i{ya). 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. Al though we have described the Dp-brane as a fixed hyperplane, it should not be thought of as a rigid non-dynamical object. The description we have just given should be understood in the context of perturbative expansions. The hyperplane represents a background configuration of the D-brane about which it dynamically fluctuates. The fluctuation modes are the modes of the attached open strings. In particular, the massless scalars ^i(ya) describe the transverse fluctuations in the shape of the Dp-brane. We wi l l return to this in the discussion of D-brane effective actions. Dirichlet boundary conditions arise naturally by considering the action of T-duali ty in a theory wi th open strings [27, 28, 29]. Compactify one direction along which the open strings satisfy Neumann boundary conditions. Now send the radius of compactification to zero R —* 0. In the T-dual description, the compactification radius goes to infinity l/(2irR) —• oo and the open strings satisfy Dirichlet boundary conditions. A s a result, T-duali ty along a direction parallel to a D-brane produces a D-brane with one less dimension. O n the other hand, T-duali ty along a direction perpendicular to a D-brane Chapter 1. Introduction 8 produces a D-brane wi th one greater dimension. In this manner, we can start wi th any Dp-brane and T-dualize unti l we obtain a space-filling D9-brane, which corresponds to open strings wi th Neumann boundary conditions in al l directions. 2 This 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 Abel ian gauge multiplet. A Dp-brane naturally couples to the R R p + 1 form potential [10, 30, 31]. We wi l l discuss this further in the next section. To match the types of R R potentials appearing in type I I A / I I B string theory, we have Dp-branes of all even p in type I IA string theory and Dp-branes of al l odd p in type IIB string theory. This is consistent wi th the action of T-duali ty on D-branes, and the fact that T-duali ty exchanges type I IA and type IIB 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. This action can be writ ten 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 in which the D-brane is represented in this context. The D-brane world-volume is labelled by the coordinates oa, a = 0 , . . .p. The embedding of the world-volume in space-time is xp(a) where the index ^ runs from 0 to 9. There is also an Abel ian gauge field Aa(a) defined on the world-volume. This description seems slightly different from the open string theory description of the previous section, but we wi l l discuss the connection later. The Born-Infeld term describes the dynamics of the D-brane and its inter-actions wi th the metric, the NS two-form, and the dilaton. In the conventions of [34], it can be expressed in the form [35] S B ? = -Tp j dp+1a e^yf-det (dax»dbx»(gixl/ - + Fab). (1.4) 2Actually, 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 the-ory, which contains 16 D9-branes, in a constant background gauge field in the 9-direction. Chapter 1. Introduction 9 Here Tp is the tension of the Dp-brane, and the gauge field strength is Fab — daAb — dbAa. The precise value of the Dp-brane tension Tp depends on the normalization conventions for the supergravity fields, particularly on the conventions for the dilaton . Once these are fixed, Tp 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 3 describes interactions with the R R potentials [37, 38]. Here, p,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 space-time 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 For p = 0, this has the same form as the coupling between an electrically charged point particle and the gauge potential in ordinary electrodynam-ics. The additional terms involving and Fab 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 Dp-brane with a non-trivial world-volume field strength Fab 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 S B I and Swz a v e separately invariant under arbitrary world-volume reparametrizations. This is a gauged symmetry that ensures q. T-duality from the DO-brane action shows that the dilaton has to be evaluated in the following manner The appearance of these Taylor expansions in the action receives some sup-port from calculations of string scattering amplitudes [83]. This gives further support to the interpretation that is in some sense the static gauge embedding functions for the multiple D-brane 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 al-ready suggested in [84, 85]) using the static gauge embedding Xti(a). More specifically, in [76] the proposed action was written using expressions of the form O O 1 {a\ X\o)) = £ - A . . . c \ > ( r y \ O ) ^ 1 (a ) . . . X^{o). (1.12) n Tl. n=0 x > ) E E ( r y \ x V ) ) (1.13) (1.14) where b. tin DaX» = (6+a,DaX% (1.15) where Da is the gauge covariant derivative. Chapter 1. Introduction 18 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 commu-tators of X1. These commutator terms describe the interactions of multiple Dp-branes with g-form potentials where q > p + 1. String scattering ampli-tudes 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 q-form 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]. 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) = 2nSijkyk- 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. The first term comes from the Born-Infeld part of the action, and the sec-ond comes from the Wess-Zumino part. Here we have set the world-volume (world-line) gauge field AQ 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 mo-tion. For K = 0, any static configuration of commuting matrices X1 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 rDO _ To (1.16) X * = («/xo/T0) J * , (1.17) Chapter 1. Introduction 19 where the matrices J1 form an iV x N representation of the SU(2) alegbra [J\Jj}=isijkJk. (1.18) 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 X1 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]. 1.3 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 X1. 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 DO-branes transform under general space-time diffeomorphisms. The matrices X1 pose a distinctive problem, because they are interpreted as spatial co-ordinates; from the known couplings between X1 and the bulk supergravity fields, there is a precise sense in which X^(t) = (t,Xl(t)) are the embed-ding 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 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 diffeomor-phisms that mix the world-volume and transverse directions. In the many D-brane case, the problem is worse because the transverse coordinates X1 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 gen-eral covariance in multiple DO-brane actions. In chapter 2, we focus on spatial diffeomorphisms, and explore in detail the problem of defining general coor-dinate 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 con-sider first. 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 X1. There are two main questions to address. First, how do the matrix coordinates of DO-branes transform under a gen-eral change of coordinates? And 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 coor-dinates 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 de-pends 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 co-variance in field theories is the tensor calculus. Accordingly, we find a matrix valued object constructed from X1 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 fol-Chapter 2. Invariance under spatial coordinate transformations 22 lowing geometric construction. For any point xl in a Riemannian space, there is an associated vector field vl defined in a neighborhood of xl such that v1 (y) is simply the Riemannian version of the displacement vector from yl to xl. We show that a matrix generalization V% of the vector field v1 exists so long as there exists a consistent matrix transformation rule. More specifically, given a transformation rule for X1 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 de-termining V1 as a function of X1 and the metric gij without making explicit use of the matrix transformation rule. In section 2.4, we use this matrix-valued vector field V1 to construct a large class of generally covariant actions for DO-branes. It is easy to form scalar fields C using V1 wi th any tensor fields constructed from the metric. The usual procedure in 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, in this case the values of V1 at different points in space do not really represent independent degrees of freedom. Specifically, in going from X1 to V 1 , we have not really introduced any additional degrees of freedom, so that in fact the value of the vector field V1 at any single point contains al l the information of X1. To take care of this redundancy in the object V\ we are led to introduce an additional ingredient: a matrix distribution function Sd(V) that is inserted into the integral wi th the effect of localizing the action to the vicinity of the branes (d is the dimension of space). The object 6d(V) reduces to a diagonal matrix of ordinary delta functions when V1 and Xx are diagonal, and it describes an extended distribution for general non-commuting matrices. The final actions take the form Here, £ is any matrix-valued scalar field constructed from V1 and tensors built out of the metric. This represents a large class of generally covariant actions. To illustrate the use of these methods, we demonstrate that essen-tial ly any generally covariant action depending only the matrix coordinates and the metric can be expressed in 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. (2.1) 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 X1 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 ds2 = -dt? + gij{y)dyidyj, (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 X1, and developing methods to write down actions S[gij, X1] that are invariant under the resulting transformations. The dimen-sion 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 sce-narios 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. 