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Holography in string theory DeBoer, Philip Albert 2005

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Holography in String Theory by Philip Albert DeBoer B.Sc, University of Prince Edward Island, 1999 M.Sc, University of British Columbia, 2001 A THESIS SUBMITTED IN PARTIAL F U L F I L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF DOCTOR OF PHILOSOPHY in T H E FACULTY OF G R A D U A T E STUDIES in PHYSICS T H E UNIVERSITY OF BRITISH COLUMBIA September 2005 © Philip Albert DeBoer, 2005 Abstract i i Abstract This thesis presents three aspects of holography, or holographic duality, in string theory. The goal of holography is to describe gravitational theories by field theories in fewer dimensions. Although recently there have been significant strides toward understand-ing generic holographic dualities between particle theory and gravity, it is not clear how string theory might implement this. In Chapter 2 the role of compact directions in holography is explored. We discover that a general-ization of Bousso's prescription is necessary. In this chapter we also discover that holography provides no clear solution to understanding closed timelike curves, unlike earlier speculation. Using holography as a tool, we are able in Chapter 3 to compute the ther-mal two-point function of the graviton in the dual to a Little String Theory. This is a strongly coupled non-gravitational theory. It exists as a limit of a certain weakly coupled string theory. We find a typical thermal spectrum with no obvious interpretation in terms of particles and bound states. We find evidence supporting a conjecture that this theory is thermodynamically unstable. Finally, in Chapter 4, we investigate the inhomogeneous decay of an un-stable D-brane. We study the average number and energy of closed strings emitted during the decay, and confirm that the space-filling D-brane decays Abstract iii completely into lower-dimensional D-branes. One of the long-term goals of this research is to understand how closed-string fluctuations appear in the open-string sector of string theory. This is directly related to the appear-ance of holography in string theory since gravity appears in the closed-string sector, while the end-points of open strings can be described by field theory. Contents iv Contents Abstract ii Contents iv List of Figures vii Preface viii Acknowledgements ix 1 Introduction 1 1.1 Holographic Duality 1 1.2 String Theory 4 1.2.1 Bosonic action 9 1.3 Outline 10 2 Compactified Plane Waves 14 2.1 Introduction 14 2.2 Entropy Bounds and Holography 17 2.2.1 History 17 2.2.2 Geodesic Expansion 19 2.2.3 Covariant Entropy Bound 20 2.2.4 Example: Collapsing Star 22 Contents v 2.2.5 Holography 23 2.3 Spacetimes 26 2.3.1 Godel-like Universes 26 2.3.2 Compactified Plane Wave 28 2.3.3 T-duality and Dimensional Reduction 30 2.3.4 Review of Length Scales 32 2.4 Geodesies 33 2.4.1 Introduction 33 2.4.2 Null Geodesies 34 2.5 Holographic Screens 38 2.6 Conclusions 42 3 Little String Theory 44 3.1 Introduction 44 3.1.1 Construction 45 3.1.2 Duality 46 3.1.3 Zero Temperature 47 3.1.4 Finite Temperature 48 3.2 Holographic Construction 52 3.3 Vertex Operators 55 3.3.1 Review of SL(2, R) Results 56 3.3.2 Restriction to SL(2,R)/U(1) 60 3.3.3 Worldsheet Supersymmetry 62 3.3.4 Mass-Shell Conditions 64 3.4 Correlation Functions 65 3.4.1 Euclidean Space 67 Contents vi 3.4.2 Minkowski Space 68 4 Inhomogeneous D-brane Decay 71 4.1 Introduction 71 4.2 Dynamical D-branes 73 4.3 Unstable D-branes 76 4.4 Sen's Conjectures 77 4.5 Bosonic Homogeneous Decay 81 4.6 Radiation 83 4.7 Bosonic Inhomogeneous Decay 87 4.7.1 Introduction 87' 4.7.2 Out-Going Closed-String Radiation 88 4.7.3 Emission Amplitude 90 4.8 Analysis 92 Bibliography 95 A Christoffel Symbols of the Compactified Plane Wave . . . . 103 List of Figures vii List of Figures 1.1 Tree-level open-string vertex: conformal transformation to disk. 5 4.1 Closed-string radiation from D-brane 74 4.2 Tachyon potential in bosonic string theory 79 Preface viii Preface Parts of this thesis describe published research. Chapter (3) was published as [1]. Chapter (2) is based on [2]. Acknowledgements ix Acknowledgements I am grateful for my supervisor Prof. Moshe Rozali for his guidance over the last number of years, and especially his willingness to answer my many little questions. I also thank the students and postdocs which I have had the pleasure of meeting here, and especially Dominic Brecher for his role in our collaboration. I thank Roze for her love and unwavering support. Know that the L O R D is God. It is he who made us, and we are his; we are his people, the sheep of his pasture. Psalm 100:3 (NIV) Chapter 1. Introduction 1 Chapter 1 Introduction 1.1 Holographic Duality Concretely denning holographic duality is possibly the most important un-solved problem in mathematical high-energy physics. Holographic duality is a duality between quantum gravity and a lower-dimensional non-gravitational theory. The gravitational theory will be referred to as the bulk theory, while the lower-dimensional theory will be called the boundary theory. The physical motivation for proposing a holographic principle is an en-tropy bound [3] which states that in the semiclassical limit the number of degrees of freedom in a quantum gravitational system must scale like an area. This is surprising, as local quantum field theory would suggest that the number of degrees of freedom is extensive, growing linearly with volume1. Mathematically, holography will build non-perturbative dualities and as such it has the ability to relate strongly coupled degrees of freedom to weakly cou-pled ones. In this sense holography could become a useful tool for studying strongly coupled theories. One immediate application of holography could be in describing the strong force of our world. Although there is a gauge theory description of the strong force, known as the SU{3) quark model (or QCD), it is exceedingly difficult 1 For more detail see Chapter 2. Chapter 1. Introduction 2 to make predictions based on this model. The gauge coupling is large at accessible energies, suggesting that it could have a useful weakly coupled gravitational description. Recent progress has been made in this direction [4]. In this paper, Nastase suggested that the "Color Glass Condensate" — a deconfined yet strongly coupled state found at the core of certain high-energy collisions — could be explained as a dual black hole. This picture is not yet quantitative, however. Perhaps the ultimate goal of holography is to describe our world as a pure gauge theory, with some gauge group G. The couplings of the electroweak sector 5?7(2) x U(l) are very small, so that the quantum field theory (QFT) description works very well. The quark coupling is somewhat larger, so that the gauge theory description has only achieved partial success. The gauge coupling for the group G should be so large that in our everyday experience we would find it most convenient to describe this sector of the theory by weakly coupled gravity, but it could nonetheless have a holographic dual. However this picture sets a rather long-term goal that is far from being realized. In fact holography has been realized concretely in only a few cases — most notably, in the AdS/CFT (Anti de-Sitter/conformal field theory) cor-respondence between Type IIB string theory on AdS 5 x § 5 and Af = 4 super Yang-Mills on § 3 x R — and even in these examples the correspondence has not been proven, although the relationships have been highly constrained. The picture that has emerged from these examples is that a radial co-ordinate in the gravitational description becomes the R G scale in the non-gravitational field-theoretic description. There is an isometry along the ra-dial direction, and the corresponding gauge theory has a constant coupling Chapter 1. Introduction 3 — so on both sides the theory is independent of this scale. More generally translation along the radial direction will not be an isometry, and the gauge theory will have a running coupling. Additional compact dimensions in the gravitational picture form an internal group manifold for the gauge theory, as already mentioned. Furthermore, bulk normalizable modes correspond to states in the Hilbert space of the conformal field theory (CFT), while bulk non-normalizable modes correspond to local probes, or vertex operators. Since holography maps a field theory to a quantum-mechanical gravita-tional theory, it must be developed within a framework that can describe both types of system. The most natural formalism for describing holographic dual-ity is string theory. String theory naturally accommodates quantum gravity, and gauge theories also arise naturally in string theory. Holographic duality might also be related to the open-closed string du-ality. At low energies and weak coupling, open strings decouple from the metric and are best described by a lower-dimensional gauge theory. So the open-string sector can be used to describe a type of boundary theory. On the other hand, the closed-string sector always couples directly to the metric and is naturally interpreted as a theory of quantum gravity. If the open string can be realized as a closed-string fluctuation one can hope to understand the gauge theory in terms of the closed-string gravitational description and thus develop the holographic relationship between them. Recent work on D-brane decay has begun to address this issue [5, 6, 7, 8]. Thus it seems useful to develop string theory as a tool for understanding holography. The next section presents a brief overview of the fundamental aspects of string theory, with an emphasis on its differences from QFT [9].. Chapter 1. Introduction 4 For a more detailed introduction to string theory, see [10, 11]. This chapter concludes with an introduction to the three research projects presented in this thesis. 1.2 String Theory Most of the recent work in string theory has been motivated by the desire to extend the enormously successful Standard Model which describes our world. Although the Standard Model has been successful where it has been tested, it is not clear how the gravitational force fits into this picture. The most promising resolution to this puzzle is string theory. There are a number of hurdles to overcome in developing string theory. It needs innumerable free parameters, many of them discrete, to fix the vac-uum, although for a fixed vacuum the Lagrangian has no free parameters. A formulation of string theory that makes no reference to classical background sources is not known. Related to this is the lack of a quantum-mechanical multi-string, or string-field, description. A string generalizes the concept of a particle, and traces out a world-sheet rather than a world-line as it moves through spacetime. This smears interactions and removes the ultraviolet divergences of Q F T . In string the-ory, classical gravitational backgrounds arise as macroscopic superpositions of gravitons. There are two important classes of string worldsheets: they can be open or closed. Open strings are described by worldsheets that look like ribbons. Closed strings are described by cylindrical worldsheets, with time running Chapter 1. Introduction 5 along the length of the cylinder. There are two useful expansion parameters in string theory — the string length y/ot' and the string coupling g. The string coupling g governs the probability for two strings to interact. The usefulness of a perturbative ex-pansion in g is known from QFT, where it plays the same role of governing the expansion in Feynman diagrams. The a' expansion is an expansion in a relevant energy scale, for example the radius of curvature of the gravitational background. Open-string diagrams are formed by gluing ribbons together. For exam-ple, the tree-level three-string vertex is a fat Y (Fig. 1.1). One important feature of open-string diagrams is that all n-point diagrams can be cut into products of three-point vertices. Thus open-string field theory has only one interaction term in its Lagrangian. Figure 1.1: Tree-level open-string vertex: conformal transformation to disk. One might expect this to work for closed strings as well. However, to describe all possible closed two-dimensional surfaces, infinitely many inter-actions are needed, rather than just one. Holography gives an appealing explanation of this: the closed-string theory dramatically over-counts the number of degrees of freedom, and it is the open-string theory which correctly counts the degrees of freedom. There could then be a symmetry relating the Chapter 1. Introduction 6 closed-string interactions, but it is not understood. Both open- and closed-string diagrams can only be calculated by using the conformal symmetry to shrink interactions to point-like insertions called vertex operators. For example, the tree-level three-point open-string diagram can be drawn as three ribbons streaming off a disk. Using conformal sym-metry this can be reduced to three point-like insertions on the boundary of the disk (Fig. 1.1). By converting them to local insertions the calculations become tractable. But this conformal gauge choice leaves behind only the on-shell theory. Rather than use a spacetime action S[O(X),T(X)] = f dAx£\a(x),T(X)], which describes the location of the string in terms of spacetime coordinates, we use a worldsheet action <S[Xp(cr, r)] = / dadr2,[X^(o, r)], which describes the spacetime dimensions in terms of the worldsheet coordinates o and T . The index p runs over the spacetime coordinates. From the worldsheet perspective, the functions X^ are fields living on a two-dimensional manifold and taking values on an internal manifold called target space, or simply spacetime2. Under worldsheet diffeomorphisms, the X^ are scalars, so from the worldsheet perspective the string action S[X^(o, r)] is a theory of scalar bosons X having an internal Poincare symmetry. In this sense string theory is a two-dimensional conformal field theory (CFT) living on the string worldsheet. Conformal transformations are map-pings that preserve angles, and in two dimensions there are infinitely many generators of these transformations. This enormous symmetry severely con-2 A point particle can be thought of as a vector field living on a world-line, although this is seldom done. Chapter 1. Introduction 7 strains the correlation functions in the theory. Sometimes it even leaves the theory exactly solvable. One of the most basic conformal transformations is the scale transforma-tion. An important part of scale invariance is that there can be no dimen-sional parameters in a scale invariant theory. Since there are no scales in the worldsheet CFT, there is no notion of asymptotes. Thus on the worldsheet there are no S-matrix elements. However there are still local operators and their correlation functions. From the worldsheet perspective this is indeed a second-quantized theory like QFT, but from the target-space perspective it is a first-quantized theory like ordinary single-particle quantum mechanics. Quantization of the classical symmetry of the 2-D worldsheet CFT gives rise to a local anomaly, often referred to as the Weyl anomaly. This renders the theory inconsistent except where the anomaly vanishes. For bosonic string theory the anomaly vanishes in 26 dimensions. One can put fermionic terms on the worldsheet to get superstring theory in the NSR formalism, which has worldsheet supersymmetry. It is anomaly free in 10 dimensions. There are two ways to put fermions on the string — they can have periodic or anti-periodic boundary conditions around noncon-tractible closed paths3. The fermionic states with periodic boundary conditions are known as Ramond (R) states, while the anti-periodic fermions are Neveu-Schwartz (NS) states. The closed strings have decoupled left- and right-movers, so 3 For open strings these.appear in loops; for the closed string they exist already at tree level. Chapter 1. Introduction 8 there are two choices to make. The four closed-string fermions are NS-NS, NS-R, R-NS, and R-R. The NS-NS and R-R states couple to target-space bosons, while the other two are fermions. The mode expansion of the Ramond states involves branch cuts, which are difficult to deal with, so most of the discussion focuses on the NS-NS states. The most important NS-NS fields are inherited from the bosonic theory; they are the dilaton, metric, two-form field, and tachyon. The new discrete quantum number in superstring theory is the worldsheet fermion number. Superstring theory has two immediate advantages over its bosonic counterpart. One is that there are now target-space fermions. The second is that, since the tachyon and all other states have definite fermion number, one can project out the states of either fermion number and still have a consistent interacting theory for the remaining states. Such a projection is known as a GSO projection after [12]. It is useful in particular to project out the sector containing the open-string and closed-string tachyons. The GSO projection can be chiral or non-chiral on the worldsheet. The non-chiral projection in the oriented closed-string theory gives the Type IIA string theory. It is invariant under spacetime parity since swapping the left-and right-chirality sectors is a symmetry in a non-chiral theory. Thus there can only be even R-R field strengths, and odd R-R potentials. The chiral GSO projection gives Type IIB string theory, which has states of only one chirality and -even R-R potentials. Chapter 1. Introduction 9 1.2.1 Bosonic action The bosonic worldsheet action is [10] 5[X"(a,T)] = j dadrVh [(habGlu/{Xfl)+ieabBtlv{Xfl)) daX^X" + a'i?