(y) has the form V»(y)  = V£ (y)  +  in  AV»(y),  where V£ (y) is the expression obtained by replacing all occurences of x in the expansion of v^(y) with X and using the completely symmetrized product of matrices. For reference, we write down the explicit expansion for the correction terms AV^(y) to fifth order in X. This was determined by imposing Lorentz invariance under the transformation law given in appendix B . l (only the fourth order terms in the transformation law play a role in this case). The expansion for the time component is l  m  1  AV°(y)  = AV ° - AV ° + . . . , 4  (B.17)  5  where Al/ ° = T^Sym (X [X\  \X , X*]] + X [X ,  J  j  4  j  [X\X*}} + P[X\  j  + X ' [ X ' ' , [ X \ X ] ] + X'[X«, [X\X% +Xl[y.  X , [X ,y  X}} + X*[y • X, [X\y  A V f = ^ S y m [Px%y i  1  X}} + 2Xi[y • X, [X*,y • X}} + X'[X>', \y X, y • X]]  j  +2X*X [(y  j  + X*[y • X , [ X ' , y • X}}  {  [y-X,y  [X ,X }}  ),  X  • X]] + X [y • X, [Xfy 3  [X'.X*]] + X ' X ^ X ' , [(y X ) , ^ ] ]  X]])  (B.18)  • X), [X'', X ]] + X'X*[X>', [(y • X), X ]] 4  1  +X'X [(y • X ) , [X'.X*]] + X ' X ' [ X \ [X\ (y • X)]] i  + X ^ X [ X \ [X\ (y • X)}} + X ^ [ x \ [X^ (y • X)]] i  +X*X [X>, i  [x\ (y • X)]] + 2 X ' X [ X , [X\ (y • X)]] i  i  + X ' X * [ X ' , [X\ (y • X)}} + X ' X ' I X * . \X\ (y • X)]] +X>X [(y i  • X ) , [X'.X*]] + X X [(y 3  l  • X ) , [X^X*]]  +2X'(y  • X ) [ X \ [ X ' , X% + X'(y  • X)[X*, [X\ X*]]  +2&{y  • X)[X\  • X)[X\  +X'{y  • X)[X\  +2X*(y • X)[X\  +X'{y  X*]] + P(y  [Xl,X<]] + X ' ( y • X)[X*, [P, X'}} + 2X'(.y • X)[P,  • X ) [ X \ [XfX ]} 1  [X>, X ]] 4  [x'.X']] [X\  X%  + X*{y • X ) [ X ' , [X\ X*]]  +2X%  • X)[(y  X), [X>, (y • X)]] + X'(y  +4Xi(y  • X)[(y • X ) , [X'', (y • X)]] + 2X\y  • X)[X>,  [(y X),(y  X)}}  • X)[(y • X ) , [X>, (y • X)]]  Appendix +X*{y • X)[(y  B.  X), [Xf  Appendix  108  to chapter 3  (y • X)}} + P(y  • X)[Xf  [(y X), (y • X)]]  +3X'(y  • X)[(y  X), [Xf  (y • X)]} + 2X'(y  • X)[Xf  [(y X), (y • X)]]  +3Xi(y  • X)[(y  • X), [Xf  (y • X)}} + P{y  • X)[Xf  [(y -X),(y  +P{y  • X)[(y  • X), [X\ (y  X)]] + X^(y  •  • X)[(y  • X), [Xf  X)}}  (y  X)}}  +X\yX)[(yX),[Xf(yX)]} +X Xi[Xf  [(y • X), X'}} + X'X^Xf  i  +{l/2)X P[X\  [Xf  i  +(l/2)X Xi[(y  (y • X)}} + ( 1 / 2 ) X ^ [ X \ [Xf  • X), [XfX*]}  i  +X Xi[Xf  [X\ (y • X)}}  + (lfflXWlX*,  [X\ (y • X)}} + X'X^Xf  i  +(l/2)X &[Xf[(y  (y • X)}}  [(y • X),X>}}  [X\ {y • X)}]  X)^ }}  i  1  +(y • X)&[X\  [XfX*]]  + (y •  +(y • X)Xi[X\  [XfX%  + (y • X)X*[Xf  + (y • X)P[X\  [XfX'})  + (y • X)X*[(y  +(y • X)Xi[Xf  [(y • X), (y • X)}} + 2(y • X)X*[(y  • X), [Xf  +(y • X)P[Xf  [(y X), (y • X)}) + (y • X)X*[(y  X), [Xf  +(y • X)X'[{y  • X), [Xf  - ( 1 / 2 ) X ^ [ X * . [Xf  • X), [XfX*]}  +XJX [Xf  [XfX^+X^y  +x>(  • x)[xf  [JCX]]  + x (y  [X'X]]  + 2X\y  +X>(y • X)[Xf  j  • X), [Xf  • X)[X\  [XfX%  • X)[X\  [X>,X*]]  • X)[(y  [(y Xl  +4X'{y  • X)[Xf  [(y • X), (y • X)]] + 2X>(y • X)[(y • X), [Xf  (y • X)]]  (y • X)]]  X), [Xf  • X)[Xf  +2X'(y • X)[(y  (y • X)}}  [(y • X),X*]]  +2X (y 3  (y • X)}}  [(y •  - {l^PX^Xf  • X)[X\  (y • X)}}  [X\ (y • X)]]  i  +Xi(y y  • X), [Xf  (y • X)]} + PX^Xf  - ( l / 2 ) X ^ [ ( y • X), [XfX'}} +XJX [(y  [X\X%  (y • X)}} + (y • X)X*[{y  i  i  Xtf^XflX^X*]]  (y • X))] + AP(y  • X)[(y  (y X)}} + 2X^(y  • X)[(y  • X), [Xf • X), [Xf • X), [Xf  (y • X)]] (y • X)}] (y • X)}}  +2X^yX)[Xf[(yX),(yX)}} +XJ(y  X)[(y  • X), [Xf  (y • X)}} + X'(y  X)[Xf  [(y • X), (y • X)]]  +XJ(yX)[(yX),[Xf(yX)}} +Xi[{y  XlX^X^X^+X^y  +P[X\  (y • X)][XfX ]  + X^X^X^Xf  +P[X\  (y • X^XfX'}  + 2P[X\  {  XlX^XfX'} (y, X)] (y • X)][Xf  X'}  Appendix +2X*[X ,X ][Xl, i  +P[P,  B. Appendix  (y • X)] + 2p[(y • X), P][P,  i  (y • X)][P,  X*] + P[P,  109  to chapter 3  P][P,  X} 1  (y • X)]  +P[x\ (y • X)][P,P] + P[x\ X % X \ (y • X)} +P[(y • X),P}[P, X ] + P[(y • X),P][P, P] +2p[(y X), (y X))[P, (y • X)\ + 2p[(y X), (y • X)][P, (y • X)] 1  •  +P[(y  (y • X)) + P[(y • X), (y X)][X ,  X), (y • X)][X\  3  (y • X)]  +P[(yX),(yX)}[X>,(yX)} +2p[(y  • X^PftX^X*]  + 2p[(y •  X\X \[X\X \ i  i  +P[P, X%P, (y • X)] + P[P, (y • X)}[P, P\ +2p[P, (y • X)}[P,P] + 2p[(y • X), P}[P, P] +2P[P, (y • X)][P,X*] + 2p[P,X ][P,  (y • X)]  i  +P [(yX),P}[P,P}+P[(yX),p [P, P] +4p[(y • X), (y • X)}[P, (y • X)] + 2P[(y • X), (y X)}[P, (y • X)} +3X'[(y X), (y • X)}[P, (y • X)] + P[(y • X), (y X)}[P, (y • X)} +P[{y -X), P]{P,P] + P[{y • X),P][P, P] +P[P, (y • P))[P,P] + P[P, P][P, (y • X)} •  +2p[(y  • X), {y •  X)][P, (y • X)} + 3P[(y  • X), (y •  X)][P, {y • X)}) .  The expansion for the spatial components is A.V (y) i  = -AVi + AV< + ...,  (B.19)  where AVI = ^ S y m  (P[P, [P, (y • X)}} +P[P,  [P, (y • X)}]  + P[X\  [X>, (y • X)}} + P[(y • X),  +P[Xi, [(y •  X),P}} + P[(y  [Px [P, [X , P]] + PP[X , +PX [P, [P, P}} + Px [P, [X , P}} +Px [X , [P, P]} + PP[P, [P,P]] +PP[P, [X , P]} + PP[P, [X ,P}} j  AV£ = ^ S y m J  j  k  k  j  k  k  k  k  +PP[X , k  [P,X ]] + PP[P, j  [P, X }} j  [P, P}]  • X), [X\  [X*,**]])  P]}  (B.20)  Appendix  B. Appendix  110  to chapter 3  +X X [X ,  [XfX*]]  +X X [X ,  [(y X), (y • X)]] + 2X X%y  +X X [X , k  [(y • X), (y • X)]] + X X%y  -X), [X , (y • X)]]  +X X [(y  • X), [X , (y • X)]] + X X%y  • X), [X , (y • X)]]  k  l  k  k  i  k  k  i  k  + X X%y  X), [X , (y • X)]]  k  k  • X), [X , (y • X)]]  k  k  k  i  k  k  k  k  +X (y  • X)[X\  [X , (y • X)]] + X (y  • X)[X\  [X , (y • X)}}  +X (y  • X)[(y  X), [ X , ^ ] ] + X {y  • X)[X ,  [ ( y X),  k  k  +2X (y  k  k  f c  k  k  • X)[(y • X), [X ,X }}+X (y  k  k  i  • X)[X ,  k  • X)[(y • X), [X , X*}} + X (y  +X (y  • X)[X ,  [\  +X (y  • X)[X ,  [X\ (y • X)]] + X (y  +X (y  • X)[(y • X), [X , X'}} + X (y  k  k  k  k  k  k  j  k  [X , (y • X)]] k  [X , (y • X)]] k  • X)[(y • X ) , [X , X*}}  k  [X\X }}  k  k  + (l/2)X X [X\  j  +  X)[X\  k  +(l/2)X X [X ,  [X , (y • X)]]  • X)[X\  k  k  k  • X)[x\  k  (y • X)}} + 2X (y  X  [(y • X),X*]]  k  +X (y k  k  k  [X ,X ]}  j  k  j  (l/2)X X [X ,[X ,X }} k  +(y • X)P[X\  j  k  j  i  [Xf (y • X)}} + (y • X)X'[X'", [X\ (y • X)]]  +(y • X)X'"[X*, [Xf (y • X)]] + (y • X)X>[(y • X),  [XfX ]}  + (y • X)P[Xf  [(y • X^X*]]  [XfX }}  +X (y  • X)[X\  [X , (y • X)]] + X (y  • X)[X ,  [X\ (y • X)]]  +X (y  • X)[X\  [X , (y • X)]] + X {y  • X)[X ,  [X\ (y • X)})  +X (y  • X)[(y • X), [X , X*}} + X (y  • X)[X ,  [(y • X), X*}}  +X (y  • X)[(y • X), [X , X*}} + X (y  • X)[X ,  [(y • X), X*}}  k  k  k  k  k  k  k  k  k  k  k  +X X [(y k  + (y • X)X'[(y • X),  k  k  1  k  k  k  1  • X), [X , (y • X)]] + X*X<[X , [(y • X), (y • X)]]  i  k  fc  +X X>[(yX) [X ,(yX)}] k  k  1  +X [X ,X ][X ,  X ] + X [X ,  X ][X\  +X [X\X ][X ,  X ] + X [X ,  X]  k  i  j  k  k  j  j  k  k  j  k  k  X]  j  k  j  [X\X ]  j  j  +X  [X , X ] [X , X ] + X [X , X ] [X ,  X]  +X  [X , X*] [X ,X ]  X]  k  k  k  1  k  j  j  j  k  j  j  {  k  + X [X , X ] [X , k  j  {  k  j  j  -X [X\X ][X ,X ] k  j  k  j  +X [X\ k  (y • X)}[X ,  (y • X)} - X [X ,  +X [X\  (y • X)}[X ,  (y • X)} + X [x\  k  k  k  k  k  +X [(y  • X), X*][X , (y • X)] + X [(y  +X [(y  • X), (y • X)}[X ,X*]  k  k  fc  k  k  + X [(y k  k  (y • X)][X\  (y • X)]  (y • X ) ] [ X , (y • X)] f c  X), X%X , k  • X), (y •  (y • X)] X^X",**)  Appendix  B. Appendix  111  to chapter 3  ^(ffjjp^l).  +X [(y k  From equation (3.38), we can see that v°(y)\  =  yi=0  ^ • • • ^ y ^ l ^ o  0(x ) 2  =  0(x ).  (B.21)  n  Using this result and equation (3.37), we also have z/(y)|  =  yi=0  O(x)  V(s/)ly<=o = S  + 0(x )  ij  ^ . . . ^ ( y ) ^  =  2  0(x ).  (B.22)  n+1  Note that any y appearing in the expansion of (V£ (y) + <5fV) must be accompanied by an X in the form of (y-X). As a result, any y appearing in A V must also be in a factor like (y-X). This allows us to obtain equations analogous to (B.21) and (B.22) for the case of the matrix-valued vector field l  m  1  l  M  V°(y)\  =  0(X )  vi=0  =  0(X )  V*(y)\^  =  O(X)  =  - ^ +  =  0(X ).  yi=0  d ...d V (y)\ 0  h  in  3;V%)|  y M )  ^•••^(yJl^o  2  n  0(X ) 2  (B.23)  n+1  Also we have the following formulae, for all integers m > 0. yi=0  =  0(X )  yi=0  =  o(x )  yi=Q  =  -d X /dt  yi=0  =  d^P/dt ™  d (-  yi=0  =  d X /dt .+  dj(-  yi=0  =  S 5  (- -d ) V° d (- -d ) V° 2 m  2 m  0  8j(- -d ) V° 2 m  (--d ) V 2  0  m  {  2  2  2m+1  j  i  m t Q  0(X )  +  3  0(X ).  Here d = rf^d^d^ where [77^] = d i a g ( - l , ! , . . . , ! ) . 2  Z  3  2m+1  i j  0(X )  + <D(X )  2  2m+1  +  2m+l  2  

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