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 yl = Fl(y), the transformation rule of the metric is familiar ^ = %%9kl{v)- ( 2- 3 ) Chapter 2. Invariance under spatial coordinate transformations 24 but the transformation rule for the DO-brane matrix coordinates Xl(t) needs to be considered in more detail. The situation is clear for a single DO-brane, since we just have ordinary coordinates xl(t) for the single DO-brane rather than matrix coordinates, and the transformation rule is simply ~x\t) = F\x{t)). (2.4) Physically, a single DO-brane is just a point-like object and the transforma-tion 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 trans-form, 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 y1 —> yl = Fl(y), we would like to know the corresponding transformation of matrices Xi = &(F,X). (2.5) Some basic points serve as an initial guide. For example, we intepret diag-onal 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 ^ ) , Fi(x2),... F(xNf), (2.6) so that the elements of diagonal matrix configurations are transformed in-dividually. Also recall that the matrix coordinates are Hermitian N x N matrices which transform under a U(N) gauge symmetry X1 —+ UXlW. For compatibility with U(N) gauge invariance, the simplest scenario is where the matrix coordinate transformations commute with the gauge transformations U$(F,X)U* = §{F,UXU]). (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 X1. In view of these general considerations, a natural ansatz for the transformation rule is just the matrix version of the Taylor series expansion O O 1 X) = ^ ... djnF*(0)X* ... X*. (2.8) n=0 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))dkdiFi(0)dmFj(0)[Xk, [X\Xm\] + G(X4). 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 X1 &(Ft X) = £ $)^jn (V(0), 0F(O), ddF(0)... )x* . . . X*"t (2.11) n=0 V ' where the coefficients of the expansion are some unknown functions of Fl 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 Chapter 2. Invariance under spatial coordinate transformations 26 under linear diffeomorphisms. More specifically, for any linear coordinate transformation Fi(y) = M) yj, we have &(F,X) = M) XK While these assumptions may appear reasonable, they are in fact incom-patible with the requirement that the transformation rule satisfies the compo-sition 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 coor-dinate transformations is linear. Together with the regularity assumptions of the coefficients functions, this already implies that the coefficient functions must have the form Then the constant is fixed by demanding the correct limit for diagonal ma-trices, 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 contradic-tion. 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 <&l(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. 2.2.2 Non-tr ivial 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 F1 and X\ but also on the metric gij. To guide our seach for such a transformation law $l(F, X, g), we rely on the composition rule as the key consistency condition. Note that the composition rule now takes the form constant x ... djnFl(0). (2.12) (2.13) Chapter 2. Invariance under spatial coordinate transformations 27 where the first diffeomorphism F1 also transforms the metric according to the usual tensor transformation rule, gij(y) = (dyk/dy"l)(dyl/dyj) gki(y) with f = F\y). Suppose we start with the general expression oo &(F,X,g) = ]T $ ; , . . j n (F(0)^(0) , . . . ..X*". (2.14) n=0 Here the coefficients of the series expansion depend on Fl, 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 yl —> yl + e1 acts on the matrices in the obvious way X1 —> X% + e\ The composition rule implies ^(FoH-1, $(H, X, g),g) = $(F, X, g). (2.15) Now take the special case where Hl 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 oo E i . J J i ? ( o ) , # ) - . . ) ^ . . . ^ n=0 oo = E ®l..Jn(F(y),9(y), • • . ) (* - v)h •••(*- 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 X1 — yl for any point y1, and not just the origin yl = 0. Also note that on the two sides of the equation, we have the same functions $}j...j n, 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 $ L i n = ^ 1 - - - ^ ^ ( 0 ) + A$j 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 X1) 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 X1 — F z(0) rather than Xx. Then at each order in X1, only a finite number of terms contribute. This gives us a set of equations for the correction terms of the schematic form A^jn(HoF(0),g(0),...) (2.18) = dhFk-(0)...djnFk»(0) A^ f c l . . . f c n (H (F (0 ) ) , ~9(H0)), •••) + 0 f c i f(F(O)) ... i n (F(0), p(0) , . . . ) + remainder. The remainder terms depend on the functions F\ H1 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 &{F,X,g) =.F*(0) + djFi(0)Xj + ^-djdkF(0)XjXk+ (2.19) +^djdkdlFi(0)X^XkXl - ^ ^ ( 0 ) ^ ( 0 ) ^ , [Xk,X1]} + 0(X4). In this expression, TJ f c is the Christoffel symbol constructed out of the metric gij. The third order discrepancy in (2.10) is cancelled precisely by the in-homogeneous 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 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 transforma-tion law should transform the elements of a diagonal matrix configuration individually. This condition will be satisfied if we find that the correction terms A $ ' . X J 1 . . . X 7 ' " can be written in a form that involves a commu-j\'••jji tator 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 F1. It is clear that the expression (2.19) satisfies all of these additional con-straints. 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 obstruc-tion 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 expres-sion for $(F, X, g) is given in appendix A.2, and we see that it is already becoming rather complicated at this order. 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 X1 —» X1 — $l(F,X,g) that satisfies the following consistency conditions: (i) it has a series expansion in X1 and therefore commutes with gauge transformations (ii) for commuting matrices, + ul - J2 - H i - i n f o ) " 7 1 • • • (2.21) n=2 n -where the extended Christoffel symbols T^ -n 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 vl by the condition expi(v(y)) = x\ (2.22) for all yl. When we need to be precise with our notation, we will write vx to indicate the dependence of the vector field on the location of the brane xl. 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 vx(y). Pictorially, we can think of vx 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 vx(y), we invert the series (2.21). The first few terms of the series are <{y) = A* + \v)k{y)^^ + i (r}w(y) + 3r}m(y)rs(y)) A'A*A' +..., (2.23) where we have used the abbreviation A 1 = x% — y1. In a different coordinate system, we can write down the corresponding expression using the Christoffel symbol in that coordinate system r*fc and the coordinates of the particle xl and the observation point yu. vl{y) = A* + if}fcG/)£>-A* + j{ (f}w(y) + 3f^ .m(i/)fS(y)) A'A*A1 + ..., (2.24) Chapter 2. Invariance under spatial coordinate transformations 32 with A* = xx — y1. Intuitively, we expect vlx (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. viiy) = f ^ ) - ( 2 - 2 5 ) For a single DO-brane, it seems that replacing the coordinate description xl with the field description vx amounts to a lot of unnecessary complication. The reason we introduce vlx 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 0 ) , S=^jdtgij(x)xi(t)x^t). (2.26) This action is generally covariant because xl(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 Xl(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 vx(y): s = \IdtIddy 5d{x ~ y^(yK(yH(y)- (2.27) The time dependence of vx(y) comes the time dependence xl(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 covari-ant. To see what this action is in the standard coordinate description, we use the series%expansion for vlx(y). Using the fact that vx(y)\y=x = 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 x (y), so we might expect that any conventional action written using the Chapter 2. Invariance under spatial coordinate transformations 33 coordinates x1 can be rewritten using the vector field vlx(y). In the following sections, we will find that it is possible to define matrix generalizations of the vector field vlx(y) and the delta function 5d{x — y). We will then use these as building blocks for constructing actions for multiple DO-branes satisfying general covariance. 2.3.2 M a t r i x vector field from a matrix transformation law To construct a matrix generalization of vlx(y), we might first try an order by order approach, similar to the one we took in constructing the matrix trans-formation law. We write down the matrix vector field as a series expansion in powers of A i = X1 - yl: oo Vx(y) = ^ + zZ vLj,An • • •AJn- (2-28) n=2 This expression should reduce to the series (2.23) when the matrices are diagonal. Also, the coefficients V£ -n must be chosen to be appropriate functions of the metric and its derivatives at y1 such that Vx(y) has the correct vector transformation property Vim = %V&V), (2-29) when y% = Fl(y) and X1 = §l(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 Fx to a function on matrices. Note that we may regard the x dependent vector field vlx(y) of the previous section also as a y dependent function of x, fy(x) = vlx(y). This suggests that we may be able to obtain the matrix generalization oivlx(y) by promoting the ordinary function f* to a function on matrices using the map = V(F,X,g) (2.38) oo = F(y) + £ Vh...3SF(y\ g(y),.. .)A* . . . A * , n=l where ^ A are functions of Fl and qu and their derivatives evaluated at 31 •••Jn J y\ The procedure seems simple enough, and it appears at first sight that we obtain a satisfactory matrix transformation rule by arbitrarily specifying the functions V£ - n . 