$(X")]. (1.1) The indices a, b refer to worldsheet coordinates o, r; and p, v are target-space indices. The worldsheet metric is hab, and R is the worldsheet curvature built from this metric. This action also depends on several background target-space fields that must be specified by hand, but are subject to certain consistency constraints. These fields are the metric G^, the anti-symmetric potential B^, and the dilaton $. One should think of these background fields as macroscopic super-positions of strings or as expectation values of the string fields. As mentioned, it is not known how to describe strings without reference to these background fields. The requirement of worldsheet conformal invariance leads to equations of motion for the background fields by requiring that their worldsheet R-functions vanish. The /^-functions are not known exactly, but can be deter-mined in an expansion in a'. These equations of motion can also be derived from a spacetime action, which in the so-called string frame is 2rt0 \ [dDx^Ge~™ \ - 2 { ° , 2 6 ) + R - ^H^H^ KQ J I 3a' 12 + Ad^d^ + 0(a')). (1.2) The Einstein-frame picture is related to this one by absorbing the overall factor e~2* into the metric. The total number of spacetime dimenions is D, Chapter 1. Introduction 10 and K0 is related to the Planck mass MP = y/hc/G by K 0 e ( * > = v ^ / M p . The spacetime Ricci scalar is denoted R . The three-form field strength con-structed from B is UWiiV = d^B^ + d^B^ — dvB^. Since strings are extended in two spacetime dimensions they can interact with such an anti-symmetric potential. Strings which wind a compact direc-tion have a conserved charge with respect to the components BZM, where z is the compact direction. 1.3 Outline This thesis considers three projects related to holographic duality in string theory. In the first, the nature of the boundary theory of a specific spacetime is examined with a view to understanding certain bulk properties. The second project uses holographic duality as a tool for calculating boundary correlation functions. Finally, some work is done to understand the instability of the open-string vacuum, with the hope of further developing open-closed duality. Most of the work that has been done to establish holography as a gen-eral principle has been done in the context of particle theories. However the known examples of holography are intrinsically string theoretic. To under-stand how holography might fit into the broader framework of string theory, there are some basic questions that must be answered. Among these is the effect of T-duality4. Since string theory is manifestly invariant under T-duality, it is appealing to build this explicitly into holog-raphy. In Chapter 2 [2] we conduct a preliminary study of this idea, in the 4 This is discussed in more detail in Section 2.3.3. Chapter 1. Introduction 11 context of the compactified plane wave. One motivation for studying this in the context of the compactified plane wave was to understand the closed timelike curves that are present in the compactified plane wave. It is difficult to study the effect of the closed timelike curves from the gravitational perspective. It is not clear how to de-fine QFT consistently on this background, but on the other hand the usual problems associated with closed timelike curves are not present here. A completely new perspective of this situation, provided by a holographic du-ality, could make these questions more clear [13]. We ask some questions about what a holographic dual might look like, and how it could change our understanding of the spacetime picture. For example, it could be that the consistent QFT is dual to a region of space too small to contain closed time-like curves. Our investigation suggests that this approach is not likely to be useful in our example. We found that the presence of compact directions could change the naive analysis. The fact that the covariant entropy bound can give different results in different limits of the geometry is an additional complication. In Chapter 3, we use holographic duality to perform calculations in a strongly coupled boundary theory. The holograhic dual of this example, called LST 5 , is already known. In this example, the duality relates a strongly coupled theory in six dimensions to a weakly coupled ten-dimensional string theory. The duality is used to conduct an indirect study of the strongly coupled system. 5 LST is short for "Little String Theory", but LST is not a familiar critical string theory and bears little resemblance to them. Chapter 1. Introduction 12 But the duality is understood only for finite-temperature LST. In fact, it is understood only for a particular temperature, the Hagedorn temperature. LST has a Hagedorn the density of states, which grows as an exponential with energy rather than as a power. Since the number of available states grows so rapidly as the energy is increased, the description of the system breaks down at the Hagedorn temperature. The natural solution [14] is that the Hagedorn temperature is the maxi-mum temperature of LST, and that it undergoes a phase transition. During a phase transition the string Lagrangian changes, since the background fields are not constant. The end result is a "new" theory — LST refers only to a particular vacuum. If this is true, then LST is simply unstable at the Hagedorn temperature. In this case tachyonic modes should appear. One can look for modes of LST which at tree level are massless. Corrections in the string coupling can make them unstable. These states of LST appear as poles in the correlation functions of the non-normalizable modes of the weakly coupled description. We indeed find evidence of these instabilities, although they are not simple poles and do not seem to have an interpretation in terms of ordinary particles. Finally, as discussed earlier, two of the remaining problems in the for-mulation of string theory are the role of the open-string tachyon, and the relationship between open- and closed-string quanta. The existence of the tachyon implies that the open-string vacuum is unstable. If there are open strings there will always be the possibility of a tachy-onic mode, so the stable vacuum shouldn't have any elementary open-string excitations — thus one candidate is the closed-string vacuum. Sen began Chapter 1. Introduction 13 to test this picture some time ago [5, 6]. He found promising results at the linearized level by considering a very symmetric situation. In Chapter 4 we describe a slightly more general process — a brane that decays at a rate that depends on one spatial coordinate. This is known as the inhomogeneous tachyon decay. Specifically, the inhomogeneity considered here is a sinusoidal oscillation in one spatial direction. We find that all of the mass of a space-filling D-brane is radiated away. In summary, this thesis presents three projects that contribute to our understanding of holographic duality by exploring both precisely what holo-graphic duality is (Chapters 2 and 4) and what it can do for us (Chapters 2 and 3). Chapter 2. Compactified Plane Waves 14 Chapter 2 Compactified Plane Waves 2.1 Introduction Einstein's equations have many interesting solutions, including some with rather bizarre properties, such as closed timelike curves (CTC's). Massive physical particles follow timelike trajectories, so CTC's allow one to travel back to, say, one's birth. Although CTC's have never been observed in our universe, this does not mean that they do not exist. Geodesies do not cost any energy to follow, so they dominate the quantum field theory path integral. Thus geodesic CTC's have infinite self-interaction already at tree-level, and they lead to divergences in field theories with geodesies CTC's. Non-geodesic CTC's, on the other hand, require an acceleration to be followed. So the contribution to the path integral would be exponentially damped by the affine length required to traverse the CTC. But Hawking proposed that the laws of physics prevent the formation of CTC's [15]; he called this the chronology protection conjecture. He supported the conjecture by showing that the formation of CTC's would cause a large backreaction. However the spacetimes considered here have CTC's from the outset and therefore are not affected by this argument. One example of a spacetime having non-geodesic CTC's is the Godel Chapter 2. Compactified Plane Waves 15 solution [16]. For matter it has only rotating pressureless dust. A related example of particular interest here is a supersymmetric analogue known as the Godel-like universe, or G L U [17, 18]. It has gauge fields rather than dust, but its CTC's are similar to the ones found in the Godel spacetime. Although there is no immediate reason to disregard these types of solu-tions, it is possible that quantum fluctuations will destabilize them. Hence it is interesting to study them in the context of string theory, which is a theory of quantum gravity. If string theory is defined as a path integral over all physical spacetimes, it seems like these spacetimes should be included as contributions to the path integral. This observation leads to many interesting questions, since it is not immediately clear what the contribution of such spacetimes will be. Perhaps their contribution will be insignificant in some way, or perhaps they must be excluded by some new physical constraint. The purpose of the research described here is to be one step towards understanding these larger questions. The authors of [13] suggested that a "holographic complementarity prin-ciple" might exist. The idea of holography is that gravitational spacetimes may be related to non-gravitational theories in fewer dimensions; the most important recent developments have been put forward by Bousso [19, 20], based on his conjected covariant entropy bound [3]. Since the CTC's in the G L U have a minimum size, the authors proposed that a holographic dual theory be constructed to describe a bounded region not containing complete CTC's. It might be more clear in such a holographic theory that the CTC's in the G L U cause no serious problems. To test this proposal further, in this research [2] we generalize it to the Chapter 2. Compactified Plane Waves 16 compactified plane wave (CPW). The C P W is related to the G L U by T-duality [18, 21, 22]. There are non-geodesic CTC's in the CPW, but they wrap a compact direction, unlike in the G L U . They do, however, have a minimum size associated with them. In this research we address how the G L U story changes under this T-duality. It turns out that to reproduce the G L U result we need to consider a delocalized observer. Our research establishes concretely the need to under-stand Bousso's bound under T-duality. If string theory is to have an entropy bound it must be a generalization of Bousso's bound that is invariant under T-duality. Bak and Yee [23] have now done this for the simplest case. We also find that there is no clear way for holographic protection to operate in the CPW. We conclude that the ideas of [13] do not have broad applicability. In Section 2.2 the covariant entropy bound is introduced and its con-nection with holography discussed. Section 2.3 introduces the spacetimes of interest here, and finally I discuss our research [2]. Lastly I draw some conclusions and discuss ideas for future research in Section 2.6. After sketching briefly in Section 2.2.1 the developements that led up to Bousso's recent work on holography, and introducing the necessary mathe-matics in Section 2.2.2, the construction of the covariant entropy bound is discussed (Section 2.2.3). An example of its application is given in Section 2.2.4, which describes the behaviour of entropy during the collapse of a star. The principle of holography as motivated by the covariant entropy bound is developed in Section 2.2.5. The spacetimes of interest here are introduced in Section 2.3.1 and 2.3.2. Chapter 2. Compactified Plane Waves 17 ^ These are related by T-duality and dimensional reduction, as explained in Section 2.3.3. In Section 2.3.4 the hierarchy of relevant length scales is re-viewed. Section 2.4 begins with a more detailed outline of the research performed here. In Section 2.4.2 the null geodesies of the C P W are found. In Section 2.5 the holographic screens are constructed from the null geodesies. Finally, in Section 2.6 the results are analyzed and some conclusions are drawn. Several suggestions for further research are also included. 2.2 Entropy Bounds and Holography 2.2.1 History Among the first to recognize how thermodynamics relates to black holes were Bekenstein and Hawking (see [24, 25] and references therein), who postulated that the entropy of a black hole is proportional to the area of its event horizon. The Bekenstein-Hawking entropy SBH of a black hole is A k c 3 (2.1) 4 Gh' where A is the area of the black hole horizon, k is the Boltzmann factor, and c3/Gh is the inverse Planck area 1/1%. This equality is satisfied by Schwarzschild black holes in asymptotically flat spacetimes. To generalize this, one might propose that the entropy of any matter system be bounded by the area circumscribing it, in Planck units [20]. This would make black holes the most entropic states, since they would saturate Chapter 2. Compactified Plane Waves 18 such a bound. However, as a spacelike entropy bound this is clearly valid only for weakly gravitating systems. For consider an observer falling in to a collapsing star. The radius of the star becomes arbitrarily small, while the entropy of the star cannot decrease during the collapse. Thus at some point a collapsing system will violate this entropy bound. But it is difficult to find a generalization that is both broadly applica-ble yet stringent enough to provide useful information. For example, even dropping the condition of asymptotic flatness cannot easily be done; without this condition one can consider closed universes, where it is not clear which volume is bounded by a given closed surface. With this motivation, Bousso has proposed an entropy bound that applies in time dependent situations, where the usual entropy bounds do not apply. This is known as the covariant entropy bound (CEB). It is a refinement of the proposal of Fischler and Susskind [26] to use light rays to define the region in which entropy be measured. Roughly, congruences of light rays emanating from some spatial surface B are used to define null volumes called lightsheets, L(B), which are bounded by the surface B. It is the entropy within the lightsheet L(B) that is bounded by the area of B, rather than entropy inside the spatial volume bounded by B. The precise prescription for constructing lightsheets L(B) that are "in-side" B, which is the main substance of the CEB, is outlined in the next two sections. Chapter 2. Compactified Plane Waves 19 2.2.2 Geodesic Expansion Given any null vector k in some region of spacetime, a null congruence, or family of light rays, can be constructed as the set of null geodesies tangent to fc. In this context the tangent vector k is also known as the generator of the congruence. For example, one can define k at each point on some two-dimensional spacelike surface B. If k is further chosen to be orthogonal to B, the resulting null congruence is also orthogonal to B. One can then define the surface B(\) as the set of points through which the congruence passes at fixed affine parameter A. The surface B(X) will have some area A(X). This area is the cross-sectional area of the null congruence. The geodesic expansion 9 is defined as the logarithmic derivative of this area with respect to the affine parameter along the geodesies; » = i £ - V . * . (2.2, (jf^2 + + ^T^k"kV) + u^v. (2.3) The geodesic expansion obeys the Raychaudhuri equation d9__ ( 92 d\ = The t w i s t - a l w a y s vanishes for surface-orthogonal light rays, which are the only kind considered here. The exact definition of the shear, denoted by o^, is unimportant here, except to note that its contribution