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* J n lead to an expression ^ ( F , X, g) wi th an overall dependence on y%. In other words . —¥(F,X,g)^0. (2.39) oy3 Therefore the coordinate independence condition is violated, and we do not obtain a matrix transformation rule that satisfies the composition rule. The good choices of V-x ^ are those for which the yl dependence drops Out. We already know of one such good choice, given by equation (2.34). Now let us derive some constraints to help identify the good choices of Vy j n . To do this, differentiate both sides of equation (2.37) wi th respect to yl. The derivative of the left hand side is 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 y1 is taken with X1 held fixed. The term in the large square brackets vanishes if and only if •i-X{ = -£-r*(F, X, g) = 0. (2.41) Combining this with the derivative of the right hand side of (2.37), we find - +%imv^- ^ if and only if (2.41) is satisfied. To put this into a nicer form, add the appro-priate 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 Vx(y) as a series in powers of A 1 using the definition (2.35), we can eliminate these factors of A 1 in favour of Vx(y) by inverting the series (2.35). The result is oo ViV&y) = + £ ^..jMvWxiv) • • • vxn(y), (2.44) n=2 where the _jn(g(y)) are functions of ^ and its derivatives evaluated at yl. Now VjVx(y) satisfies (2.43) if and only if these functions jn(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 VjVx(y) is of the form (2.44) with tensor coefficients Tj t - n. It is clear that Vx (y) transforms as a vector field under the resulting matrix transformation rule by construction. 2.3.4 A differential equation for V1 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 V1 for the matrix vector field. Now the prescription is the following. Write down the most general ex-pression of the form oo X 1 = y* + V{ + £ jnVjl ... V*», (2.45) 71=2 that reduces for diagonal matrices to the expression for the exponential map exp^(V). Here -n is a function of the metric and its derivatives at y\ This implicitly defines V1 as a series expansion in powers of A 1 = X% — yl oo V1 = A1 + Y, vL-inAJ1 • • • A J n • (2-46) n=2 Use this latter expression to calculate the covariant derivative of V1 as a series in A1. Then eliminate all factors of A1 by substituting back the starting expression (2.45) to obtain a differential equation of the form 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 V1 in the form of a differential equation. Note that the Tj 5 -n can be completely expressed in terms of the original coefficients A ^ -n are their derivatives. Therefore this differential equation constraint on V1 actually constitutes a set of recursion relations for A* „• . Finally any solution of these recursion relations will define a matrix vector field. And if desired, the matrix transformation rule under which V1 trans-forms as a vector can be explicitly obtained. 2.3.5 Solving the constraints The recursion relations for A ^ -n 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 condi-tions (v) and (vi) specified in section 2.2.3. Here we present the third order Chapter 2. Invariance under spatial coordinate transformations 39 result. We find the general solution A* = V* - \ v ) k V 3 V k - (Ir*w + aR)kl + bR\^ V>VkVl + 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 V1 allows us to construct (to third order) a two parameter family of objects V* + (aR)kl + bRijt) V'VkVl (2.49) 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 V1 should satisfy Vl(yo) — X1 — yl0 when one is working in Riemann normal coordinates about the point yl0. 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. 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 DO-brane 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 8d(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 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 ddk 6d(X -y) = j (2TT)C oMx-v) (2.50) 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 Sd(V) = J -0^d exp (ihV\y)) . (2.51) To see if this definition is sensible, we check the behaviour under general coordinate transformations: dy r ddk J (2^p (2TT)C exp (ikiV\y)) dy dy •dy* 5d(V). (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 invari-ant under general coordinate transformations. For the purpose of carrying out computations, we can express the matrix distribution function in an alternative form. Define Wi(y) = Vi(y) + yi. Then the Fourier expansion for the distribution function becomes r ddk / \ 5d(V) = J —-d exV(ikWl(y))exp(-ikiyl) (2.53) E n=0 (2ir)d ( - l ) n nl W»(y)...W^(y)}djn...djn5d(y). (2.54) Chapter 2. Invariance under spatial coordinate transformations 41 This form is useful for computing integrals with the matrix distribution func-tion 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 Vl(y) and tensors built from th.6 metric £(y) = r (g(y), v{y), v(y), R(y), VR(y),...) (2.55) and use this to form the Lagrangian L = j ddy Tr (Sd(V)C) . (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 num-ber of covariant derivatives of curvatures. In fact, we will demonstrate that essentially any generally covariant action (depending on g^ and X1 only) can be expressed in this form. 2.4.3 Expansion in powers of X1 To make contact with known results, it is desirable to re-express the La-grangian as a series in powers of X1. This can be done by integrating over space using the form of the matrix distribution given in (2.54): O O 1 L = £ 3 [dh • • • d^Tc (£(y)Wh(y). • • W^(y))] . (2.57) n=U We have the explicit expression (2.34) for V1, and therefore W\ so this gives a straightforward algorithm for obtaining the generally covariant Lagrangian L as a series in Xx 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 X1 is finite. Chapter 2. Invariance under spatial coordinate transformations 42 2.4.4 How large is the large class? We have discussed a method for generating a large class of generally covari-ant 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 X1) of the Lagrangian. Denote the leading terms of the Lagrangian by L n , and suppose that these terms are of n-th order in X1. If we perform a change of coordinates by an infinitesimal shift of the origin yl — yl + s\ then the variation of the these leading terms gives an expression of order (n — 1) in X1. 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 For example, if any X1 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 diffeo-morphism FL 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 where Ai...i„ are functions of the metric and its derivatives evaluated at the origin yl — 0, and Tn-Zn(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) w i t h F l (0) = 0, any such term becomes dxiLo = —Lo(X + e) = 0. (2.58) Ah..AMV),dm,---)T^{X) (2.59) A 1 . . . i»W) ,^ (0 ) > . . . )T i l - i - (X) . (2.60) Chapter 2. Invariance under spatial coordinate transformations 43 Since Fl(0) — 0, we also have Tii-in/x) = dhFil{0)... djnFin(0)Tjl-jn(X) + D(Xn+1). (2.61) The total Lagrangian is generally covariant, so if we match (2.59) wi th the leading contribution to (2.60) using (2.61), we find that A ^ . . . ^ must be a tensor. To summarize, the above considerations tell us that the leading terms of the Lagrangian can be written in the form Lo = Tr (Co), where CQ is constructed from tensors built out of the metric such that its trace satisfies dXiTr(£o) = 0. In appendix A . 3 , we argue that in fact the following stronger statement holds. We can choose the expression Co appearing in (2.62) such that it satisfies dXiC0 = 0. In other words, the terms occuring under the trace can be rearranged so that (2.58) holds even without taking the trace. From these properties of Lo, it follows that there exists an integrated Lagrangian L of the form (2.56) whose leading terms are precisely L0- We simply choose the scalar C(y) in the integrand to be Notice that the function To appearing in the integrand is the same function as that appearing in (2.62) but evaluated on different arguments. To see that this integrated Lagrangian has leading terms identical to L 0 , observe that the condition dXiCo = 0 implies that none of the terms in (2.57) involving the expressions . . . djk£,(y)\y=0 for k ^ 0 can contribute to the leading terms of the action. In other words, the leading contribution to the Lagrangian L is contained in the expression Tr ( £ ( 0 ) ) = Tr ( ^ ( O ) , V(0),..., g(0), VJ2(0), . . . ) ) • (2.64) A n d since V l ( 0 ) = X1 + 0(X2), the leading terms of L are given precisely by LQ. W i t h this result, we can argue that any generally covariant Lagrangian that can be expanded in powers of X% wi th coefficients that are functions of the metric and its derivatives at the origin, and which contains a single overall trace, can be re-expressed in the integrated form (2.56). The argument is Co = Fo(g(P), X, X, R(0), Vi2(0),...), (2.62) c(y) = MM, (^y), v(y), R(y), VR(y),...) (2.63) 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 Li that has same leading terms as L using the prescription given above. Then the difference between these two Lagrangians L — Li is another generally covariant Lagrangian, but with leading terms of order n +1. Now we can find another integrated Lagrangian L 2 that has the same leading terms as L — L i by the same procedure as before. Then the difference L — Lx — L 2 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 = ^2kLk. Incidentally this also shows us how to characterize the most general covariant Lagrangian up to terms of order m in X1. 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. 2.4.5 Relation to the base-point independence approach of de Boer and Schalm We have just described a method of writing down generally covariant La-grangians 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 yl, dC(y)/dyl = 0, then we would have C(F-1(y)) = C(y). (2.65) In this case, there is no need to integrate over space. We can simply choose an arbitrary reference point yl0 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 non-vanishing 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 Chapter 2. Invariance under spatial coordinate transformations 45 and Schalm [91]. These authors developed a formalism for writing down gen-erally covariant DO-brane actions directly in terms of X1, based on the use of Riemann normal coordinates about some base-point along with the require-ment of "base-point independence". What we just outlined in the previous paragraph is essentially a coordinate-independent description of their pro-posal. This can be seen more clearly by observing that our canonical choice of Vl(y) satisfies Vl(y0) = X% — ylQ in Riemann normal coordinates about ylQ. 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 La-grangians 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 La-grangian. To see this, consider any generally covariant Lagrangian defined by the integral expression (2.56). The expansion of this Lagrangian in powers of X1 gives us L(X,X,g(0),dg(0),...). (2.66) Now choose an arbitrary point yl0 and transform to a new system of coordi-nates y1 = Fl (y) such that yl0 is mapped to the origin of the new coordinate system, Fl(y0) = 0. Furthermore, we can choose this new system of coor-dinates to be a set of Riemann normal coordinates about the origin yl = 0. Now general covariance of the Lagrangian means that we must have 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 X1 = Vl(0). Making these substitutions, the Lagrangian becomes L(X, 1, g(0), dg{0), • • •) = L'{V{0), V(0)tg(0), £(<>), 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~x so that 0 is mapped to ylQ and L'(V(0), V(0),~g(0), R(0), •..) - L'(V{y0), V(y0),g(y0), R(y0) • • .)• (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(y0), V(y0),g(y0), R(yQ)...), (2.70) and since the choice of y0 was arbitrary, the scalar expression on the right hand side of the above equation must be independent of yl0. Therefore, we have expressed our original integrated Lagrangian as a base-point indepen-dent 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 C0(y) taken to be the starting point of the base-point independence method. Instead of trying to find order by order a series of higher order corrections £\, £2 , • • • needed to achieve base-point indepen-dence, we can simply form 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 re-written as a base-point independent Lagrangian. If £ 0 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. 2 A . 6 Examples Here we present some examples of Lagrangians that are invariant under gen-eral coordinate transformations. We consider a fixed background geometry satisfying the equations of motion Rij = 0. The DO-brane coordinates X1 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^in (terms with exactly two time derivatives) to fourth order is given (2.71) by Chapter 2. Invariance under spatial coordinate transformations 47 Here, we have suppressed the overall factor of To in the action. There are two arbitrary constants ax and a 2 . After performing the spatial integral, the result can be written as a base-point independent Lagrangian Lidn = (2-73) T r ( ^ g i j V ' V i + R . . k l a i ) v * y ' V ' V f c - Q + a, + 2a 2 ) V ^ V ^ W +0 (V r 5 ) This Lagrangian can be expanded in powers of X1 if desired, by using expres-sion (2.34) relating V1 and X1. One observation is that second order leading term completely determines the th i rd order terms (in X 1 ) of the Lagrangian, because these come from the uniquely determined order X2 term in series expansion of V1. The most general expression for the (minus) potential energy L p oti to sixth order is given by £ P ° t i = (2-74) /ddy Tr [5d{V)^gijgkl[y^ VkWJ, Vl}+ +9rjRkimn(h[V\ Vk)[V^ Vn]{Vl, Vm] + b2[{v\ V% ^ [ [ V ™ , V% V"'])}) . Aga in we have suppressed an overall factor of T0. There are two arbitrary constants b\ and b2- A s a base-point independent Lagrangian, this becomes -kpoti = (2.75) +9ijRkimn {(8b2)ViVjVkVnVlVm + (&! - 86 2 + v V J V f c V n V m V ' + + (bi + j^j ViVkV:iVnVlVTn - (2b2 + ^ ViVkViVnVmVl -- + 8b2 + viVkVnVjVlVm + (fci + 8b2 - -^j v < K * V n W V , B V ' J +0(V7)). Chapter 2. Invariance under spatial coordinate transformations 48 Expanding in powers of X1 we find that, similar to the case for the kinetic en-ergy, the fifth order terms of the potential energy are completely determined by the fourth order leading term. 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 re-sults. 2.5.1 Emergence of the geodesic equation For a single DO-brane, if we take the kinetic energy term (l/2)gij(x)xtx3' 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 xl(t) = alt is a solution for any constant vector a1. 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 k i n 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 Xx(t) = Axt for any constant vector of matrices A1. A direct calculation shows that this condition is satisfied if the constants a\ and a 2 in (2.73) are chosen to satisfy a\ + 4a 2 = —5/12. 2.5.2 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 / k i n + L p o t i of the Chapter 2. Invariance under spatial coordinate transformations 49 kinetic and potential terms given in (2.73) and (2.75), we can consider T-duality 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 0 ,**] (2.76) 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 ds2 — g^dy^dy" = (dy°)2+gijdytdy:'. ' Demanding that these two procedures give the same answer and implies that the constants oi, a^, &i, and 62 must satisfy a\ = b\ and a2 = —462. 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 ex-panding the action around a background configuration X1 = Dl + Yl, where D1 = diag(x\ . . . xlN) corresponds to branes with well-defined positions at xla, a = 1... N, and Yl 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 pro-portional to the geodesic distance between the corresponding branes at xla and x%p. For the Lagrangian L = L k i n + L p o t i , we can constrain the arbitrary con-stants 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 coordi-nates about Xp. Then we find (Mlp)ij = dkapdlap (gki(xp)gi3ixp) - gik(xp)gji(xp)j (2.77) -dkapdlapd^pd^{2a2 + %b2)gkl(x0)Rimjn{xp)Sj + ..., where dlap = xla — Xp. Note that there is no summation over the repeated a and j3 indices in this expression. Since we are working in Riemann normal Chapter 2. Invariance under spatial coordinate transformations 50 coordinates about Xp, the geodesic distance between xla and xlp is given simply by \Jgij(x/3)dzQpd3ap- Therefore the geodesic distance criterion holds (to this order) if and only if we choose a2 = —4b2. Interestingly, this is necessarily true if the T-duality constraint is satisfied. 2.5.4 Agreement with known results The effective action for DO-branes in type IIA string theory in ten non-compact 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 trans-verse metric gij — rjij + hij are (to linear order in hij), i / d 9 y My)Str( + {X\Xk}[X3,Xk}) S9(X - 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 ax = - 7 / 3 6 , a2 = - 1 / 1 8 , &i = - 7 / 3 6 and b2 = 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 Jd9yCQ(y)Tr(59(X-y)). (2.79) Then general covariance tells us this term should be generalized to Jd9y C0(y)Tr(69(V)(l + £ ' ) )> (2.80) where CJ is some scalar field involving commutators of V1 of higher order in the background fields. Another example is the generally covariant gen-eralization of the term that gives rise to the D-brane dielectric effect. The Chapter 2. Invariance under spatial coordinate transformations 51 linearized coupling is in this case The generally covariant term that reproduces this to lowest order is 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. (2.81) (2.82) 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 transforma-tions present an additional complication, since they look complicated even in the Abelian (single D-brane) case if we restrict to the static gauge. Be-fore attempting to analyze the full group of space-time diffeomorphisms, it is natural to begin with the simplest case for which the additional complica-tion 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 transfor-mation 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 X1, 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 trans-formation rule exists to all orders.1 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 X1 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 (X2) kinetic term and to the simplest potential term Tr [X\ X 7 ] 2 to obtain (independent) Lorentz invariant results (we work up to 0(X6)). We 1Oi course, this should be guaranteed if string theory is consistent and Lorentz invariant in flat space. Chapter 3. Invariance under Poincare transformations 5 3 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 Gali lean invariance (including parity) in the Abel ian case. Finally, we show that for any choice of field there is at most one invariant action depending on X1 and not X1 or higher derivatives of X1. Unfortunately, we find that any invariant generalization of the Abel ian kinetic term may be written in 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. In section 3 . 