tends to decrease 9. The component of the energy-momentum tensor TM„ along the congruence is the force imparted by the change in geometry due to matter, according to Einstein's equation. It is important that the null energy condition be satisfied, so that this term is always nonnegative; essentially this condition requires that matter always be gravitationally attractive. Chapter 2. Compactified Plane Waves 20 One result of this equation is known as the focusing theorem, which states that when 6 < 0 it remains negative and reaches 9 = — oo at some finite affine parameter A 1 . Points where 9 = ±oo are known as caustics. Away from singularities, dA/dX will be finite. In this case the caustic is the point where neighboring geodesies in the congruence converge, sending A (A) to zero. 2.2.3 Covariant Entropy Bound The conjectured covariant entropy bound (CEB) can be stated as follows [20]: The entropy on any lightsheet L(B) will not exceed the area of s l l { B ) ] s m ± * _ . ( , 4 ) This bound measures entropy not in spatial volumes but rather in null volumes which he called lightsheets. The essence of the prescription given below for constructing lightsheets from a surface B is that it determines precisely which entropy is bounded by the area of B — only the entropy inside the lightsheet itself is to be counted. Given a spatial surface B, there are always four choices for the light rays leaving it; the light rays may go towards or away from the surface, and they may go forwards or backwards in time. The covariant definition of "inside" is defined by choosing the lightsheets L as congruences with non-positive lrThe light rays may hit a singularity before focusing if the space is geodesically incom-plete. Chapter 2. Compactified Plane Waves 21 geodesic expansion. This leaves at least two possibilities for L, and up to four. The CEB applies separately to each admissable lightsheet. To form a lightsheet, congruences must be terminated by hand at caus-tics, since their expansion is positive when they emerge on the other side. Lightsheets can also be terminated by boundaries since geodesies never travel through them. Singularities which are reached in finite affine parameter also terminate lightsheets. It is evident now that this choice has been made, that one is not restricted to open universes, or even to patches of space which enclose a volume — one need not consider the entire black hole horizon, for example, but one could consider just one hemisphere. This prescription for choosing L also eliminates the need for asymptotic conditions, since it depends only on the local behaviour of light rays. The presence of the energy-momentum tensor in Raychaudhuri's equation (2.3) shows that the light rays converge more rapidly in the presence of matter. Therefore if entropy is present in some spatial volume, this entropy will be accompanied by energy that will make the light rays passing through it converge more rapidly. The resulting lightsheet is correspondingly smaller than it would be in the absence of said entropy. Hence increasing the entropy in a spatial volume does not result in a proportional increase in the entropy in the lightsheet. This relationship between entropy and geometry is the main motivation behind the bound. The CEB has been proven in classical gravity under general conditions by Flanagan, Marolf, and Wald [27]. To prove the CEB, they assumed that entropy can be realized as a fluid; that is, they describe entropy as a vector Chapter 2. Compactified Plane Waves 22 field living in spacetime. They also require some energy theorems, since negative energy trivially allows violation of the bound. One set of assumptions conjectures the existence of a global entropy flux s, which satisfies two conditions. The first condition limits the entropy asso-ciated with a given energy density. It is satisfied in thermal equilibrium by both Bose and Fermi g clSGSj clS long as their temperature is below the Planck temperature. The second condition puts a restriction on the gradient of the entropy density, bounding it by the energy density. This latter condition is necessary for a thermal or fluid description of the entropy, since it requires that the entropy density not fluctuate too wildly. The null energy condition is also implied here, since the energy density is used as the upper bound of a non-negative quantity. Since this set of assumptions is satisfied by a wide variety of systems, it establishes that the CEB has at least some general applicability. At the very least it eliminates many candidates for counter-examples of the CEB. In fact, given that no counter-examples to the CEB have been found at all, this proposal should be taken very seriously. 2.2.4 Example: Collapsing Star As an example of the application of the CEB, consider the collapse of a spherically symmetric star. Choose the surface B to be the surface of the star, and consider the lightsheet formed by the geodesies traveling forwards in time towards the center of the star. By the symmetry of the problem the caustic will be a single point at the center. Since all the entropy in the star must pass through this lightsheet, a Chapter 2. Compactified Plane Waves 23 spacelike projection can be used. The entropy measured in the star's spatial volume will be equal to the entropy in the lightsheet. As the star collapses, its area continues to decrease although its entropy does not. However the star has a spacelike singularity in its future, signaling its collapse into a black hole. Thus lightsheets formed near the end of the life of the star will not reach all the way into the interior of the star, but will be terminated by the singularity. At this point the spacelike projection will no longer be valid. Instead, the entropy on the lightsheet will decrease and the CEB will hold. One could try to violate the bound before the collapse by increasing the entropy of the star by breaking the spherical symmetry. But this would pro-duce inhomogeneities in the energy-momentum tensor and would cause the geodesies to deviate from their straight-line path to the center. New caustics would form, and not all geodesies would reach the center of the star. Thus as the entropy of the star at some fixed time is increased, the lightsheets change their shape so that they continue to probe a constant entropy. 2.2.5 Holography Holographic Principle The existence of the CEB suggests that the number of degrees of freedom in a quantum theory of gravity should scale like area, and not like volume. This is at odds with the estimate from local field theory. A gravitational analysis demonstrates that the field theory estimate is too large. One analysis is simply based on counting the number of states in field Chapter 2. Compactified Plane Waves 24 theory. We can approximate a local field by a harmonic oscillator in each Planck volume. Each oscillator has a ground state and can be excited up to the Planck energy. So there are a finite number of states n in each Planck volume. The total number of states of some spacelike volume V is then N ~ nv. But according to the conjectured entropy bound nothing can be more dense than a black hole, so the number of states in V is bounded the number of states of a Schwarzschild black hole of the same size. The mass of such a system in field theory grows like its volume, so most of these states are far too massive to be gravitationally stable. These massive objects should thus not be counted as independent states. Another problem with the field theory counting comes from the unitarity of quantum mechanics. If a heavy state requiring V degrees of freedom to describe it collapses to a black hole of surface area A, the state suddenly requires only A/4 degrees of freedom. The unitarity of quantum mechanics forbids the loss of degrees of freedom in this manner. Thus local quantum field theories always seem to overcount the degrees of freedom in a gravitating system. Note that the role of quantum mechanics in the above analysis is to provide a natural ultra-violet regulator at the Planck scale. Quantum states are also nice because they give a discrete number of degrees of freedom. Altogether these arguments strongly suggest the existence of a holo-graphic dual, which would by definition have no extra degrees of freedom. However there is no reason to believe that this holographic dual should be a local theory; nor is it clear that it will have a timelike coordinate. Chapter 2. Compactified Plane Waves 25 Despite these potential obstacles, examples of holographic duality are known. One example is LST, which I discuss in Chapter 3. The most well-studied example is the AdS/CFT correspondence [28, 29, 30]. This is a con-jectured duality between supergravity on AdSs x § 5 and AT = 4 S Y M on the conformal boundary of the space. Unfortunately this example enjoys several special features which make it difficult to generalize. The most prominent of these is that the field theory is local. In addition the field theory lives on the boundary of the space, not the interior as a generic dual might; and it has a conventional time coordinate. Holographic Screens As mentioned, the holographic dual should live on a codimension one surface, termed "screen." One can imagine considering the holographic dual that captures the physics of all the entropy accessible to some particular observer. To do this, first recall that lightsheets typically end on caustics, where initially parallel null geodesies meet. Rather than begin with a surface, one can begin with the caustics. By emitting light rays from the caustics one can recover the original congruence. So the idea is to define the caustic to be a line in spacetime — it will be the worldline of the observer. The holographic screen is constructed by building a codimension two surface at each point along the worldline of the observer. So choose the observer to lie at a caustic point by sending off light rays from the observer in all directions, and following the lightsheets outward. Each value of the affine parameter defines a surface on which the CEB applies, as long as the geodesic expansion does not flip sign. But for the surface to Chapter 2. Compactified Plane Waves 26 be relevant for holographic duality it should describe as much as possible. So follow the lightsheet outward from the observer until it reaches the point where 9 = 0 immediately before it flips sign; this will be called the preferred surface. One such surface is constructed at each point along the observers world-line. The holographic screen is constructed by taking the union of all of these surfaces. The holographic dual living on this screen should describe all the spacetime that is foliated by the constructed lightsheets. Generically such preferred screens will be observer-dependent, but this need not be the case. In fact, for the AdS/CFT example, the screen lives on the conformal boundary of the spacetime, and this is clearly independent of any observer. 2.3 Spacetimes 2.3.1 Godel-like Universes A l l supersymmetric solutions of minimal 5-D supergravity have been clas-sified in [17]. They found that one class of solutions had the form of a Godel-like universe (GLU) Here u = £2 Pipfdcfri and Bt is a positive real number with units of inverse length. The time coordinate is t, and the spatial dimensions are the two rotating planes, shown here in polar coordinates. This solution also has a 2 (2.5) Chapter 2. Compactified Plane Waves 27 two-form potential B = dz A u. The positive energy density of this field is balanced by a negative cosmological constant A . 2 The coordinate system used here describes the space as seen by an ob-server sitting at the origin of this coordinate system.. The topology of the spacetime is that of R 5 , but the one-form u causes the spacetime to rotate. This means that all inertial reference frames have angular momentum about the origin; this is known as the dragging of frames. The rotation of the plane i is parametrized by The G L U is also homogeneous. [17] This form is reminiscent of the rotating space that Godel studied; hence the name. The Godel solution was a spacetime with rotating pressureless dust and a negative cosmological constant. This new supersymmetric version has no dust, but rather a gauge field. However the gauge field is still pressureless and has the same density; in fact, the gauge field and the dust have the same energy-momentum tensor. This spacetime has CTC's generated by K = _^ a ify,; , whose length is 1 — _^ aiP?Pi- K generates closed curves as long as the af's are integers, since the <j>i are both angles with periodicity 2ir. These closed curves are timelike when the radial directions are large. Note that these CTC's are not geodesic; it will be shown below that none of the geodesies are CTC's. The homogeneity of the spacetime has some important ramifications that must be kept in mind. Since there is one CTC, there are CTC's through every point, including the observer himself. Also, there is no fixed center of rotation — if the observer moves, the observer still sees himself at the centre of rotation. 2 So A has the sign it would have in an AdS space. Chapter 2. Compactified Plane Waves 28 To fit this into string theory one can add a circle3 S 1 — call it z — and 4 other directions x\ So the string metric is 9 ds210 = ds\ + dz2 + ^2 dx{dx\ (2.6) i=6 A constant four-form potential extending in the transverse directions must also be introduced. In this form the spacetime is T-dual to the compactified plane wave. 2.3.2 Compactified Plane Wave The compactified plane wave (CPW) metric is given by ds2 = -{dt + UJ)2 + Y,(dP2 + + (dz - w)2 + E d x 2 (2-7) where again u = ^0iP 2dfa. The coordinates have the same interpretation here as for the Godel-like universe (GLU). However, notice that the T-duality converted the B field into a new term in the metric. There is still a back-ground four-form field however. Timelike geodesies rotate in the positive fa directions as they move for-wards along t. But, at constant time, spacelike geodesies are twisted in the opposite direction along z. So the null geodesic traveling forwards in time towards negative z is not twisted at all 4 . The sign convention here — travel to negative z to see stationary space — is a result of the relative minus be-tween dz and u in the metric. This sign was chosen to correspond to the sign convention for B in the G L U . 