4 , we look for a more natural way to write Lorentz invariant actions. A s in our previous studies, we look for matrix-valued covariant objects defined as fields over space-time from which we can bui ld manifestly invariant actions as integrals over space-time. We find (at least up to fifth order in X1) that there exists a covariant matrix vector field V^(y) built from X1 but transforming simply under a Lorentz transformation = S}1vyv as V»(y) = A " , r ( i / ) . In the Abel ian case, V1 is the derivative of the proper distance to the tra-jectory along a geodesic which intersects the trajectory orthogonally. In addition, discuss the existence of a covariant matrix distribution function ®(y) which reduces in the Abel ian case to 0 ( y ) = jdr J-dTx»dTx» 5d+\xv{r) - y»). In section 3 . 5 , we show that all Poincare invariant actions may be written using these two covariant objects as S = j dd+1y Tr (C(V(y))Q{y)), where £ is a scalar built from V. The independent Poincare invariant struc-tures may be characterized by their leading terms, which may either be / dt Tr {X2) or may be written as the integral of a Lagrangian L(X, X, X,...) with an even number of Xs and time derivatives satisfying 6\L(X + e, X,...) = df,L(X, X + /?,...) = 0, i.e. a term wi th al l Xs and Xs appearing in complete commutators. This is precisely the necessary condition we found in 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 associ-ated with the branes (e.g. the stress-energy tensor or Dp-brane currents). Throughout this chapter, we focus only on the DO-brane matrix coordi-nate degrees of freedom. In particular, we neglect the fermionic degrees of freedom, and assume that the gauge field A0 has been set to zero by gauge transformations. 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 Xl(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. 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 yi1 = (t,y%). In static gauge we specify the spatial coordinates of the DO-brane as a function of time xl(t). Clearly, this description is inconvenient for the purposes of discussing Lorentz invariance. For example, consider chang-ing to another system of inertial coordinates by using a general Poincare transformation. f = A V + a" (3-1) and let xl(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. Chapter 3. Invariance under Poincare transformations 55 The Poincare transformation (3.1) allows us to write down the relation i = A V + A V W + «°. x\t) = A y + Ay j ' ( t ) + a\ (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°0t + A ° j O ; l ( £ ) + a 0. 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 invari-ance transparent. Introduce an arbitrary worldline parameter o and describe the brane by a set of embedding functions xM(a) whose Poincare transforma-tions are simply #*(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 ma-trices 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 transformations 56 have discussed the Poincare transformations as passive coordinate transfor-mations. 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 Sax1' = a1 5aox% = ~a°xl Swxl = uijxj S0xi = 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 The 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 X1 and generalize unambiguously to the non-Abelian case without modifi-cation. 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 (fipSp ~ SpSp)^1 — $uii=ppi-pifrX% (3.7) (SsSp - SpSg)Xl = 5ao=p.aXl (3.8) (5aoSp - SpS^X1 = S^pX* (3.9) {SpSw-5Jp)Xi = dpi^jflX* (3.10) Chapter 3. Invariance under Poincare transformations 57 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 transfor-mation law at higher orders in X * . We wi l l therefore write the putative Poincare transformation rules for the non-Abelian case as 5sXl = a} (3.11) 6aoXi = - a 0 ^ (3.12) S^X' = ]XkXk) +0(X8), 2 8 / Chapter 3. Invariance under Poincare transformations 64 independent of the choice for the 0(X4) and higher order terms in the trans-formation law. It turns out that these correction terms reproduce known terms in the DO-brane effective action obtained [65] by T-dualizing the sim-plest (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(X4X2) terms in S = - Jdt STr v / de t (Q i 0( l - X^XJ), (3.31) where Qij = Sij + i[X\Xj] and Qij is the inverse of Qtj. On the other hand, the required correction to (3.30) at order X8 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 X4, we find that adding the symmetrized version of terms in the Abelian relativistic kinetic term suffices up to 0(X5) to make the action invariant, but this breaks down at 0(X6). 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 S = -jdt S t r ( l - \x2 - \{X2)2 - ^ ( X 2 ) 3 (3.32) (xixk[xi,xi][xk,xj] - ZXiXiXk[Xk, {xfx1}} -3XiXk[Xi,Xi][Xk,X>] + 2XjXk[Xk,Xi][Xj,Xi}) ) + 0(X8). is zero up to 0(X7) terms that would presumably be cancelled by the leading order variation of 0(X8) corrections to the action. The corrections here are not among the known terms appearing in (3.31). We will see in the next sub-section 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 al-ways add with arbitrary coefficients any of the higher order invariant struc-tures 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. Chapter 3. Invariance under Poincare transformations 65 3.3.3 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.7 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 Ln of order Xn. According to the con-straints of the previous subsection, Ln 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\Ln = Tr ( S y m ^ X ' X * + fi X3 X^C^X)). 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}Ln + 5°Ln+2 = jtTv (fiX3Qn(X)). Note that we cannot have terms where 0 is contracted with a derivative of X on the right side since this would produce fiX3 terms which are not present on the left side. Comparing all terms containing a second derivative of X , we have Tr ( S y m ^ ' X O Q ^ X ) ) = Tr (fiX3jtQn(X)). (3.33) Now, on the left side, the fiX3 always appears adjacent to the X in the trace. On the right side, Q is of order X n - 4 , so there will certainly be terms for which the X is not adjacent to fiX3. 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. 7This 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. Chapter 3. Invariance under Poincare transformations 66 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(^X2 + • • • ) . (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 varia-tion 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 i F i ( A ' ) ) ) (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 Fs ap-pearing in commutators.8 Also, since the total number of time derivatives and Xs was assumed to be equal, F will contain at least one undifferenti-ated 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 Xi X1 + 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). At 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. 8Terms 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. Chapter 3. Invariance under Poincare transformations 68 3.4 Covariant objects The naive order-by-order approach to writing down Poincare invariant ac-tions discussed in the previous section is cumbersome to say the least. Fol-lowing 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,xl(t)). We define an associated space-time vector field in the following manner. At any space-time point sufficiently near the DO-brane world-line, there exists a geodesic that passes through y M and intersects the DO-brane world-line orthogonally (with respect to the Lorentzian metric). Call the point of intersection x^(ty) — (ty, xl(ty)). 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. In other words, v»{y) = afty - y", (3.37) where the time ty is implicitly determined by the condition - 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 vfX{y) is not globally well-defined.9 However, this is enough to ensure that there is a well-defined expansion for vfi(y) in powers of the static gauge coordinates 9For a uniformly accelerating trajectory, v will cease to be well-defined beyond the Rindler horizon. Chapter 3. Invariance under Poincare transformations 69 xl(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 trans-form as a four-vector under Lorentz transformations, In particular, for an infinitesimal boost we have Sv°(t,y) = P-v(t^-0-Wv°(t,y)-P-ydtv°(t,yj, 5v(t,y) = 0v°(t,y^-tP-Vv(t,y)-P-ydtv(t,y). (3.39) 3.4.2 Generalization to the case of multiple DO-branes We would now like to see whether v11 generalizes to the non-Abelian case. That is, we would like to construct a set of matrix-valued functions V^(y) de-fined as a formal expansion in terms of Xl{t) which transform as a space-time vector field and which reduce to d i a g ( ^ ( y ) , . . . , v^N(y)) when the matrices Xl(t) are diagonal. To ensure the latter condition, we may write Wt(y) = V*m(y) + AV>(y), (3.40) where Vs^m(y) is the expression obtained by replacing all occurrences of xl in the expansion of v'x(y) with X1 and using the completely symmetrized prod-uct of matrices and AVp,(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 X5, it is possible to choose AV^(y) so that the covariant transformation rules hold. At fifth order in X\ our explicit expression for VfJ,(y) in terms of X1 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 M + dpVw[V^, dvVp]. There may be some canonical choice for V^, as we found for the spatial matrix Chapter 3. Invariance under Poincare transformations 70 vector-field in chapter 2, but we do not know the additional constraints that would select this. 1 0 On the other hand, we will see that any choice for V 4 (assuming one exists) will allow us to construct the most general Poincare invariant actions. 