3 In general z need not be compact, but it is the only form considered here. 4 Recall that the /3, are positive. Chapter 2. Compactified Plane Waves 29 Now closed timelike curves are generated by K = dz + Y^&id^. Since the fa are angles they are of course periodic, and since z is also a compact coordinate if is clear that K generates a closed curve. Its length is given by K2 = 1 — YJp?o;i(2/?j — ai), so it is timelike when cu < 2/3; and K is closed only when the at are rational multiples of 1/R, where R is the radius of the 2-circle. The causal structure is very similar to that of the G L U . In neither case is there a global time function. A global time function is a timelike coordinate whose orthogonal hypersurfaces are spacelike everywhere. The lack of such a coordinate renders the Cauchy initial-value problem ill-posed. This issue is not believed to be important for the quantities calculated in this study, since these quantities are defined locally5. Since the Cauchy problem cannot be defined, quantum field theory also has no definition on this background. One could consider restricting to some finite region of the geometry, but one then needs to find boundary conditions that are consistent with the geometry. It is not clear what the correct choice of boundary conditions should be. Thus quantum field theory is not at present a useful tool for studying these geometries. This is unfortunate, since string theory, with its infinitely many fields, is much more difficult to work with. 5 In the case of the C P W , the lack of a well-defined Cauchy problem is inherited from the pp-wave. However the CTC's and the lack of a global time function are new; they can be traced back to the compactification. Chapter 2. Compactified Plane Waves 30 2.3.3 T-duality and Dimensional Reduction String theory enjoys a duality known as T-duality when the target space has a circular dimension z. This duality, although evident already from perturbative calculations, is an exact duality of the full theory. The T-dual of a spacetime is obtained by replacing the radius R of the circular dimension by a'/R. To be a duality, the eigenvalues of Hermitian operators must not change, although generally the interpretations of operators will change. In particular, the energy spectrum must be invariant. In so-called natural units (GJV = h = c = 1), the eigenvalues of the Hamiltonian of the bosonic closed string are given by a'(Pz)2 w2R2 ~ c + c H = ^ - J - + ^ - + N + N- (2.9) where Pz and w are the momentum and winding respectively in the z direc-tion. The left and right level numbers N and TV are independent of R, as is the central charge c + c = 48. For the mass to be invariant under T-duality one should identify a new set of momentum modes as the old set of winding modes, and vice versa. If the metric has components gZfl along the circular dimension, the new metric exchanges these components with a two-form gauge field BZil. This can be understood by considering, in the original geometry, a string with momentum Pz but no winding. This string will certainly interact with the metric components gzll. However in the T-dual geometry the string will be stationary along z but will have nonzero winding. Thus, it must interact with a gauge field analogous to gzli. Likewise considering a closed string in the original geometry with no momentum but with winding, one can see that Chapter 2. Compactified Plane Waves 31 the original gauge field must become part of the metric. In general for a spacetime with a metric g u v and an NS-NS two-form B M I / y in a normalization where gzz = 1, T-duality along z generates the metric and B-field g and B given by [18] 9nzQvz + B ^ B vz (2.10) and B U u B ^ z g u z -\- g u z B u z B » z = g » z (2.11) respectively. The 10-dimensional G L U and C P W presented earlier are related by T-duality along z. When a compact dimension z is present, one can construct a lower-dimensional effective theory by taking the radius R to zero. This is called dimensional reduction, or sometimes compactification. Only the modes Pz = 0 should be considered; since the energy due to momentum scales like Pz/R2, states with nonzero Pz are very massive and hence not excited. On the other hand, the winding modes all become mass-less. In fact the winding number becomes a charge with respect to a one-form field Au = Bzlt. One can start with an 11-dimensional C P W by tacking on sufficiently many flat transverse directions, and then dimensionally reduce to get a 10-cjzz = 1 9uz = Buz Chapter 2. Compactified Plane Waves 32 dimensional G L U . In this case the G L U is supported by a R-R one-form, rather than an NS-NS two-form. Both 10-dimensional solutions can be dimensionally reduced to get a 9-dimensional G L U . The 9-dimensional theory should be thought of as a Lan-dau theory since it has a background one-form gauge field. This dimensional reduction is thus also useful for gaining insight into the higher-dimensional theories. 2.3.4 Review of Length Scales There are at least three important length scales in the type of analysis per-formed here. They are the curvature scale R, the string length y/~o7, and the Planck length lp. At the Planck length quantum gravitational effects are dominant and the geometric interpretation of spacetime breaks down. Thus we will always require R^$> lp. The string length \fa' is an independent parameter that determines where stringy effects become important. Although one would certainly expect stringy effects to be crucial at the Planck length, they can become important long before then. For now we take R ^> y/o7 ^ > lP. Since R and y/a1 are both much greater than lp we can trust a classical treatment of the spacetime backgrounds. And because R \fa' the strings behave for the most part like point particles. Stringy effects will be important when we consider T-duality which leaves a new hierarchy yfa' ^> ^ ^S> lp. Chapter 2. Compactified Plane Waves 33 2.4 Geodesies 2.4.1 Introduction There are two major ideas that make the study of holographic screens in the C P W interesting. There has appeared in the literature [13] the suggestion that holographic screens might be used to help understand spacetimes containing CTC's. The basic idea is that the holographic dual on a screen might describe the physics on only one of its lightsheets. So if a screen encloses some region that does not contain any CTC's there is the hope that the holographic dual will not "see" any CTC's and will provide a clearer description of the physics. However this idea has not been realized concretely, and we provide evidence that generically a holographic dual will still need to contend with the presence of CTC's. The second important aspect to this research is that it explores holo-graphic screens in the presence of compact directions. Since string theory contains objects which can wrap such directions, it is sensitive to them in a way that particles cannot be. Furthermore string theory enjoys T-duality which implies that the G L U and the C P W are one and the same. This should be reflected in all physical results, including the construction of holographic screens. However, as argued above (2.3.4), the prescription will require mod-ification for use in string theory. Related to this is the treatment of matter fields as opposed to the met-ric. Bousso's approach, being based on a particle description, distinguishes between the metric and the matter since particles live on a gravitational Chapter 2. Compactified Plane Waves 34 background. But string theory interchanges these under T-duality. This would suggest that Bousso's prescription should also be generalized to treat matter fields on the same footing as the metric. Our work suggests how Bousso's prescription might be generalized, at least in the case where metric and matter fields are related by T-duality. Once the null geodesies are found, their expansion is calculated; this defines the location of the preferred screen. Then the region of spacetime interior to the screen is studied to see if it can contain CTC's. Finally, the smeared screen is constructed, and it is related by T-duality to the preferred G L U screen consider. To simplify the calculation, concentrate on the case where all the fa are equal, fa — 8. It is then convenient to define an overall radial coordinate P ~ V_3 Pi a n d the angle x between the planes. The metric can now be written as ds2 = -(dt + tof + dp2 + p2dX2 + Y,pWi + (dz - U)f + d x 2 - (2-12) 2.4.2 Null Geodesies First the translation symmetries and corresponding conserved charges are identified. This leads to a single nontrivial differential equation. The geodesies have both linear and oscillatory components, resulting in a helical motion. It is also found that the result of this calculation, and especially the calculation of the screen locations (Section 2.5), are substantially simplified when the momentum in the transverse directions xl vanishes. Finally, it is simple to show that none of the null or timelike geodesies are closed. Chapter 2. Compactified Plane Waves 35 The quadratic Lagrangian is l=W*°)x»i" = \(f^ (2.13) where the dot denotes the derivative with respect to the affine parameter A. The constraint £ = — e can be imposed; taking e = 1 yields timelike geodesies, while e = 0 gives null geodesies. The Euler-Lagrange equation of motion is the geodesic equation F + Y%xvxP = 0. (2.14) The Christoffel symbols T$p are listed in Appendix A. So solve (2.15) Here the dot refers to derivatives with respect to the affine parameter. The Lagrangian has a number of symmetries, described by Killing vectors. Thus there are a number of conserved Noether charges, dt ^ = p dz ^ = u dp Here E is the energy of the point particle following the geodesic. The mo-mentum along the z direction is Pz; it is an integer since z is compact. The Chapter 2. Compactified Plane Waves 36 angular momentum in the rotating planes is given by Lj . Finally, the mo-menta in the transverse directions x% is P\. E = i + 6^2p24>i Pz = z-d^pUi h = pUi-Bfiii + z) PI = x. (2.17) The conserved charges can be put back into the Lagrangian to get 2 £ = _£2 + p2 + p2^2 + YJ{ji+ SPi(Pz + E)j + Pi + YlPt= ~2£-(2.18) The equation £, = —e can now be solved to find the geodesies. If e is positive the geodesies will be timelike, while for e = 0 they are null. The geodesic equations for the radial coordinates are Pi — — (— + QpiiPz + E)) - 8Pi(Pz + E)) = 0. (2.19) Pi \Pi J \Pi J From this it is clear that with nonvanishing L , the geodesies can never reach the origin, so since only geodesies that do pass through the origin are needed, set Li to zero. The solutions are p i = (p 7 r i) isin((P z + JB)/3A) (2.20) where pm, the maximum radii, are functions of the various momenta as de-termined by (2.18) and discussed below. To simplify this, only geodesies that move in one of the two planes will be considered. When Lj = 0 there is a conserved charge J corresponding Chapter 2. Compactified Plane Waves 37 to rotations along %. Fixing J = 0 holds the (pm)i constant. Thus setting (Pm)2 = 0 is consistent. The remaining equations of motion can be found from (2.17); the geodesies are given by x*(A) = x0 + pi\ <P{\) = 4 + (P, + £)/3A t(A) - t0 + EX- ^(2(Pz + E)B\-sm(2(Pz + E)B\)) Bp2 z{\) = zQ + PzX + ^ [2{PZ + E)8\ - sin (2(PZ + E)B\)). (2.21) Notice that it is possible for the time coordinate to decrease even though the affine parameter always increases. This is why the time coordinate t is not a global time function. The geodesies spiral out from the origin, and reach a maximum distance P m = V—WiKTsJ2— ( } at affine parameter A = 7r/(2/3(Pz + E)) and then in the same affine distance they go back. This equation and subsequent analysis for null geodesies will be much simpler if Pt vanishes. In this case 1 l E - P ' - 1 ^ (2.23) r m 0\E + Pz B — 2 where rj is the launch angle of the geodesic between the p and z axes. Since the C P W is homogeneous, having the equations for all geodesies through one point is the same as knowing all the geodesies in the spacetime. With these equations in hand, it can be proven that there are no closed Chapter 2. Compactified Plane Waves 38 timelike or null geodesies in the spacetime. To arrive back at p = 0 one clearly requires A = im/8(Pz + E). Plugging this into t(X) - t0 = 0 yields This requires e = Pt = 0 and Pz + E = 0. The latter condition implies A —> oo, so that null and timelike geodesies are never closed in finite affine parameter. 2.5 Holographic Screens For simplicity we keep P t 2 = 0. As explained earlier (Section 2.2.5) the screen is found as follows. Consider an observer fixed at the origin of our coordinate system. At some moment he emits lightrays in all spatial directions; hence he lives along the caustic of the resulting congruence. The equations for these geodesies were found in the previous section. The expansion of this congruence is calculated, and the place where it vanishes is found. This defines the location of the surface as a function of time. The screen is then the union of the surfaces along time. : Of particular interest here is the region bounded by the screen. We find that this "holographic region" includes the entire spacetime, as a result of the compactness of z. Now that I have found the geodesic equations y^ (2.21) I can easily con-struct the generator of the geodesic; it is given by E + Pz + e + P? Pz + E = 0. (2.24) (2.25) Chapter 2. Compactified Plane Waves 39 So y gives the equation of the curve and k is the tangent vector. Then the geodesic expansion is 6 = V • k = 3„fc" + T»aka (2.26) where the T are the Christoffel symbols (Appendix A). It is convenient to rescale A, A -» A/(P Z + E). So we find 0 = 3/?cot(/?A) (2.27) so that the screen is located at affine parameter = P.28) The screen is parametrized by angles and t0. The angle between the p and z direction is given by 77, but it will be useful to continue to express this as pm. The screen is located at p = pm = ^ t a n | , and along x = Xo and <j>i = 4>i + ^ . It is along the straight line z = z° + ^ in the compact direction. To find the metric on the screen it is simplest to first rewrite (2.12) in terms of x>* = (t°, A, pm, x, 4>l, 4). ds2 = -(l + pm)2dt°dX-(dt0)2 - 2 ^ sin(/3A) (coS(BX)dpm + pm sin(/3A) £ ^dtf + ^ sm2(BX)(dp2m + p2mdnl) (2.29) where d$l\ parametrizes the solid angle of the two rotating planes. The metric on the screen is now given by substituting A = A s ; ds2\screen = - ( d i 0 ) 2 - 2_°i° + dp2m + p2mdn23 (2.30) Chapter 2. Compactified Plane Waves 40 where w° = 0 £ p ? m d # . From the description of the screen it would seem as though the region of large z was excluded from the holographic region. That is, it seems to lie beyond the screen. However, z is a compact direction, so large values of z are equivalent to small values — and therefore all points in the spacetime can be reached by a geodesic that starts at the origin and terminates at A = A s! Thus the screen does not partition the space into two disjoint sets as one would expect, even though all the geodesies are of some finite affine length. This last idea implies that the holographic dual will be nonlocal, since two distinct points on the holographic screen can be correlated with a shared point in spacetime. Although not a necessary condition, it seems that holo-graphic duals will always be nonlocal when the full spacetime has compact directions. This also implies nonlocality of the dual when the spacetime has a magnetic field arising from T-duality or dimensional reduction. One can see from 9 = V • k that the expansion in orthogonal directions is additive. In the Godel directions the geodesies first expand then contract again, as can be seen from (2.27). But in the flat transverse directions xl the expansion is always positive. Thus including nonzero transverse momenta will only serve to push the screen farther out in p. So there does not seem to be any sense in which a screen complementarity principle could be useful for understanding the CTC's that appear in this spacetime. Since the screen found above is not translationally invariant along the compact direction, it is difficult to see how it can be connected to the di-mensionally reduced geometry. This relationship is particularly interesting Chapter 2. Compactified Plane Waves 41 as string theory suggests that our own world may have additional undiscov-ered compact directions. We would like to understand how an effective lower dimensional theory would be affected by these compact directions. Rather than consider an observer that is localized in the compact direction z, we can consider a delocalized observer that sits at all points along this direction. Such an observer would not send off geodesies with a z-momentum, but would rather send off geodesies from all points along z. So z° will become a new screen coordinate, replacing 77 which is fixed at 7r/2 now. Since this screen is constructed by a delocalized observer "smeared" along z we call the resulting screen a smeared screen. Recalculating the geodesic expansion one now finds that 0= 374^2 . (2.31) PVI - P v This vanishes at "•• = t < \ < 2- 32> so the holographic screen lies within the critical radius where CTC's appear. This screen is much more intuitive than the unsmeared screen. A l l points at p > pss are excluded from the holographic region. The induced metric on the smeared screen is (2.33) Unlike the unsmeared metric, this metric can be T-dualized along z in which case it exactly reproduces the result of [13]. There is a nice way to understand this. Under T-duality, momentum modes become winding modes, and vice versa. Since winding modes are Chapter 2. Compactified Plane Waves 42 massive, they were not considered by the authors of [13] for the G L U . How-ever, the winding modes correspond to momentum modes in the CPW, where they are massless modes that should be considered. 2.6 Conclusions The hope in this research was that the preferred screens would indeed shield each observer in some way from the large closed timelike curves. But it was found that the entire spacetime is in the interior of the preferred screen. However, the difficulty does not seem to be directly related to the presence of CTC's.