3.4.3 A covariant matrix distribution Assuming that the covariant object V,i(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 yfv. To obtain an invariant action, we should integrate over space-time, but we still need some analogue of the 8d(V) term in chapter 2 that would localize the action to the well-defined positions of the individual branes in the case of diagonal X1. We have not been able to construct such a distribution directly from the covariant object V M . 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 X1. Our goal is to construct from Xl(t) a matrix valued field Q(y) such that actions of the form S = J dd+1y 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, 5d+\xv(r) - y») . (3.42) We will call it the covariant matrix distribution. Under Lorentz transformation a density should transform as 0 ( A y ) = 6 ( y ) , (3.43) 1 0One constraint that we might impose is that V should satisfy d^V^ =dvVtl. This holds in the Abelian case, since VM = — ^ d^V2. In the non-Abelian case, given any definition of V * we can take = - j ^ f V " ^ ) which ensures that is covariant and that d^Vv is symmetric. Chapter 3. Invariance under Poincare transformations 71 or specifically under an infinitesimal boost 50e(t, y) = -tp • ve(t, y)-p-y dtS(t,y). (3.44) Defining the moments of the distribution as @ ( n - i n ) ( t ) = J d d y $)y^... tf" , (3.45) we find that the constraints (3.44) become SQ(il-in) = ntp(hQi2...in) _ p^Qdh-in) (3 4 6 ) dt To zeroth order in the commutators a solution to this constraint is given by eg^") = Sym (Vl -X2 X(h ... Xin^ . (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 esym(t,y) = Sym ( V l - * 2 Sd(X(t) - 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 transfor-mation rules (3.11). A lengthy calculation in [2] using a transformation rule valid to second order in commutators and all orders in X1 gives the necessary corrections terms up to second order in commutators and to all orders in X1. 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 Vfl(y) and 0(y) exist to all orders, and proceed to discuss the Poincare invariant actions. Chapter 3. Invariance under Poincare transformations 72 3.5 Manifestly Lorentz invariant DO-brane actions Given the vector field Vfi(y) and the covariant matrix distribution Q(y), it is now manifest that any action J dd+1y 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 Q^V) = E ^ p ^ ' ^ W di,... dlp6d(y), (3.50) of 0 in terms of its moments. Then the action takes the form S = fdtf:Tr(^dh...dipC(V)\yi=0ei^(t)). (3.51) J p=o \P- J Since 0 ( l l - * p ) = 0(XP), the leading term in the action will come from the set of all terms for which n + order(d^ • • • dinC) 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 . 3.5.1 The most general Poincare invariant action In section 3.3.1, we showed that the leading term S0 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 2 ) 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 2 ) , we can write a Poincare invariant completion as - j dd+1y Tr(0(y)). Chapter 3. Invariance under Poincare transformations 73 Otherwise, the leading order Lagrangian L0 may be written as a sum of terms for which all X s and Xs appear in commutators. Now L0 = 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 (X1)^ will pair with the index on some other matrix (X1)^ where m and n are the number of time derivatives on the first and second matrix respectively.11 We now define a matrix object C(y) built out of V M 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 (X*) ( 2 f c ) • • • ( X { ) ( 2 ° — ( - d 2 ) k V ^ • • • {-d2)% ( v ' ) ( 2 W ) . . . ( v i ) ( 2 W ) _ ^ -\(-&)kd,yv---(-di)ldvV>i (3.52) 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 ( 2 f c ) • • • (X^+V... (Xj)i2p) • • • (XJY2q+V (3.53) __> -(d2)kV^...dil(-d2)lVa---{-d2)pVv---du(-d2YVa After these replacements, we are left with an object C that transforms as a scalar field, so the action / dd+1y 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 duV^ appear in commutators (since we assume all X s and X s do), so that possible lower order terms from the leading yx in V1 vanish. As a result, the leading order term in £(y = 0) is exactly £ , and all of the y-dependent terms in C(y) 1 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. 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 L0. 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 — J2ciSi there must a subset whose leading terms have maximum order. Choose an action Smax in this subset, and suppose that Smax has leading term So at order Xp. 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 Smax — S' is of the form S — XI CiSi and has a leading term of higher order than Smax, 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. 3.5.2 Examples To close this section, we discuss as examples the Poincare invariant comple-tions 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 1 2 Jdd+lyTv(e(y)[-l + C4(V(y))}y where £ 4 is an arbitrary scalar built from Vs and an equal number of deriva-tives, which may without loss of generality be taken to be a term with at least 1 2Note that any choice for £4 may be absorbed into a redefinition of Q(y). The arbi-trariness in 0 corresponds to the freedom to make such redefinitions. Chapter 3. Invariance under Poincare transformations 75 two commutators of order V4 or higher. 1 3 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 con-strained, 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 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 4n time derivatives. Then 4n < 2n + 4, so we have n < 2. It is then easy to write down all possible leading terms con-taining 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 coef-ficients 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 i s r C J dd+1y Tr (e(y)([V„Vl/}[V^V} + C6(V(y))). 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, 1 3 This follows since the leading term in any higher order invariant action will be at least of order X4 and by the construction of the previous subsection, we may construct such an action using an C with terms of order V4 and higher. 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 Jdd+ly Tr ( e s y m ( y ) ( [ v ; y m M , v ; y m j [ v ; y r n ^ v ; y m t ' ] ) ) . where 0 s y m and Vsym 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 STr(bn[X\Xj}{X\ Xj]X2n + cn[X\ Xj][X\ Xk]XjXkX2n), n so in this case, the additional constraint of boost invariance fixes the infinite series of coefficients bn and c„ completely. To close this section, we note that our structures V*1 and 0 provide an alternate way to write invariant actions even in the Abelian case. For example, using our prescription, the Galilean invariant term \x2 has Lorentz invariant completion Jdd+1y d2v»d\9{y). 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 con-straints on the form of the effective action. In this section, we note that similar constraints arise in the expressions for the conserved space-time cur-rents 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 consider-ations 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 = pjdwyC^(y)J^y). (3.55) Chapter 3. Invariance under Poincare transformations 77 Since is a Lorentz vector, Jp(y) must be some expression built from Xl(t) transforming as a vector under Lorentz transformations. At low ener-gies/small velocities, the leading order expression for J^(y) — (p(y),Jl(y)) (ignoring fermions) is a simple generalization of the Abelian expression [75], / \ r d9k p(t,y) = Tr (Sd(X(t)-y)) = J - j L L . Tr ( e i f c l ^ ) , (3.56) r d9h J%y) = Tr (X*(t) 5d(X(t) - y)) = J ^ Tr ( X V f e J ^ ) . It is easy to check that current conservation, c V = 0, (3.57) is satisfied with these definitions. However, we will now see that J M does not transform as a vector under Lorentz transformations (without additional correction terms). A Lorentz vector field J M should transform under Lorentz transformations as > (Ay) = A"vJ"(y). (3.58) This implies that under an infinitesimal boost we have 5pp{t,y) = P-J{t,y)-0-Vp(t,y)-P-ydtP{t,y), 60J(t,y) = Pp(t,fl-tP-VJ{i,fl-p-ydtJ(t,y). (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 fypfr-'-) = p. J I M •••<») + n£/? (V2" in ) - & JtP{Jiv'in)' fyjfr-*»> = ^ p ( i l " - ^ ) + n t / ? ( i l J ' 2 " ^ ) - / ^ ^ J ^ 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 ^ = STv(X^---X^), ^ s y m ' i n ) = S T r C X ' 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 X1 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. 79 Chapter 4 Summary and outlook In this thesis, we have presented some progress towards understanding gen-eral covariance in multiple DO-brane actions. For the restricted problem of general covariance with respect to spatial diffeomorphisms discussed in chap-ter 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 free-dom. Specifically, we discovered that if we could find a matrix-valued field Vl(y) (constructed out of the matrices X1) that satisfies certain well-defined constraints, then it is possible to explicitly determine the transformation rule for the matrix coordinates X1 of the DO-branes under arbitrary spatial diffeo-morphisms. Furthermore, this V1 can be used to write down essentially any generally covariant action for X1 coupled to a spatial metric in a form that makes the covariance obvious. While we were not able to prove the existence of Vi(y), we were able to construct it as an expansion in powers of X1 — yl to fourth order. This approach can be extended to include more general coordinate trans-formations. 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 t = t, $ = ^{1^), (4.1) together with a position dependent transformation of the time coordinate, leaving the spatial coordinates unchanged i=F°(t1yi)> 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 hy-persurfaces 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 transforma-tion (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 co-ordinate in a non-trivial manner such as (4.2)), we have considered the prob-lem 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 outlook 81 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 direc-tions 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 Vtx(yu) replacing the space-time embedding coordinates X^. 