-Rather, the essential problem here is with the prescription for finding Bousso's holographic screen. His construction is based on point particles, which do not see the G L U and C P W as equivalent. But the presence of a compact coordinate in the C P W changes the structure of the screen when viewed from the perspective of string theory. Particles with Pz = 0 follow trajectories analogous to those found in the G L U , and when considered in isolation give rise to the smeared screen. The problematic particles are those with Pz < 0. As indicated in Section 2.3.2, these modes see less twisting of the spacetime. Hence it is not surprising that these modes, which see a "flatter" spacetime, can travel to large radial distances. Since Bousso's bound seems to give inconsistent results when applied in this way, we are inspired to ask what the stringy version of Bousso's bound might be. Of course, it might not exist at all; it is possible that these bounds Chapter 2. Compactified Plane Waves 43 can only be recovered in a low-energy particle limit — and even then, it might not work in all backgrounds. A simpler question to ask is whether the prescription can at least be made consistent with T-duality, and whether null geodesies are always sufficient. A modification to take this into account has already been suggested in [23]. Another question to ask is what happens to the entropy of the different modes under the smearing. It would be interesting to relate this to the change in screen areas. Finally, some recent work has been done to understand the specific geome-tries we considered here, especially in the language of the G L U . By adding a soliton to the background it has been suggested that a more reasonable version of this background can be constructed [31, 32, 33, 34]. However more work is needed to understand how to construct a consistent holographic screen for the G L U . Chapter 3. Little String Theory 44 Chapter 3 Little String Theory 3.1 Introduction Little String Theory (LST) is a non-gravitational theory that lives in six or fewer dimensions [35, 36]. Thus one is led to think of LST as the boundary theory in a holographic duality. The bulk dual is known and even better understood than LST. Here the bulk theory will be used to construct and study LST. It is interesting to study LST and compare its behavior to conventional field theories and to critical string theories. This allows for an examination of nonlocal theories without having the complications associated with quantum gravity. Many restrictions of local field theories are lifted by nonlocality, and it is interesting to see how this is realized in a concrete sensible example. For example, the density of states normally grows like a power of the energy in local theories, whereas in nonlocal theories it can grow as the exponential of a power of the energy. Little String Theory can also studied at finite temperature. This is per-haps a more useful thing to do, but you might expect it to be more compli-cated. Here it turns out to be easier because the bulk theory simplifies. Chapter 3. Little String Theory 45 3.1.1 Construction The bulk theory can be defined as the decoupling, or near-horizon, limit of a stack of NS 5-branes in Type IIB closed-string theory. An NS 5-brane is a solitonic solution of gravity; that is, it is a nontrivial classical solution to the low-energy effective action of perturbative string theory. As its name suggests, it extends in five spatial directions. It is the source for the NS-NS two-form potential B^v. An NS 5-brane has a constant energy density along its world-volume. The spacetime has an event horizon parallel to the brane, so the brane looks like an extended black hole. In the near-extremal limit, in which the horizon approaches the singularity, a naked singularity is not left behind. We can stack k NS 5-branes on top of each other. The qualitative features of the solution do not change, but the analysis simplifies in the large-fc limit. To define LST as the world-volume theory of the NS 5-branes, we want to take Newton's constant GN to zero to decouple the brane from the rest of the spacetime. We also want to keep the world-volume coupling gNS finite so that we are not left with a trivial theory. Since GN = g1tringa' while gNS = a', the limit sends the string coupling to zero at fixed energy. This decoupling limit is also known as a near-horizon limit since it moves the asymptotic structure to infinity faster than anything else, thus zooming in on the region near the event horizon. Propagation from the near-horizon region to the original asymptotia is suppressed; the near-horizon geometry has a new asymptotic region. Perturbative string theory cannot probe the region close to the NS 5-branes, since the string coupling grows along the linear dilaton direction. Chapter 3. Little String Theory 46 Although many questions are left unanswered, there are some basic things one can discover at zero temperature. It is convenient to think of the LST spacetime as that of the world-volume of the NS 5-branes1, while letting the symmetry of the transverse angular dimensions be implemented by internal degrees of freedom. The dilaton direction, along which the string coupling runs, is the holographic coordinate. 3.1.2 Duality Since LST appears on the boundary side of a holographic duality, its off-shell Green functions can be studied. S-matrix elements of nonnormalizable states in the dual theory correspond to off-shell correlators in LST. Since S-matrices are always accompanied by a mass-shell condition, the off-shell correlators of LST depend on one fewer parameter. Bulk non-normalizable modes perturb the boundary; they change the boundary conditions there. Thus from the LST perspective they are local operators. Normalizable modes in the bulk, that is, local excitations with particle properties, correspond to states of the Hilbert space of LST. The Green function of some operator O can be calculated at arbitrary momentum p by calculating the S-matrix element of the associated nonnor-malizable mode (f>o\ Gbdyip) = Sbulk(p, 3)^-^11 • (3-1) Here p is the 6 - D momentum along the world-volume of LST, j is the momen-1 Further compactifications are possible, but those examples will not be considered here. Chapter 3. Little String Theory 47 turn corresponding to the linear dilaton direction, and the angular momenta around the § 3 are not excited. The bulk mass-shell condition is j = f(p). More explicitly then, Gbdy(p) = (0(P)0(-p)) = SbulkipJ = f{p)) = t=oo(<f>o(pJ = f(p))\<f>o(-pJ' = f(-p)))t=-oo- (3-2) The fact that the S-matrix cannot be calculated off-shell is not a problem. 3.1.3 Zero Temperature It is interesting to ask whether a sensible holographic interpretation of the S-matrix elements is possible. For example, if calculations in the bulk predicted non-unitary evolution of LST then there would be no sensible interpretation of LST as a quantum mechanical system. Fortunately this does not happen. However LST is certainly not an ordinary quantum field theory. It ex-hibits T-duality invariance when compactified, so is a non-local theory. The nonlocality of LST is gentle and subtle. You might see it because of a dif-ferent fall-off in G at large energy. LST also has a Hagedorn density of states. As these properties are common in string theory it is natural to try to understand LST in string-theoretic terms. But since perturbative string theory must live in ten dimensions, LST is nonperturbative in the string coupling. That is, the amplitude for splitting and joining strings is of order one, rather than being a free parameter that can be taken to be arbitrarily small. This means that the sum over Riemann surfaces in the worldsheet path integral must be done exactly. As such it is Chapter 3. Little String Theory 48 not clear to what extent a worldsheet has meaning in LST. Perturbative string theory always has a massless graviton; this is another way to see that this cannot be an ordinary string theory. One can see this by looking for poles corresponding to the exchange of massless spin-2 particles in the correlation functions of the holographic dual of LST. Another way to see this is to observe that the bulk 10-D graviton has a mass from the 6-D point of view, due to integrating out the dilaton. The mass of the graviton is m2 ~ l/k. In the strict limit k -> oo the graviton does become massless, but one is left with ordinary flat space in this limit. This is why it is important to keep k finite, although large. 3.1.4 Finite Temperature Since the bulk theory has an event horizon, one way to cut off the running of the dilaton is to consider the Euclidean solution. The time coordinate is Euclideanized and then compactified. This keeps only the spacetime outside the event horizon of the NS 5-brane, and corresponds on the boundary to studying LST at finite temperature. Studies of the thermodynamics of LST, using this holographic duality [37], reveal qualitatively different features from more familiar theories [14, 38, 39, 40, 41, 42, 43]. The construction of the NS 5-brane geometry and finite temperature con-verts the dilaton and time directions of the near-horizon geometry into a two-dimensional black hole known as the CGHS cigar [44]. It can be visualized as a warped disk. The tip of the cigar is mapped to the center of the disk. This corresponds to the location of the event horizon; it is the place with largest string coupling. Chapter 3. Little String Theory 49 The fact that the string coupling is bounded from above by its value at the tip is what makes this scenario tractable. As one moves away from the tip the string coupling decreases monotonically, until at the boundary of the disk it vanishes. The circumference of the cigar at the boundary is the inverse temperature. It is easier to use the canonical ensemble, which fixes the temperature but lets energy vary about its mean. Thermodynamic quantities can be studied by finding the partition function Z(3) or its logarithm, the Helmholtz free energy F. To first order in gtip = k/p, F vanishes. This implies that the entropy grows linearly in energy rather than logarithmically as string theory would suggest. It is more reliable to calculate thermodynamic quantities in the micro-canonical ensemble. The basic quantity in this approach is the density of states p(E) — the Laplace transform of Z(B) — or its logarithm the entropy. Since the area of the black-hole horizon is linear in the black-hole mass by the first law of black-hole thermodynamics, the Bekenstein-Hawking entropy yields the Hagedorn density of states. Although the energy of this system is fixed, one can calculate an expected temperature as a function of the energy. At tree level the average temperature is independent of energy, which means that the specific heat is infinite. In [38, 39] it was pointed out that the infinite specific heat found in the tree-level approximation of the bulk geometry is an artifact of examining it in weak string coupling. The point is that one is expanding in inverse energy, which is valid at infinite energy; we should use the microcanonical description and write B(E) = BH + OLJ'E + 0(E~2). Including stringy corrections, we can expect the density of states Chapter 3. Little String Theory 50 to take the form p = Eae & h E . This gives the more reasonable specific heat C{E) = ZB*H + 0{E-i). The specific heat away from the limit a —> 0 was argued later to be negative [14, 45]. Negative specific heat implies that the thermodynamic equilibrium of the system is unstable. For suppose you put a cold gas in contact with an ordinary hot heat bath in which the particles have a large average energy. The gas will tend to absorb energy from the bath, but this will decrease the temperature of the gas if it has negative specific heat. Hence the gas never reaches thermal equilibrium with the bath; in fact that bath drives it farther away! Negative specific heat is a generic feature of black holes in asymptotically flat spaces, so perhaps one should not be too surprised. This is not an unphysical result; it is merely inconvenient. It would be very interesting to understand how to describe the stable thermodynamic equilibria of LST, or even whether such equilibria exist. For a nice review of these issues the reader may consult [46]. It would be useful to find a clear interpretation of the implied instability of the thermal ensemble. In [14] it was speculated that some mode, found earlier in the context of zero temperature "double scaled" LST [47], could become tachyonic at one loop level. Moreover, the analysis in [14] predicts an interesting dependence of the instability on the spatial volume of the system2. The tachyon signals a phase transition; it should condense. But on the black hole side the temperature is low and you don't really think there should 2 Also there may be an interesting dependence on k, the number of 5-branes. However, here and in [14] we are working in leading order in the large k limit. Chapter 3. Little String Theory 51 be a phase transition. However there could be important states localized in the strong-coupling region of the dilaton whose contribution becomes impor-tant, but which you don't understand. What likely happens is that LST is no longer dual to the weakly coupled bulk; you need to properly understand the strong-coupling region of the bulk to properly understand LST. However, the unstable mode discussed in [14] was originally found in [47] under quite different circumstances. In [47] the NS 5-branes were separated in transverse directions but were studied at zero temperature. So it is not clear that these unstable modes actually exist in the finite-temperature scenario of interest. One of the goals of this work is to examine the conjecture of being on the verge of instability in the context of LST at finite temperature. Indeed we find that the system has modes which could destabilize it, given the correct mass shift at one loop. Furthermore we find complicated dependence of this phenomenon on the spatial momenta of the modes, which is reminiscent of the claim in [14]. I calculate the S-matrix element of bulk graviton-graviton scattering in the finite-temperature Euclidean background, utilizing the results of [48]. The mass-shell condition is solved to yield a Matsubara two-point Green func-tion of LST. Then I continue it to get the time-dependent finite-temperature Green function. I do not explicitly discuss the Lehmann representation, but the Green function does not have simple poles and branch cuts. I know the quantum numbers of the states that lead to poles, but these are multiparticle states in the sense that the divergences have branch cuts and are not simple poles. So I don't know anything about the particle content per se — and I Chapter 3. Little String Theory 52 can't really compare with the Atick-Witten scenario. In the next section I go through the construction of LST in more detail. Then I spend some time setting up the correct vertex operators, after which I calculate their correlation functions and analyze the result. 3.2 Holographic Construction Consider a stack of k NS 5-branes. Adding k copies of the soliton to the same place in flat space leaves the string-frame metric [49] £=- i1 - il) * 2 +i 1 +h) {v^+r,dai)+** <3 3 ) These coordinates are dimensionless; lengths should be measured in units of the string length y/o7. The time direction is t. The branes are extended along the R 5 ~ dx\, and the transverse directions R 4 are parametrized by the radial coordinate r and the solid angle CI3. The event horizon is at rn. The string coupling depends on r, which is sometimes referred to as the dilaton coordinate. Explicitly, + (3-4) The string coupling at r = oo is = e^r=0OK The NS-NS H-field is an exterior derivative of the dilaton; H^P = -e^/a,*. (3.5) From (3.3), (3.4), and (3.5) one can see that there are three independent dimensionless couplings, k, g^, and r 0 . In terms of these quantities the Chapter 3. Little String Theory 53 string-frame3 energy per unit five-volume, or tension, [50] is E 1 / fc + p (3.6) V5 (2n)W V<& where p = 4- (3-7) Too Here p/(2ir)5 [50, 42, 40, 46, 36] is the dimensionless energy density above extremality. The decoupling limit g^ —> 0 that defines LST is taken at fixed p. It is convenient to employ the field redefinition r = r 0 cosh cr = pg2^ cosh cr, which restricts attention to the region outside the horizon. Note that this does not eliminate the horizon although it hides it; the horizon does not disappear in the decoupling limit. In terms of these coordinates the metric and dilaton are ds2 — = - tanh 2 odt2 + {pg^ cosh2 o + k) [do2 + dOf) + dx\ (3.8) and e2* = g l + -sechV. • (3.9) p Now the decoupling limit —> 0 is clear. It leaves behind ds2 = - tanh2 odt2 + kdo2 + kdSl\ + dy\ (3.10) for the metric, and k e2* = — sechV = gtipSecb.2o (3-H) for the dilaton. The region of large o, pg2^ cosh2 cr fc, is decoupled by this limit. It is in this sense that the limit is a near-horizon limit. The maximum string coupling is now gtip. 3Hence it does not include the contribution of the dilaton. Chapter 3. Little String Theory 54 For string perturbation theory around a fixed classical background to be a good approximation one must have k 1 to keep the radius of the sphere dfl3 large. For the quantum string corrections to be small p 3> k is also required, so that gup, and thus the coupling everywhere, is small. The LST boundary operators correspond to non-normalizable modes in the bulk. In the original coordinate r these non-normalizable modes are supported on the boundary, where the string coupling is small, but they are exponentially suppressed in the bulk. So they can be thought of as fixed background sources or operators localized on the boundary. The zero-temperature (extremal) limit is p —> 0. In this limit the dilaton diverges and the string loop expansion, which is in powers of the exponenti-ated dilaton, does not generate a useful asymptotic expansion. At