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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. Appendix A Appendix to chapter 2 92 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 + dj9kl + dk9jl). ( A . l ) Under a general coordinate transformation yl = Fl(y), the Christoffel symbol transforms as « (~, = dldy^dTrl , , _ dymdyn d2f ljk\V) dy1 dyi d y k m n { y > dyi dyk dymdyn' 1 ' ' The solution to the geodesic equation with £ l(0) = yl and d£,l/d\(Q) = u\ is given by 0 0 \ n C(\) = y i + \ui-Y: -rl.,n(y)uh... (AA) 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 sym-bol, but acting only on lower indices in the above equation. The round brackets denote complete symmetrization of the indices. A Riemann nor-mal coordinate system about the point y%0 is a coordinate system in which Tjj jn(yo) = 0, so that geodesies which pass through the point yl0 are given by £* = yl0 + Xu\ Finally, our conventions for the Riemann tensor are R)H = dkT)t - dtrjk + r^r™ - r ^ r ™ . ( A . e ) Appendix A. Appendix to chapter 2 93 A.2 Solving the constraints for V1 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* + VJkAjAk + VjklAjAkAl..., (A.7) where once again A 1 = Xx — y1. To reiterate, our objective is to find a suitable set of coefficients VI „• such that the covariant derivative of Vl(y) can be expanded as a series in Vl(y) with tensor coefficients. The series for Vz(y) can be inverted so that we have A * = Vi + A)kVjVk + A)klVjVkVl.... . (A.8) The lowest order expansion coefficients for the two series are related as follows vjk - ^jk Vjkl = -A)kl + (A*-aA£j + AJ a A" f c ) Vjklm = ~A)klm + (AjaAklm + AamAVkVlVm (A.15) + ^ijnRklrn(ym[y{^ , V]] + ^ ( V ) ) + 0(V5). Here, the expression Cljkl(X) indicates any complete commutator expression, -^—Cijkl(X + e) = 0. dem y 1 There are six independent terms of this type, Cijkl(X) = a[X\Xj}[Xk,Xl] + b[X\Xk][Xj,Xl] + c[X\ [Xj, [Xfc,X*]]] +d[Xf [X\ [Xk, X1}}} + e[Xk, [X\ [Xf X1}}} + f[Xk, [Xf [X\ X1}}}. With this definition, the covariant derivative of V is a tensor given by V . y i = _5{-(lR{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 Fl, the metric and all of its derivatives are evaluated at y = 0 ) : ¥(F, X, g) = Fl + djFiXj + \djdkFiXjXk + ^-djdkdiFiXjXkXl -^dmdkF T™,[Xf [Xk,X1]} + ^djdkdldmFiX^XkXlXm +^-djdkdnFi Tfm(XlXmXjXk + XjXlXmXk +XjXkXlXm - XjXlXkXm -XlXjXmXk - XlXjXkXm) Tnklm{XkXlXmXi - XkXlXjXm Appendix A. Appendix to chapter 2 9 6 -XkXjXlXm + XjXkXlXm) +^-dndpFi TnjkTplm(XjXlXkXm - XjXlXmXk) +^-djdnFi YnkpTpm{XjXlXmXk + XjXkXlXm -2XlXmXjXk + XkXlXmXj +XlXmXkXj - 2XkXjXlXm) -^dnF* Rnklm(Xm[X^k, [Xl\X3]) + Vklm(X)) + 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--in = ^dtr • • diAK + y%=o in the expansion of Vy + y in powers of y. Using (A. 15) find w} = -r;.fcxfc-i(rjfc^^ 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 to chapter 2 97 useful to have expressions for Ws in Riemann normal coordinates. Going to this system, we have for example, 4 -3-"(fci)j ~Q R\lm)(j;k) + so that we obtain Wl = Xl + 0(X4) wjk = ™*U*' -1 v,j%fc)mx<^ ra> - ivyHflHfc)A:('x™) + 0 ( X 3 ) w;klm = 0(VVRX). (A. 18) A.3 A technical result about commutator expressions In this appendix, we argue that for any expression £ — A^.^X11 • • • Xln with more than three Xs such that Tr(£) satisfies condition (2.58), that is dxiTr(£) = 0, (A.19) there is another expression £ ' , equivalent under the trace (Tr(£') = Tr(£)) satisfying dXiC = 0. For simplicity, we discuss terms without X s , but our argument can easily be extended to include them since they trivially satisfy dXiX = 0. Note first that any symmetry properties of the coefficients A can be trans-ferred 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 Q h - i u = ^ a j M D . . .XM«). Appendix A. Appendix to chapter 2 98 It now follows that (A. 19) will be satisfied if and only if a x«Tr(C? i l- i») = 0, (A.20) 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 X11 appears first, Tr(Oh-in) = TrpC^Oi 2 -*") . (A.21) Then (A.20) implies that 0 - y'dxiTriO*1-*") = yhTx{0\2-in) + X ^ ^ T ^ X ' 1 ^ ^ - " * " ) . m In order that the right side should vanish, the expressions involving ylm must vanish independently for each m, therefore Tt(Oi) = 0 dxid = 0. The second condition implies that 0\ is a sum of expressions built from products of nested commutators.1 Since the trace of 0\ must also vanish, we should have a where Aa and Ba 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 On"mln contained at least four X s , at least one of Aa and Ba must contain more than a single X for each a (we may assume, without loss of generality that it is Ba). Then, inserting this expression into (A.21) and rearranging the commutator, we find Tr(O) = T r ( 0 ' ) , where Q' = Y:[Xh,Aa]Ba. (A.22) a 1 Equivalently, 0\ must be the dimensional reduction to 0+0 dimensions of some gauge covariant expression. Appendix A. Appendix to chapter 2 99 Further, it is clear that dXiO' = 0 since O' is a sum of complete commutator expressions. Finally, if we define £' = ^ ,.,n(o'r-in, then Tr(£ ' ) = Tr(£) and dXiCJ = 0 as desired. Note that for terms at order X 3 , both A and B must be single X s , and as a result our assertion fails. 100 Appendix B Appendix to chapter 3 B.1 A consistent transformation law to sixth order in X1 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 60Xi = (3H- pjSym{XlXi) + ftT'i. (B.1) If we define the matrix Clki to be Cikj = [Xk, Xj]} + [Xk, [Xj,X1}} - [Xj, [Xk, X% (B.2) then a consistent choice for the matrix T u to sixth order in X1 is Tj = ^Sym(xkCikj + XlXlXkCikj - x k x l (xl[xl, [xk,xj]} + xk[xl, [x\xj\] + ( 1 / 3 ) x l [Xk, [Xl,Xj}}^j -XkXl (\[Xi,Xl][Xk,Xi] + [Xl,Xk][Xl,Xj} + [Xi,Xj}[Xl,Xk]) - ( l / 1 2 ) X f c {[X1, [Cklj,X1]] + [Cklj, [Xl,X1}} + [X1, [Cilj,Xk]\ +[Xk, [Cil\ X1}} + [X\ [Cilj, Xk}} + [Cilj, [X1, Xk]} +3[Xl,[Cilj,Xk}] +[x\ [[Xk, [X\X%X1]] + [[xk, [xl,x% ix\x1}} +[Xk, [{x\ [Xl,X%X1]] + [[x\ [X\X% [Xk,X1]} +[X\ [[Xk, [X1, X%X1}} + [[Xk, [X1, X% [X\ X1}} +[Xk, [[X\ [X\X%X1]} + \[X\ [X\X% [Xk,X1]} +[X\ {[Xk, [X1, X%X1}} + [[Xk, [X1, X*]], [X\ X1}} +[Xk, [[X\ [X1, X%X1}} + {[X\ [X1, X% [Xk, X1}} Appendix B. Appendix to chapter 3 101 +[Xl,[[Xi,[Xk,Xj +[Xk,[[X\[Xl,Xj +[Xk,[[Xi,[Xl,Xj -[Xl,[[Xk,[Xj,Xi -[Xk,[[Xl,[Xj,Xi -[Xl,[[Xk,[Xj,Xi -[Xk,[[Xl,[X',X\ -[Xi,[[Xl,[Xj,Xk -^^[xWxfx1 -[X\[[Xl,[Xj,Xk -[Xk,[[Xl,[Xj,Xi +[Xj,[[Xk,[Xl,Xi +[Xk,[[Xj,[Xl,Xi +[Xl,[[Xj,[Xk,Xi +[Xk,[[X*,[Xl,Xi +[Xk,[[X',[Xl,Xi +[Xi,[[Xj,[Xl,Xk +[Xk,[[Xj,[Xl,Xi +[[Xi,Xl],[[Xk,X' +{[Xl,X^[[Xi,X -\[Xl,Xj],[[Xk1Xi -[[XfX%[[Xl,X -[[Xj,Xk],[[Xl,Xl +[{X\X%[[X\Xk +[[Xl,X%[[XfX +[[Xi,X%[[XlX +0(X8). },xl}} + ],*']] + },x1}}-},x1}}-IX1}} IX1}}-},x1}}-ix1}}-},x1}}-},x1}}-IX1}} + },xl}} + },x1}} ixl}} + IX1}} + ],*']] + ],*']] + ],*']] + },xl}} + ),x1}}-IX1}}-IX1]}-IX1}} + },xl]} + %Xl}} + [[xW.x^ux",*1]] \[X\[X\X%[Xk,X1]] [[Xk,[XfX%[Xl,X1]} [{Xl,[XfX%[Xk,X1}} [[Xl,[X',X%[Xk,X1]] [[X\[XfXk}l[X\X1}] l[Xl,[XfX%[Xk,X1}} \[X\[XfX%[X\X1}} [[Xl,[XfX%[Xk,X1}} l[Xk,[Xl,X%[XfX1}} [{X\[Xl,X%[Xk,X1}} [[Xf[Xl,X%[Xk,X1}} [[Xf[Xl,X%[Xk,X1]} [{xf{xl,xkux\xl\} [[Xf[Xl,X%[Xk,X1}} [[Xk,X^[[X\Xl},X1}} [[X\X%[[Xl,XJ},X1}} l[Xk,X%[[Xl,X>],X1}} [[Xl,Xk},[[XfX%X1]} [[Xl,X%[[XfXk},X1}} [[XfX%[[X\X%X1}} \[X\X%[[X\X%X1]} [[X\Xk],[[Xi,X%X1]] (B.3) Appendix B. Appendix to chapter 3 102 B.2 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 2 ) or can be written in a way such that all Xs and Xs appear inside commutators. We first show that any action satisfying 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 La-grangian L. Then employing the symmetrized expansion discussed in section 2, we may write 1 L = L+J2 ^ T S T r ( L ( i , . . i m ) X i l . . . X * - ) , (B.5) m=l m -where the various terms in L and L(t1...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 SeS to vanish under a translation 5X — e, 5L must be a total derivative 5eL = J^-UK (B.6) Generally, Ul may be written ^ = E ^ S T V ( ^ 1 . . , m ) X - . . . X - ) , 1-We axe using the fact that any product of matrices may be written as a sum of com-pletely symmetrized products, where the individual terms in a product must be individual matrices or complete commutators of the form [x£1\[x£>\[...,[x^\x£r)]...}}} where (n) represents the nth time derivative. Appendix B. Appendix to chapter 3 103 so (B.6) becomes E 7 ^ T ) j S T r ( L ( u . . ^ ) 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 + ^ " i l - l n > where C is a commutator, C ~ [(Xj)^\ Aj1""1"]. Since S T r ( [ ( ^ ) ( ' ) , ^ j 1 - i ' l ] X i l • --Xin) = n S T r ^ 1 - " * " ^ 1 , (Xj)®] • ••X1"), the term in (B.5) involving C can rewritten such that it has only n — 1 free Xs2 and therefore can be absorbed into a redefinition of Lix...in_x. With this redefinition, we now have kn-in) = U t , n . (B.8) In particular, U must be completely symmetric on all its indices, so we can W l i t e Uh-in = U(m-3n)-Comparing the terms in (B.7) with (n — 1) free Xs, we find where Cj 1...j n_ 1 is a commutator. As before, by rearranging terms in the trace, we may eliminate C in favour of a redefinition of £ ( i 1 . . . i n _ 2 ) - Thus, and it must be that t/j l 2 1... i n_1 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(ii-ik) = UUh-ik)XJ + U(h-ik) (B-9) 2This assertion would be incorrect if A were [X*,Xj], but this is impossible, since the i and j indices would have to contract with the indices on two other Xs which are symmetrized. Appendix B. Appendix to chapter 3 104 for all k>l. Substituting (B.8) and (B.9) into (B.5), it follows that £ = 4 Z ^ S T r ( L 7 ( i l . . , n ) J ^ • • • X*-) - Tr (tyX*) + L, dt „n\ where we have integrated by parts to get the second term. Since L and Ul 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, deL(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 5fiL=jtPi&. (B.10) 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{ll...lm)X^...Xi-), ( B . l l ) 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 $ l oc X\ any free X in ®\ i x .