finite temperature p 0 and the event horizon shields observers from the diverging dilaton and cuts if off at a finite value. The dilaton reaches its maximum k/p, at the tip of the cigar in the center of the disk, and it runs to <7oo = 0 at the edge. The conformal field theory describing the finite-temperature theory is exactly solvable. In field-theory language, what we have now is a Lorentzian finite-temperature situation. The temperature is generated by the horizon. Direct calculation of the Green function is not possible, but the time-independent Matsubara Green function Q can be found by Euclideanization and compactification of time. The Lorentzian Green functions can be generated by analytic contin-uation of Q. To calculate Q in the bulk dual of LST, Wick-rotate t —> its and then compactify tE. Under this compactification, the full two dimensional ( £ E , C T ) Chapter 3. Little String Theory 55 part can be recognized as the black hole solution first studied by Witten [51]. He showed that it can be described by a gauged SL(2,R)/U(l) W Z W model. Backreaction and Hawking radiation were further analyzed in the 1/fc expansion by C G H S [44]. The manifold of SL(2, R)/U(l) is the cigar manifold discussed earlier. It is conformally equivalent to a disk, with time running along the angular direction and the dilaton running down the radial direction. The part of the vertex operators living on the cigar are constructed in [52]. A t small o the metric is ds2 = o2dt\ + kdo2 (3.12) from which it is clear that to avoid a conical singularity only one possible compactification is allowed, namely tE ~ t^ + 2nVk. However there can be string-theoretic corrections to this, of the order gtip. 3.3 Vertex Operators We calculate the two-point function of the bulk graviton at tree level. Since Qtip — k/p our result is valid to zeroth order in 1/p and thus there is no explicit p dependence. The goal is to search for poles which could be desta-bilized by loop corrections. The graviton is created by a vertex operator which is roughly of the form V ~ ev~^ip^ip„)eiqX$jm. The state created by $ lives on the cigar; it must be non-trivial, since we want to probe temperature dependence. Along the world-volume is a plane wave with momentum q and the SU(2) part is just an identity matrix that I have suppressed. I excite fermions tp to get a graviton Chapter 3. Little String Theory 56 and survive the GSO projection. The Faddeev-Popov ghosts tp implement superconformal symmetry. To construct the vertex operators on the cigar, we begin by construct-ing the vertex operators of SL(2,R) including their spectrally flowed im-ages. These correspond to zero-temperature vertex operators. We consider only the discrete representations of SL(2,R) as these correspond to the non-normalizable modes, which are operators in the LST. We take the modulus with respect to a specific U(l) subgroup to construct the finite-temperature operators. Then we review the supersymmetrization and the relevant GSO projec-tion. We enforce a mass-shell condition so that the correlation functions will have an interpretation in LST. 3.3.1 Review of SL(2, R) Results The worldsheet C F T describing the 2-D black hole (£E,CT) is a field theory with SL(2,R) gauge group. SL(2,R) is the multiplicative group of 2 x 2 invertible matrices with real entries and positive determinant. They generate special4 linear transformations. To begin, we need to set up the vertex operators of SL(2,R). We use the notations of [53, 52]. The SL(2,R) level k Wess-Zumino-Witten (WZW) where g is a worldsheet gauge field taking values in SL(2, R) , and Twz is the 4 That is, they have positive determinant and thus exclude reflections. action is wz, (3.13) Chapter 3. Little String Theory 57 Wess-Zumino term FwZ = 247 L ^ di9 9~ 1^ 9 9~ ldk9) •  (3- 14) The three-manifold E has the worldsheet <9£ as its boundary. Since SL(2,R) has no non-trivial 3-cycles, the level k is not quantized5. Because the CFT decomposes into independent holomorphic and anti-holomorphic fields, the gauge symmetry is enlarged to SL(2, R)L <8>SL(2, R ) R . The action is left-invariant under holomorphic left-moving SL(2, R) transfor-mations, and right-invariant under anti-holomorphic transformations, that is, S [g(z, -z)] = S [n(z)g(z, fJJT 1^)] (3.15) where fl and fl are independent elements of 5L(2, R) . Corresponding to the SL(2, R ) L (g> SL(2, R ) R gauge symmetry, the con-served currents of this action are As a function of the left- and right-moving world sheet coordinates z and z the currents are given by J^(z) = kTr ((T^Yg-'dg) , J^(z) = kTr (T**dg g'1) , (3.16) where T 3 , ± are the generators of SL(2,R). The currents can be expanded, oo oo Jt*{z) = £ 4'^-^, JtfW = £ JFe*". (3.17) n=—oo n=—oo The coefficients obey the Kac-Moody algebra k r 7 3 7 ±] _ , j ± [ J + J - ] = -2J3n+m + knSn,-m. (3.18) 5 In the full geometry k appears as the level of the W Z W model on S 3 , and as the number of NS 5-branes, and therefore is quantized. Chapter 3. Little String Theory 58 — 3 i ~"3 i The same algebra is satisfied by the modes of J. The zero modes J 0 ' , J0' obey the SL(2, R)L © SL(2, R)R algebra, appropriate for a particle moving on the SL(2,R) group manifold. The left-moving Virasoro generators Ln are 1 L k-2 ^(JQJQ + J0 JQ+) JO JO "f" /\^(J-mJm "f" J-m^m 2J_mJm) (3.19) and for nonzero n 1 °° L" = yZ2- J2(Jn-m+ K-mJl ~ ^l-A (3-20) m=l likewise for the right-moving generators Ln. These obey the Virasoro algebra with the central charge c = The unflowed representations are generated on top of representations of the zero mode algebra. We concentrate on the discrete representations — those are lowest- and highest-weight representations of SL(2,R). States in the discrete representations are described by two quantum num-bers j and m. The quadratic Casimir operator is given by C2C7) = \ {Jo Jo + JoJo) ~ (Jo)2 = — 3 {J ~ 1) a n d the eigenvalues of JQ are given by m. The quantum number m takes on the values j + n where n is any non-negative integer. The state \j; m = j ) is annihilated by JQ~ as needed for a lowest-weight state. Unflowed representations of the full Kac-Moody algebra, based on these representations, are built in the usual way by the action of creation operators J 3 , ± , n < 0. These representations are unitary, after imposing the Virasoro constraints, for | < j < ^ and k > 2 [54]. Additional representations of 51/(2, R) can be built using the operation of spectral flow. Unlike in the SU(2) case, the spectral flow operation does Chapter 3. Little String Theory 59 not simply permute the representations, but rather gives new representa-tions. These representations gw have nontrivial topology on the manifold of SL(2, R ) , measured by a winding number w.e We generate winding by defining [53] gw(z, z) = e-wln^Tig{z, z)e-whl{z)T\ (3.21) Then I want to define equivalence classes, [52], g(z,z)~hL(z)g(z,z)hR(z) (3.22) where h(z, z) = hL(z)®hR(z) is an element of the U(1)L<8>U(1)R subgroup of SL(2,R) generated by the current J3(z,z) = J|(z) + JR{z). Thus h(z,z) = ei{4>L{z)+4>R{*))Tz a n c j c l e a r l y the winding generated by spectral flow is removed by the identification. We conclude therefore that without loss of generality one can start with the unflowed representations of SL(2, R) — the winding will be generated using the compact scalar cf>.7 We will concentrate on vertex operators which are tachyons on the cigar, namely states which are created only by the zero modes of the SL(2,R) currents. The vertex operators of such states will be denoted $ J T O m -6 We consider here flow which acts the same way for both left and right movers. The flow which acts with opposite signs will create winding around the timelike direction of SL(2,E), when it is compact [55]. 7 This construction of vertex operators of SL(2, R) was sketched in [56] using the lan-guage of parafermions, where the last fact is more transparent. Chapter 3. Little String Theory 60 3.3.2 Restriction to SL(2,R)/U(1) To obtain the Euclidean cigar geometry, we need to take the modulus of SL(2,R)L <g> SL(2,R)R with respect to the U(l) subgroup generated by J = (Jl + JR) [51]. The gauging is described in detail in [52], and we follow their notation. So in the path integral I want to integrate over only those configurations of g for which J vanishes. Since this is a conformal field theory J decomposes into independent holomorphic and antiholomorphic parts JL and JR which must vanish separately. To implement this constraint I could put appear as Lagrange multipliers — they are auxiliary fields with no kinetic term. Alternatively, I could couple J to an external source A by hand, and then use the source as a knob to turn off the U(l) current. It is necessary to implement this using Gaussian-averaged gauging to preserve conformal invariance [52]. Also note that the current associated with the group element h = e 1 ^ 3 is (3.23) under the path integral. From this perspective the gauge fields A and A A = Tr ((T 3)*/i - 1d/i) = d<t>L (3.24) and similarly A = dcpR. Thus the gauging can also be described through a propagating scalar field <f>(z, z) — 4>L{Z) + 4>R(Z). This field has a wrong-sign kinetic term from the Gaussian averaging. It is useful to interpret this scalar Chapter 3. Little String Theory 61 field as subtracting the desired 17(1) field. From this construction it is also clear that <f> is periodic, tfi(z, z) ~ <f>(z, z) + 2im. The original gauge symmetry is manifested by the requirement that all physical operators must commute with the BRST charge Q = j dz c(4 + l-kd(f>L) + jdz c(JR - l-kdcf>R). (3.25) The vertex operators in £L(2,R)/T/(1) are of the form y = eiiL4>L{z)+i<m4>R(z)§Jm_i_ (3.26) The BRST invariance then sets qL = —m, qR = m. Since <j> is compact we also have q L ~ q R = n e Z , (3.27) k where n is identified in [52] as the winding around the asymptotic circle of the cigar. The conformal weights with respect to the unflowed L0, L0 of the vertex operator are h - i O " - 1 ) i JP + nkf i p n ~ k-2 + Ak ' h j p n - ~ 1 ^ 2 - + 4k ' ( 3 - 2 8 ) where we denote p = m — fh to be the momentum around the asymptotic circle. This is the spectrum found in [52]. Note that it is the Casimir operator with restricted domain which appears in the mass formulas, Jti - 1) P2 kn2 L0 - L0 = pn. (3.29) Chapter 3. Little String Theory 62 This justifies the interpretation of p, n as momentum and winding, respec-tively, around the Euclidean time direction. We are interested in exploring correlators as a function of Lorentzian mo-mentum. To do this we need to construct a series of allowed vertex operators with the different Euclidean momenta, to be interpreted as Fourier modes of a single object. It is therefore essential not to restrict the possible values of p, and for that reason we must choose the solution n = 0 to the level matching condition. This restriction means that However, we will see that the set of observables with n = 0 does not allow exploration of all possible momenta at finite k. This is essentially due to the same bound on the allowed off-shell momenta found in [47]. In the present context, this puzzle may eventually be resolved by making use of observables with n ^  0. We comment further on this point below. 3.3.3 Worldsheet Supersymmetry To supersymmetrize the theory, ten Majorana-Weyl fermions are added to each sector of the model; they are free after a suitable chiral rotation. Ma-jorana fermions are real spinors, and Weyl fermions are chiral. The chiral anomaly induced by the rotation changes the scaling dimension of vertex operators. The zero-point energy on the worldsheet is altered; the term ^ ~ 2 ^ is added to both hjpn and hjpn so that this scaling becomes m = —m = p/2. (3.30) 'jpn j(j-l) (p + nkf k 4k (3.31) Chapter 3. Little String Theory 63 and likewise for hjpn. The GSO projection used here would, in flat space, project out the tachyon. We are using the flat space result since we will be mainly inter-ested in the leading results in the large k limit, and in the limit k —> oo one recovers a flat space. This projection will break spacetime supersymmetry because it imposes different periodicity conditions on spacetime bosons and fermions around the compact Euclidean time direction. Spacetime fermions are anti-periodic while bosons are periodic. Requiring spacetime fermions to be anti-periodic, together with modular invariance, introduces new phase factors into the sum over spin structures. This was done by Atick and Witten [57] using the functions Ui(p,n), which modify the usual phases of Type II strings at zero temperature. Explicitly, these four functions are KM = ^(-i+(-ir+(-ir + (-ir"), U2(p,n) = + + U3(p,n) = + + ( - ! ) » - ( - 1 ) ^ ) , U4(p,n) = ^(l + ( - i r - ( - i r + (-ir"). (3.32) Using the definition Uj = Ui(p,n) for states with momentum p and winding n the desired one-loop partition function is given by the modular-invariant trace Tr (eirU [U3 - (- l)*^ - ( - 1 ) ^ 2 - (-l)F'+FTUi U3 - (-1)FtU4 - (-l)F°U2 ± (-l)F'+FrUi]) , where r are the left-moving fermion number operators acting in the cr, r Chapter 3. Little String Theory 64 directions and Ji = LQ + L0 — 2 is the Hamiltonian. The signs between the terms have been chosen to correspond to the Type IIA,B string. The vertex operators in which we are interested survive the zero-temper-ature GSO projection. The analysis in the previous section confirms that all their momentum modes survive, enabling us to discuss the two point function as an analytic function of frequency. 3.3.4 Mass-Shell Conditions Each one of our vertex operators carries quantum numbers corresponding to its dependence on various bulk coordinates. Since only the relationship between on-shell bulk quantities and off-shell boundary correlators is under-stood, it is necessary to solve the bulk mass-shell conditions for each value of the boundary quantum numbers. Even in the zero-temperature analysis of [47] this can only be done for a restricted range of boundary quantum numbers, and we will see similar issues arise here. In our case the boundary quantum numbers are the spatial momenta qi, and the Matsubara or finite-temperature frequencies p. The graviton vertex operator in the full target space is given by QiMuP) = e - ^ i ^ e * * * ' * ^ . (3.33) where <p, <p are the bosonized superconformal ghosts and X1,^,^ are the free bosons and fermions in the R 5 component of the full CFT. The in-dices i, j in this expression are symmetrized. The vertex operator $ j P is the SL(2, R)/U(l) vertex operator discussed above, with the winding set to zero. In writing gij(qi,p) as a function of qi and p only,, we imply that other Chapter 3. Little String Theory 65 quantum numbers are to be regarded as implicit function of those, given by the mass-shell condition. The mass-shell condition arises as the requirement that gij be a conformally invariant vertex operator. The conformal weight of the ghosts exactly cancels that of the fermions and so does not affect the mass-shell condition, which is £ - ^ + £ = 0. (3,4) This will be solved to give j as a function of the quantum numbers c/j and p. In particular, the bound on j translates here to a bound on those quantum numbers, exactly as in [47]. 