„ i m) 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>l oc X1 or all X s and X s in ^(ii-im) a P P e a r m commutators. Inserting the expansions ( B . l l ) in (B.10), we have E ^ ^ y S T r ^ , . . ^ ) ^ ^ . . . X ^ ) = (B.12) E ^ S T r ( $ ; i ] . . , m ) X - . . . X - ) + ^_l i y [ STr (4 i . . , m ) X- . . . X - ) . This equation is exactly analogous to (B.7) above, and the rest of the proof proceeds in parallel to that above.3 This time, we end up with the statement 3The only change is to the comment in the previous footnote, which should now deal Appendix B. Appendix to chapter 3 105 that L = Tt E ^ S T r ( $ ( n . . , n ) X ^ • • • X<") - Tr ( ^ X * ) + L The special case that oc X1 corresponds to L0 = Tr (X2). 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 Vy,{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°2 - v°3 + t , ° - v°5 + ..., (B.13) where = y x v% = x-x + (y-x){y-x) i>3 = (y • x)(x • x) + (x • x)(y • x) + (x • x)(y • x) +(y • x)2{y • x) + (l/2)(y • x)(y • x)2 (B.14) v® = (x-x)(x-x) + (x-x)(x-x) + 2(y-x)(y-x)(x-x) +2(y • x)(y • x)(x • x) + (y • x)2(x • x) + (y • x)(y • x)(x • x) +(l/2)(x • x)(y • x)2 + (3/2)(a; • x)(y • x)2 + (y, xf{y • x) with three special cases, C oc [Xl,Xi], C oc [X%,Xi], and C oc [Xl,Xj], for which it is apparently not true that rearranging the commutators T r ( C I L . . . I B X I ' . - - X I - ) leads to an expression with no free Xs and less free Xs . 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 Xs , 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 to chapter 3 106 . +(3/2)0/ • x)(y • x){y • xf + (l/6)(z/ • x)(y • x ) 3 t»5 = (x • if(y • 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)2(x • x)(y • x) +3(y • x)2{x • x){y • x) + (y • xf(x • x) + (3/2)(y • x)(x • x)(y • x)2 +(3/2)(y • x)(x • x)(y • x)2 + (9/2)(x • x){y • x)(y • x)2 +{3/2){x • x)(y • x)(y • x)2 + 3(y • x){y • x)(y • x){x • x) + (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 • x) + (y • x)\y • x) + 3(y • x)(y • xf(y • xf +(2/3)(y • x)(y • x)(y • xf + (l/2)(y • xf(y • if +(l/24)(yx)(yxf. Note that all xs and ys appearing on the right hand side of the above equa-tions are the spatial xl and y1. Also, all factors of xl and its derivatives are evaluated at y°. The expansion for the spatial components is vl(y) = -yi + v\-vi2 + vi-vi + vi + ..., (B.15) where v\ = xi v\ = xl{y • x) vl = xi{x-x)+xi{y-x){y-x) + (l/2)xi(yxf (B.16) v\ = xl(y • x){x • x) + xl(x • x)(y • x) + xl(x • x)(y • x) +xl\y • xf(y • x) + (l/2)xi(y • x)(y • xf + xl(y • x)(x • x) +xi{y if (yx) +(1/6) x* {y if v\ = i l ( i • i)(x • i) + il(x • x)(x • i) + 2il(y • x)(y • x)(x • i) +2xl(y • x)(y • i)(x • x) + il(y • xf(x • i) + i\y • x)(y • x)(x • i) +(l/2)xi(x • x)(y • if + (3/2)^(0; • x)(y • if + il\y • xf{y • i) +(3/2)ii(y • x)(y • x){y • if + ( 1 / 6 ) ^ • x)(y • if -\-2x\y • i)(y • x)(x • i) + xl(y • xf(i • i) + xl(y • if(x • x) +{3/2)xl{y • if(y • xf + {l/2)x\y • x)(y • if + {1/2)3*{x • if +{l/2)xi(y • if(x • i) + (l/2)x\y • if(y • x) + (l/2A)x\y • if. Again, all xs and ys appearing on the right hand side of the above equations are the spatial x1 and yl, and all xs and derivatives are evaluated at y°. The Appendix B. Appendix to chapter 3 107 matrix generalization V^(y) has the form V»(y) = V£in(y) + AV»(y), where V£m(y) is the expression obtained by replacing all occurences of xl in the expansion of v^(y) with X 1 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 AV°(y) = AV4° - AV5° + . . . , (B.17) where Al/4° = T^Sym (XJ[X\ \Xj, X*]] + Xj[Xj, [X\X*}} + P[X\ [Xj,X1}} + X ' [ X ' ' , [X\X { ] ] + X'[X«, [X\X% + X*[y • X , [X' ,y • X}} [y-X,y X}} + 2Xi[y • X, [X*,y • X}} + X'[X>', \y X, y • X]] +Xl[y. X , [Xj,y X}} + X*[y • X, [X\y • X]] + X3[y • X, [Xfy X]]) A V f = ^ S y m [Px%y X), [X' .X*]] + X ' X ^ X ' , [(y X),^]] (B.18) +2X*Xi[(y • X), [X'', X 4]] + X'X*[X>' , [(y • X), X1]] +X'X i[(y • X), [X' .X*]] + X ' X ' [ X \ [X\ (y • X)]] +X^X i [X\ [X\ (y • X)}} + X ^ [ x \ [X^ (y • X)]] +X*Xi[X>, [x\ (y • X)]] + 2X ' X i [ X i , [X\ (y • X)]] + X ' X * [ X ' , [X\ (y • X)}} + X ' X ' I X * . \X\ (y • X)]] +X>Xi[(y • X), [X' .X*]] + X3Xl[(y • X) , [X^X*]] +2X'(y • X ) [ X \ [ X ' , X% + X'(y • X)[X*, [X\ X*]] +2&{y • X)[X\ X*]] + P(y • X)[X\ [X>, X 4 ]] +X'{y • X)[X\ [Xl,X<]] + X ' ( y • X)[X*, [x'.X']] +2X*(y • X)[X\ [P, X'}} + 2X'(.y • X)[P, [X\ X% +X'{y • X)[X\ [XfX1]} + X*{y • X ) [ X ' , [X\ X*]] + 2 X % • X)[(y X), [X>, (y • X)]] + X'(y • X)[X>, [(y X),(y X)}} +4Xi(y • X)[(y • X), [X'', (y • X)]] + 2X\y • X)[(y • X), [X>, (y • X)]] Appendix B. Appendix to chapter 3 108 +X*{y • X)[(y X), [Xf (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 X)}} +P{y • X)[(y • X), [X\ (y • X)]] + X^(y • X)[(y • X), [Xf (y X)}} +X\yX)[(yX),[Xf(yX)]} +XiXi[Xf [(y • X), X'}} + X'X^Xf [X\ (y • X)}} +{l/2)XiP[X\ [Xf (y • X)}} + ( 1 / 2 ) X ^ [ X \ [Xf (y • X)}} +(l/2)XiXi[(y • X), [XfX*]} + (lfflXWlX*, [(y • X),X>}} +XiXi[Xf [X\ (y • X)}} + X'X^Xf [X\ {y • X)}] +(l/2)Xi&[Xf[(y X)^1}} +(y • X)&[X\ [XfX*]] + (y • Xtf^XflX^X*]] +(y • X)Xi[X\ [XfX% + (y • X)X*[Xf [X\X% + (y • X)P[X\ [XfX'}) + (y • X)X*[(y • X), [Xf (y • X)}} +(y • X)Xi[Xf [(y • X), (y • X)}} + 2(y • X)X*[(y • X), [Xf (y • X)}} +(y • X)P[Xf [(y X), (y • X)}) + (y • X)X*[(y X), [Xf (y • X)}} +(y • X)X'[{y • X), [Xf (y • X)}} + (y • X)X*[{y • X), [Xf (y • X)]] - ( 1 / 2 ) X ^ [ X * . [Xf (y • X)]} + PX^Xf [X\ (y • X)]] - ( l / 2 ) X ^ i [ ( y • X), [XfX'}} +XJXi[Xf [(y • +XJXi[(y • X), [XfX*]} - {l^PX^Xf [(y • X),X*]] +Xi(y • X)[X\ [XfX^+X^y • X)[X\ [XfX% +x>(y • x)[xf [JCX]] + xj(y • X)[X\ [X>,X*]] +X>(y • X)[Xf [X'X]] + 2X\y • X)[(y X), [Xf (y • X)]] +2X3(y • X)[Xf [(y Xl (y • X))] + AP(y • X)[(y • X), [Xf (y • X)]] +4X'{y • X)[Xf [(y • X), (y • X)]] + 2X>(y • X)[(y • X), [Xf (y • X)}] +2X'(y • X)[(y • X), [Xf (y X)}} + 2X^(y • X)[(y • X), [Xf (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 XlX^XfX'} +P[X\ (y • X)][XfX{] + X^X^X^Xf (y, X)] +P[X\ (y • X^XfX'} + 2P[X\ (y • X)][Xf X'} Appendix B. Appendix to chapter 3 109 +2X*[Xi,Xi][Xl, (y • X)] + 2p[(y • X), P][P, X1} +P[P, (y • X)][P, X*] + P[P, P][P, (y • X)] +P[x\ (y • X)][P,P] + P[x\ X % X \ (y • X)} +P[(y • X),P}[P, X1] + P[(y • X),P][P, P] +2p[(y X), (y • X))[P, (y • X)\ + 2p[(y X), (y • X)][P, (y • X)] +P[(y X), (y • X)][X\ (y • X)) + P[(y • X), (y X)][X3, (y • X)] +P[(yX),(yX)}[X>,(yX)} +2p[(y • X^PftX^X*] + 2p[(y • X\Xi\[X\Xi\ +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,Xi][P, (y • X)] +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.Vi(y) = -AVi + AV< + ..., (B.19) where AVI = ^ S y m (P[P, [P, (y • X)}} + P[X\ [P, (y • X)}] +P[P, [X>, (y • X)}} + P[(y • X), [P, P}] +P[Xi, [(y • X),P}} + P[(y • X), [X*,**]]) AV£ = ^ S y m [Pxj[P, [Xk, P]] + PP[Xk, [X\ P]} (B.20) +PXJ[P, [P, P}} + Pxj[P, [Xk, P}} +Pxj[Xk, [P, P]} + PP[P, [P,P]] +PP[P, [Xk, P]} + PP[P, [Xk,P}} +PP[Xk, [P,Xj]] + PP[P, [P, Xj}} Appendix B. Appendix to chapter 3 110 +XkXl[Xk, [XfX*]] + XkX%y X), [Xk, (y • X)]] +XkXi[Xk, [(y X), (y • X)]] + 2XkX%y • X), [Xk, (y • X)]] +XkXi[Xk, [(y • X), (y • X)]] + XkX%y -X), [Xk, (y • X)]] +XkXi[(y • X), [Xk, (y • X)]] + XkX%y • X), [Xk, (y • X)]] +Xk(y • X)[X\ [Xk, (y • X)]] + Xk(y • X)[X\ [Xk, (y • X)}} +Xk(y • X)[(y X), [ X f c , ^ ] ] + Xk{y • X)[Xk, [ ( y X), +2Xk(y • X)[(y • X), [Xk,Xi}}+Xk(y • X)[Xk, [(y • X),X*]] +Xk(y • X)[(y • X), [Xk, X*}} + Xk(y • X)[x\ [Xk, (y • X)]] +Xk(y • X)[Xk, [X\ (y • X)}} + 2Xk(y • X)[X\ [Xk, (y • X)]] +Xk(y • X)[Xk, [X\ (y • X)]] + Xk(y X)[X\ [Xk, (y • X)]] +Xk(y • X)[(y • X), [Xk, X'}} + Xk(y • X)[(y • X), [Xk, X*}} +(l/2)XkXj[Xk, [X\Xj}} + (l/2)XkXj[X\ [Xk,Xj]} + (l/2)XkXj[Xk,[Xj,Xi}} +(y • X)P[X\ [Xf (y • X)}} + (y • X)X'[X'", [X\ (y • X)]] +(y • X)X'"[X*, [Xf (y • X)]] + (y • X)X>[(y • X), [XfX1]} + (y • X)P[Xf [(y • X^X*]] + (y • X)X'[(y • X), [XfX1}} +Xk(y • X)[X\ [Xk, (y • X)]] + Xk(y • X)[Xk, [X\ (y • X)]] +Xk(y • X)[X\ [Xk, (y • X)]] + Xk{y • X)[Xk, [X\ (y • X)}) +Xk(y • X)[(y • X), [Xk, X*}} + Xk(y • X)[Xk, [(y • X), X*}} +Xk(y • X)[(y • X), [Xk, X*}} + Xk(y • X)[Xk, [(y • X), X*}} +XkXi[(y • X), [Xk, (y • X)]] + X*X<[Xfc, [(y • X), (y • X)]] +XkX>[(yX)1[Xk,(yX)}] +Xk[Xi,Xj][Xk, Xj] + Xk[Xk, Xj][X\ Xj] +Xk[X\Xj][Xk, Xj] + Xk[Xk, Xj] [X\Xj] +Xk [Xk, X1] [Xj, Xj] + Xk [Xj, X{] [Xk, Xj] +Xk [Xk, X*] [Xj ,Xj] + Xk [Xj, X{] [Xk, Xj] -Xk[X\Xj][Xk,Xj] +Xk[X\ (y • X)}[Xk, (y • X)} - Xk[Xk, (y • X)][X\ (y • X)] +Xk[X\ (y • X)}[Xk, (y • X)} + Xk[x\ (y • X ) ] [ X f c , (y • X)] +Xk[(y • X), X*][Xfc, (y • X)] + Xk[(y X), X%Xk, (y • X)] +Xk[(y • X), (y • X)}[Xk,X*] + Xk[(y • X), (y • X^X",**) Appendix B. Appendix to chapter 3 111 +Xk[(y ^ ( f f j j p ^ l ) . From equation (3.38), we can see that v°(y)\yi=0 = 0(x2) ^ • • • ^ y ^ l ^ o = 0(xn). (B.21) Using this result and equation (3.37), we also have z / ( y ) | y i = 0 = O(x) V(s / ) ly<=o = Sij + 0(x2) ^ . . . ^ ( y ) ^ = 0(xn+1). (B.22) Note that any yl appearing in the expansion of (V£m(y) + <5fV) must be accompanied by an X1 in the form of (y-X). As a result, any yl appearing in AV M 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 V°(y)\yi=0 = 0(X2) dh...dinV0(y)\vi=0 = 0(Xn) V*(y)\^ = O(X) 3 ; V % ) | y M ) = - ^ + 0(X2) ^ • • • ^ ( y J l ^ o = 0(Xn+1). (B.23) Also we have the following formulae, for all integers m > 0. (- -d2)mV° yi=0 = 0(X2) d0(- -d2)mV° yi=0 = o(x2) 8j(- -d2)mV° yi=Q = -d2m+1Xj/dt2m+l + 0(XZ) (--d2)mV{ yi=0 = d^P/dt2™ +