3.4 Correlation Functions We are now ready to exhibit the thermal two-point functions of the operators we are interested in. The ingredients of this calculation were performed in [48], and reviewed in [47, 56]. In the notation of [56] the two-point correlator of functions (tachyon ver-tex operators) on SL(2,C)/U(1), the Euclidean version of SL(2,R), can be calculated in the x-space defined below, as <*,-(*i)*;ora)> = , * j £ ~ J ? f ° V ( 3 - 3 5 ) \zi2\ \zn\ F12I 3 where h, h are the conformal weights, and Xi2 = x\ — x2. The x basis is related to the m basis used earlier via r = / J^xi~mxJ~™$i(x)- (3-36) Chapter 3. Little String Theory 66 This gives [56] ird2(m + m')5(j - j')B(j) T(j + m)r(j - ra) <**»**>'mw> - \z12\2>>\zl2\2^(2j) T( l - j + m ) r ( l - j - m)' (3.37) The functions B(j) and 7(2.7) are defined in [48, 47]: B{j) = (3-38) and T M = ^ . (3.39, The constant v and the function B(j) both equal 1 in the large k limit we are taking. In calculating the spacetime two point function, one has to divide by the volume of the conformal group. This normally makes the two point function vanish, but in this context it is compensated by the volume of target space [58, 14, 56]. This leaves a finite piece, which was calculated in [56]. In making the correlation function of our vertex operators the addi-tional ingredients make little difference. They modify the conformal weights in the denominator of (3.37), and multiply the correlator by a polarization dependent tensor in the flat spatial directions. The two point function of the vertex operators gij(qi,p) is then given by (9ij{Qi,p)gM,p')) oc (8ikSji + 6u5jk)55(q + q')S(p + p') x x ^ 1 r ( j + m ) r ( j - m ) 2j - 1 7 ( 2 j ) T( l - j + m ) r ( l -j-fh) v ' ' Here ra, ra, and j are regarded as functions of the quantum numbers c/j, p. We now proceed to comment on the structure of this result and its analytic continuation to real frequencies. Chapter 3. Little String Theory 67 3.4.1 Euclidean Space The temperature Green's function can be written as a function of the Mat-subara frequencies p as i r2(| + | y i W T ? + | ) ( 3 4 1 ) * 7(1 + V/ T W + ? ) r 2 ( i - ! v / l + 2fcg2+p2 + £)-This is found using the conditions m = p/2, m = —p/2 (3.30) and the solution • 3 = \ + \ v / l + 2fcg 2+p 2 (3.42) to the mass-shell condition (3.34). A second solution to the mass-shell condition is discarded as it violates the lower bound on j. The upper bound imposes a constraint on a combination of the Euclidean energy .p and the transverse momentum q, y/l + 2kq2 + p2 < k - 2. (3.43) There is an additional condition m = j — n = p/2, for n an integer of definite sign, since those are the allowed values in the discrete representations. Positive n corresponds to a highest-weight representation, while negative n corresponds to a lowest-weight representation. This translates, using the mass-shell condition, to ka2 ^ - + n 2 = j ( 2 n - l ) . (3.44) Since j is positive it is clear that n must be positive as well, so only the highest-weight representations are allowed. So we see that only a discrete set of points in the frequency-momentum space is obtained by the Euclidean string theory calcualtion. Our approach Chapter 3. Little String Theory 68 is to analytically continue the result as needed, and attempt interpretation of the correlator as a function of Lorentzian frequencies8. 3.4.2 Minkowski Space The temperature Green's function Q{p) can be analytically continued to the retarded Green's function GR(LO) using p —> —iio + 5. The analytic continu-ation of Q is not unique since it can, for example, be multiplied by a phase which vanishes at the poles. Typically one requires a fall-off condition for large Euclidean frequencies in order to fix the analytic continuation uniquely. These fall-off conditions require knowledge of the correlators at arbitrar-ily high Euclidean frequencies. As our frequencies are a priori bounded for any finite k, we are unable to discuss those fall-off conditions. Since LST is only defined for finite k, taking k —> oo would not give reliable results. Fur-thermore, because LST is non-local, it is not clear what the fall-off constraint should be. Even without modifying the correlator found by Teschner, which is ex-act in tree level string theory, it is possible that small modifications to the mass-shell condition (3.34) will result in satisfying the asymptotic fall-off conditions. We assume this to be the CcLSGj clS mentioned a real time calcu-lation of the retarded Green function will be useful to confirm or refute this point. We assume therefore that the form of the correlator (3.41) applies to 8 We note that the original derivation of the correlators [48] utilizes analytic continuation as well. It would be interesting to explore alternative Lorentzian continuations, along the lines of [59]. Chapter 3. Little String Theory 69 Lorentzian frequencies as well. So the retarded Green's function is taken to be R ~ T ( l + .y/l + 2kq2 - w 2 ) r 2 ( | - + 2fcg2 - w 2 + f ) ' In a retarded Green's function an instabilty will manifest itself as a pole (or other type of singularity, for more complex instabilities) below the real axis. This will be a process which grows, rather than decays, as a function of time. As we are looking for modes which may destabilize the system upon including arbitrarily small corrections, the interesting structure for our purposes occurs on the real axis. This structure represents modes that are stable in the thermal ensemble, an unusual feature. The retarded Green's function found above has only one set of singu-larities, coming from the factor T(—y/\ + 2kq2 — u2). The singularities are located at u = ±y/l + 2kq2 - a2, (3.46) where a is a non-negative integer. The positive root is discussed below; the negative root corresponds to the existence of anti-matter. The singularities are located along the real and imaginary axes. The points along the real axis are bounded (for generic9 spatial momenta u < y/l + 2kq2. (3.47) We see that a large number of possible instabilities is present for non-zero spatial momenta q2, a number of order kq2. We note that LST contains a 9There are some more possibilities at special kinematic points, but as those will be sensitive to small corrections, we concentrate on generic kinematics. Chapter 3. Little String Theory 70 continuum of modes above a gap, whose role in the theory is unclear. A typical momentum of those modes, just above the gap, is q2 ~ Therefore there are order k possible "nearly destabilizing" modes, and nearly all of them are above the gap. The frequencies of those modes are much smaller than k, so we expect them to be reliable. For zero spatial momentum and no winding there is one possible mode, at oo = l . 1 0 The retarded Green's function also has a set of zeros. These are on the imaginary axis, so in particular they can eliminate a possible divergence only for special values of qi, the spatial momentum. Thus we find evidence that LST could indeed be unstable at the Hagedorn temperature. We see a similar picture to the one suggested in [14], where the instability depends on the spatial volume, or the spatial momentum in our case. This picture is one of being more likely to be unstable at small volume, or large momenta. However the locations of the poles we find could be sensitive to our choice of analytic continuation, as discussed earlier. It would be useful to study the choice of continuation in more detail. It would also be interesting to study the corrections in 1/p to see if they shift the poles below the real axis and confirm that LST really is unstable. An additional potential pole at w = 0 is eliminated by a zero at that frequency. Chapter 4. Inhomogeneous D-brane Decay 71 Chapter 4 Inhomogeneous D-brane Decay 4.1 Introduction As described in the introductory chapter, one of the outstanding questions in string theory is whether open strings and closed strings are independent degrees of freedom. If we start with closed strings, we can build D-branes as solitonic solutions of gravity — so D-branes appear as non-perturbative states built from closed strings. The D-branes are dynamical, and their low-energy excitations are open strings. But how can we see these open strings explicitly in the closed-string sector? Similarly, we can start with the open-string worldsheet. The D-brane living on the boundary of the worldsheet decays, leaving behind only closed strings — how do these closed strings appear on the open-string worldsheet? This question is the counterpart to the one in the previous paragraph. D-branes seem to provide the connection between open and closed strings. D-branes were first understood as sets of endpoints of open strings. Strings ending on a D-brane have Dirichlet boundary conditions in the directions transverse to the brane, so that the strings cannot leave the brane. They have Neumann boundary conditions in the directions parallel to the brane, so that they are free to move along the brane. The special case of Neumann Chapter 4. Inhomogeneous D-brane Decay 72 boundary conditions in all directions is given by a space-filling brane. Open strings only appear in this way, and they are the low-energy excitations of the D-brane. Although one can use the GSO projection to remove tachyonic modes from the superstring, it is not possible to remove all tachyonic modes. Since open strings always end on D-branes, one can say this another way: the GSO projection can be used to stabilize only some types of D-brane. So the question of understanding the open strings becomes one of understanding decaying D-branes. D-brane decay is perhaps most clearly understood in the boundary C F T (BCFT) formalism. The open string is described by a C F T on a two-dimensional worldsheet with a boundary, unlike the closed-string worldsheet, which has no boundary. The Dirichlet boundary conditions can be imposed through a boundary interaction term. The changing state of a D-brane as it decays is then understood as a time-dependent boundary interaction. We can also try a field analysis. The string field is a wave-function ex-panded in the free-string Hilbert space. However there are an infinite number of fields so it is not clear that perturbative string field theory is well-defined. There are contributions from an infinite number of Feynman diagrams even at first order in the string coupling, so one expects this term to be large even at small string coupling. In the language of string field theory, the tachyonic character of the low-est string mode is interpreted as saying that this spacetime field is being expanded about a local maximum in its potential. Based on this analysis, Sen proposed several computations that test vthe reliability of string field Chapter 4. Inhomogeneous D-brane Decay 73 theory, given some basic assumptions of the tachyon potential. Sen conjectured [5] that the height of the local maximum of the tachyon potential is given by the D-brane tension, and that the minimum of the tachyon potential is the closed-string vacuum. He suggested that lower-dimensional D-branes be thought of as solitonic solutions of string theory on the background of a space-filling brane. More recently, he proposed an explicit form for the boundary interaction. This chapter considers a more general interaction than the simple case that Sen studied. Before delving into the details, it is useful to review what is already known about the dynamics of stable D-branes. Then I introduce unstable branes and discuss Sen's conjectures. I review the radiation emitted during the homogeneous decay, and lastly I calculate the radiation emitted during an inhomogeneous decay. 4.2 Dynamical D-branes D-branes are dynamical objects that act as sources for closed strings [60]. This can be seen as follows. The one-loop open-string diagram is equivalent to a closed-string tree-level diagram. The one-loop open-string diagram is a cylinder, with the length corresponding to the length of the open string. Since an open string ends on a D-brane it is clear that the cylinder can also be attached to a D-brane. In the time direction one sees the loop as the cross-section of the cylinder. On the other hand, viewing the length of the cylinder as time, one sees the tree-level propagation of a closed string. But then this represents the emission of a closed-string from the brane. Chapter 4. Inhomogeneous D-brane Decay 74 time space 1 •space open string 1 •time closed string Figure 4.1: Closed-string radiation from D-brane. D-branes appear in this context as boundary states, which are sources for closed strings. A finite tension, or mass per unit volume, can be defined for a D-brane; it is calculated to be inversely proportional to the closed-string coupling. Thus at weak string coupling the D-branes are essentially static objects. The open strings ending on the branes are the quantum excitations of the brane. Since the scalar mode of the open string is tachyonic, D-branes are generically unstable. In superstring theory, the GSO projection eliminates strings that couple to certain RR fields. This stabilizes some D-branes in Type II superstring theory by removing the tachyonic fluctuations. There is another important way to see the existence of stable D-branes. Closed strings couple to the RR field strength, and thus carry off multipole moments from the brane. Polchinski recognized that this shows that stable D-branes in superstring theory must be sources of RR charge. The RR charge must be conserved, and since other objects in string theory do not carry RR charge one would not expect these D-branes to decay. Chapter 4. Inhomogeneous D-brane Decay 75 In fact, stable D-branes are BPS states. In the presence of a supersym-metric Lagrangian, all states are subject to the BPS bound, which restricts the charge of the state to be less than its mass. States that preserve some su-persymmetry saturate this bound and are known as BPS states; their charges and masses are equal. Stable D-branes and their bound states are important examples of BPS states. BPS states are special because their masses are protected by supersymmetry; the spectrum calculated at weak coupling is also present at strong coupling. One way to understand that a stable D-brane is a BPS state is as follows. The D-brane relates to each other the left- and right-moving supersymmetry generators on the string worldsheets, so there are fewer independent genera-tors — an isolated D-brane breaks half the supersymmetries. If the unbroken supersymmetries are not projected out by the GSO projection, then the D-brane is stable and the configuration preserves half the maximum number of supersymmetries. Since the D-brane is massive and carries only a RR charge, the RR charge must be equal to the mass, which as mentioned above goes like the inverse closed-string coupling. If, on the other hand, the unbroken supersymmetries are removed by the GSO projection, then there are no remaining supersymmetries to stabilize the D-brane, and it does decay. The BPS bound is trivially satisfied but not saturated, since unstable branes are massive and neutral. In this case one can also see the decay by noting that open-string tachyonic modes exist. Stable D-branes are an important part of the string spectrum. They are required by many dualities, including T-duality and S-duality. At large string coupling there is no clear sense in which D-branes are the endpoints Chapter 4. Inhomogeneous D-brane Decay 76 of open strings. Of course, open strings themselves are not defined at strong string coupling — they are the weak-coupling quanta. But we still know the D-brane masses and their low-energy excitations, or dynamics. 4.3 Unstable D-branes Now that we know of the existence of stable D-branes, we can ask what there is to learn about unstable D-branes. Unstable D-branes certainly exist in superstring theory, since pairs of oppositely charged branes are not protected by the BPS bound. There are also uncharged branes, which are intrinsically unstable — the bosonic D-branes are also in this category and serve as useful toy models. Depending on the dimensionality of a D-brane in superstring theory, there may be no RR form for it to couple to, since the GSO projection removes either the even- or the odd-form RR potentials. A Dp-brane1 couples to a (p + l)-dimensional RR potential and thus a (p+ 2)-form field strength. Thus in Type IIA theory odd D-branes must be neutral, and their masses cannot be protected by the BPS bound; likewise in Type IIB string theory the even D-branes are neutral. The tachyonic open-string mode living on these D-branes is not projected out by the GSO projection. These D-branes are unstable and do decay. DD pairs2 are always unstable when the distance between the branes is 1 This is a D-brane with p spatial direcions, along which the string satisfies Neumann boundary conditions. The open string is free to move along the D-brane but cannot move transverse to it; the Dirichlet boundary conditions appear in the remaining 9— p directions. 2 A D-brane is simply a D-brane with total charge opposite to that of the other D-brane Chapter 4. Inhomogeneous D-brane Decay 77 small. We can see this immediately, because the pair is massive but neutral. The instability is realized by the open string stretched between the D-branes. The scalar modes that live on just one brane are projected out, but the scalar mode of the string stretched between the two branes is not. The zero-point energy of an open string is the unstretched tachyon mass. When the D-branes are far apart the contribution to the mass of the stretched strings from the tension dominates and the D-branes are stable, although they attract one another. But when they are close together the negative zero-point energy dominates, and the lowest open-string mode is tachyonic. In all cases the unstable D-branes do not satisfy the BPS condition.. This is because there is no RR flux for these branes to couple to, and thus they do not carry any RR charge — but they still have a nonzero mass. One can see that the tachyonic open-string mode coupling to these D-branes is not projected out by the GSO projection. Thus generically one should expect them to decay to the lowest energy state having the same quantum numbers. Once the D-brane has decayed we do not expect to find open-string excitations, so we expect the unstable D-brane to decay to an excited state of the closed-string vacuum. 4.4 Sen's Conjectures Sen conjectured that the result of D-brane decay will be the closed-string vacuum [5, 61, 6, 62, 7]. As a corollary he noted that the potential difference in question. This is an oriented string theory, so opposite charge is realized by opposite orientation in spacetime. Chapter 4; Inhomogeneous D-brane Decay 78 between the open-string and closed-string vacuum energies must then be the D-brane tension. He also proposed that lower-dimensional D-branes should be thought of as solitonic solutions on the background of a space-filling brane. These conjectures are straightforward, but their importance lies in the sug-gestion that there is a field-theoretic formulation of string theory that can test these conjectures. We can also use open-string field theory to see how much energy is ra-diated away. Since the mass of the D-brane is 1/g the available energy is infinite to zeroth order in g. Hence a finite answer would imply that not all energy is being radiated away. Using compact D-branes, Sen showed that the height of the tachyon ef-fective potential is universal [5] — it is independent of the precise form of the boundary interaction describing the decay. The effective potential of T is a function whose minima gives the exact value of (T) [9] (Fig. 4.2). The height of the potential is determined strictly by zero-momentum correlation func-tions. Such correlation functions on a space-filling D-brane can only probe the volume of spacetime, which is certainly independent of the interactions in the theory. It would be useful to see this from the perspective of open-string field theory. The worldsheet action is in terms of quantized oscillators living on the worldsheet. This gives a quantum mechanical single-string theory. From this we can try to construct a string field theory, based on a field \f/ whose modes are strings, * = j d26p [<fi{p) :eipX: + AM(p) :dX»eipX: + . . . ] . (4.1) The coefficients of the vertex operators represent spacetime fields — the Chapter 4. Inhomogeneous D-brane Decay 79 Figure 4.2: Tachyon potential in bosonic string theory. tachyon <f>(p), the vector A^, and so forth. Classical string field theory is first-quantized in the sense that [X, P] is non-zero, but the spacetime fields themselves are not quantized. The string field ^ takes values in the Hilbert space of quantized strings. To describe more than just the endpoints of the decay process, Sen pro-posed an explicit form for the boundary interaction [6]. Formulating this in both string field theory and the worldsheet boundary CFT allows the study of the transition in real time. He considered the simplest case, the spatially homogeneous decay, in which all points on the spatial extent of the D-brane decay in precisely the same manner. Rather than solve the equations of motion nonperturbatively in the string coupling, Sen considered only the linearized equations of motion. The closed strings emitted from the brane interact with one another, and this can change Chapter 4. Inhomogeneous D-brane Decay 80 the background NS fields such as the metric. The linearization ignores this back-reaction of the closed strings. Normally to construct a Wilsonian effective action we would integrate out all the higher modes by solving their equations of motion. Here we don't know explicitly the equations of motion for all infinitely many fields so we just truncate the action. This gives some effective action for the tachyon, Seff = J d2epd>(p) (• - m2eff) 4>{-p) + Veffifo)), (4.2) where Veff is the cubic effective potential of the tachyon. There is a prescrip-tion for finding Veg. Using this one finds the minimum <j)min and calculates Veff((f>min) = —m-D-brane- An additive constant that is important for coupling to gravity has been dropped from the action. In [61] Sen was able to compute the effective potential in level-truncated open-string field theory, and he found amazing agreement with the conjecture that the depth of the tachyon potential is the D-brane mass. Level-truncation is a truncation that utilizes only the first few excited states of the string and only the first few interaction terms. This analysis was later refined by Taylor [63], and by Giaotto and Rastelli [64]. It is not completely obvious why level truncation works. An infinite number of terms contribute to the coefficient of the cubic term in the effective potential, and level-truncation only considers a finite subset of these. The effectiveness of level truncation might be rationalized by reasoning that very massive modes contribute only a small amount to the coefficients of light modes. Corrections in the string coupling would be very nice to study, but no-one has be able to calculate these corrections yet. The major obstacle here Chapter 4. Inhomogeneous D-brane Decay 81 is that the string loop corrections must be calculated off-shell [65]. 4.5 Bosonic Homogeneous Decay Now that D-brane statics are better understood, let us consider the dynamics of a decay. To do this, we will first find the classical form of T[x°], and then use this to do a field-theory analysis of the closed-string radiation emitted during the decay. The bosonic worldsheet action can be written as d<TdTZbulk[X»(o,T)]+ / d r £ 1 [ ^ ( c r = 0,r)]+ / d r£ 2 [X^(a = T T , T ) ] (4.3) where £buik is the worldsheet bulk Lagrangian and £ 1 , 2 are the boundary interactions on the two open-string boundaries. The worldsheet has a Eu-clidean metric, but we think of r as the time coordinate and a as the spatial coordinate. To model the decay of a single D-brane there will be an interaction on only one boundary, so we can set £ 2 = 0. In the simple case that Sen considers, the bulk Lagrangian is the free Lagrangian, Zbuik = Glxv{X)&iX'>d0lXv, (4.4) and £ x is simply £}[X°{r)]=T[X0(r)]. (4.5) For simplicity a possible two-form has been omitted. This is the same T that appears in the expansion of ^ , where it was called <f>. So T is the string field corresponding to the spacetime tachyon Chapter 4. Inhomogeneous D-brane Decay 82 mode, which lives on the boundary. In the simple homogeneous case it is not given the freedom to depend on the spatial X1. Now the question is what form we should use for the source term T. For this, we go back to the open-string field theory action (4.2), and consider a small perturbation around the vacuum. Around the maximum of the effective potential, the equation of motion for the tachyon has the form and the corrections of order T2 are ignored in the linear approximation. Since this is an inverted harmonic potential, the solution grows with time, instead of oscillating. The classical solution of (4.2) is then T[X°(T)] = A smh(kX°) + B cosh(fcX°). (4.7) Let the initial condition be T(X° = 0) = A, and take the initial gradient to vanish. Also take A > 0 so that the tachyon rolls to the local minimum and does not run away3. Thus T[X°] = Acosh(fcX°). (4.8) But this is only valid to first order in A, so when it appears in the path integral as e^fdrcosh(kx0) t h i g should really be written as 1 + A / drcosh(fcX°). The Wick-rotated version of this perturbation is the sine-Gordon pertur-bation, which is known to be exactly marginal at the frequency k = 2/\/o7 3 I f A < 0 the tachyon rolls down a bottomless potential. This solution develops a singularity in finite time. Since the bosonic string is a toy model of the superstring, which does not suffer from this instability, there is no need to consider further the A < 0 behaviour. Chapter 4. Inhomogeneous D-brane Decay 83 since this solves the equation of motion. Therefore this theory can be solved to all orders in A with only a renormalization of the coupling [66, 67, 68]; no new operators are introduced. The full solution to the linearized equation of motion is thus T[X°] = Xcosh(kX°) (4.9) where A is the renormalized coupling, which can be found perturbatively in A. This acts as a source term for closed strings, and induces radiation. The one-loop open-string vacuum amplitude gives the closed-string radiation at tree level. The boundary state is the unique closed-string state that satisfies the open-string boundary conditions. With an interaction on the boundary, |B) — e s ' n i[*°J \N) where \N) is the boundary state of the static space-filling D-brane4. So knowing the form of the source T[X] lets us find |J5). This was done by [67, 68]. 4.6 Radiation The one-loop open-string diagram is the disk amplitude with the boundary state at the boundary of the disk; this is a tree-level closed-string diagram. Thus closed strings emitted from the D-brane satisfy an equation of motion with the boundary state as the source. One can calculate the radiation emitted from a D-brane during its decay. The radiation is preferably emitted in a single string, since multiple-string 4 The boundary state |JV) imposes Neumann boundary conditions in all directions. Chapter 4. Inhomogeneous D-brane Decay 84 emission, that is, successive single-string emission, is suppressed by a power of the string coupling. It is helpful to first consider the much simpler case of an ordinary free scalar field in flat space. The equation of motion for a free scalar field <^o takes the form [9] •0o = 0 (4.10) where • is the flat-space d'Alembertian. The second-quantized solution to this equation in D spacetime dimensions takes the form where at, hp are the creation and annihilation operators respectively for the mode p on mass-shell, and they satisfy the usual continuum harmonic oscillator algebra [hp, at] = (2TT)d~16d~1(P — q). The energy of the particle with a given spatial momentum p is Ep. In the presence of a classical source term J the equation of motion is modified to be U4> = J. (4.12) This modifies the quantization of <f). One way to find the new creation op-erator is by using the method of Green's functions to write the new field as m = ux) + ifdDyf £ ^ w d { x ° ~y0) ( e _ i P ( x _ y ) - eiP(x~v)}J{y) (4.13) where <J>Q is the solution to the equation of motion without a source. The step function 6(x° — y°) preserves causality. If the source J is non-zero only from Chapter 4. Inhomogeneous D-brane Decay 85 y\ to y°f then when x° > y°f the step function is always one. Then writing the Fourier transform of J(x) as J(p) and expanding <f)0 as in (4.11), one finds (4.14) from which it is clear that the new operators are i dp ->• ap = dp + —^==J(p) ^ -+ o t = a t _ • J . ^ ) . (4.15) a p p p ^2EP The number of particles created by the source J from the vacuum is given by the expectation value of the number operator N = a^pap, because (0| a)pap |0) = 0 and the cross-terms also vanish. So the probability density for creating the mode with momentum p from the vacuum is just In general the field <f> might transform nontrivially under a group action, in which case it will carry additional indices, as will ap and at; these must be contracted with corresponding indices on • . The field <j> will also be a function of the quantum numbers that identify it as a unique physical state. The momentum integral generalizes to include integrals over all continuous quantum numbers and sums over the discrete quantum numbers. Since the source depends on the quantum numbers as well, the first phys-ical question to ask is how the probability density depends on the quantum numbers. This is the question that this research explores, for a particular example of D-brane decay. Chapter 4. Inhomogeneous D-brane Decay 86 Normally one might think of this dependence as the physical input for a calculation, but in this case the source is already defined in the theory, at least in terms of the open strings. The question is how the source depends on the quantum numbers of the closed strings. Alternatively, the question can be seen as asking how to construct such a source from physical closed-string states. Either way, the calculation amounts to calculating one-point functions for the physical closed-string states created by the source. The interaction between the boundary state and the massive closed-string states grows exponentially in time. Thus it is necessary to understand the radiation of closed strings of all mass levels. This analysis was performed in [69] for the homogeneous decay. The au-thors found an exact cancellation between an exponential suppression of the amplitude at large y/~N and the Hagedorn density of states. The resulting average particle number and energy have logarithmic and power-law diver-gences for small-dimension D-branes as expected. But the emitted energy is finite for large-dimension D-branes, and this presents a puzzle. It means that the homogeneous decay of a space-filling brane does not radiate all its energy into closed strings at leading order in the string coupling. But the homogeneous decay is not a very likely scenario for the space-filling brane. It would be nice to see that an explicitly inhomogeneous decay does bet-ter. The generic decay would certainly be very difficult to study, but as a step in this direction, we consider a decay that is inhomogeneous along just one spatial direction [7, 8]. This should leave behind an array of D(p — 1)-branes. This analysis is accurate for Dl-branes, and hopefully improves on Chapter 4. Inhomogeneous D-brane Decay 87 the situation for larger branes. We hope to see divergent energy being re-leased by large-dimension branes. This would lend support to the idea that space-filling branes decay completely. 4.7 Bosonic Inhomogeneous Decay 4.7.1 Introduction The homogeneous decay is a very special fine-tuned case. But the most general decay will be much more difficult to describe. To gain a better un-derstanding of the general case, a simple inhomogeneous decay is considered here — the decay of a space-filling brane with one inhomogeneous direction. Others [7, 8] have discussed a simple inhomogeneous decay at the massless level. Sen finds that the end-point of the decay is similar to an array of D-branes. But the energy density of the new state does not quite match that of an array of D-branes, and in fact has additional energy. Sen has called the additional energy "tachyon matter." In the closed-string language this matter is simply the closed-string radiation that has not reached infinity. This description has been made concrete in two-dimensional string theory. In this research I calculate the emission amplitude, in bosonic string the-ory, of massive closed strings radiated from a D-brane due to an inhomoge-neous decay. The boundary interaction can be described as a perturbation on the back-ground of a space-filling brane. Neumann boundary conditions must be im-posed along all directions, to model the space-filling brane. The boundary Chapter 4. Inhomogeneous D-brane Decay 88 interaction describing the perturbation along the X° and X1 directions is given by SM = ~^ J dreuX° cosuX1, (4.17) where A is the renormalized coupling, UJ is the inverse length scale of the boundary interaction, and r is the coordinate on the boundary of the world-sheet. The time factor of the interaction, euX°, differs from C O S 1 I ( W A L 0 ) in that it describes the decay of a D-brane but not its formation. In this inhomogeneous example only the decay can be studied to all orders in the coupling. The boundary interaction can be separated into ~-e^u + ~-e^v, (4.18) where U, V = (X° ± iXl)/\/2. As a result of this structure the boundary state factorizes between the Euclidean light-cone directions, and it will be convenient to choose light-cone coordinates. The boundary interaction is exactly marginal, and these operators commute with each other, when u = 1/v^cV. 4.7.2 Out-Going Closed-String Radiation The emitted radiation is created by some vertex operator V. For V to create a physical state \4>) ~ V(0,0), the state \(f>) must be annihilated by the positive-frequency Virasoro generators L„, n > 0. These constraints impose conservation of worldsheet energy and momentum, at least in expectation values. You can also see these constraints as arising from the. equation of motion due to variation of the worldsheet metric. Chapter 4. Inhomogeneous D-brane Decay 89 Using the conformal symmetry of the worldsheet, a gauge condition can be imposed on the oscillators. It is convenient here to remove all the U oscillators, so that U(z, z) = u; only the zero-mode piece u is left. This is a Wick-rotated version of the more common light-cone gauge. With this gauge choice, the positive-frequency Virasoro generators de-compose as Ln = L\ + Lnp where acts only on the oscillators in the V direction, and Lnp acts in the remaining spatial directions. The states that are annihilated already by take the particularly simply form V = eiquUeiqvVeiqvVVsp (4.19) since these states can have no V oscillator insertions. The operator Vsp is a Virasoro primary. The Virasoro constraint removes one degree of freedom, so Vsp has 23 and not 24 independent modes. There are 24 independent polarizations for the out-going modes. The missing polarization is given by states which are not annihilated by but are nonetheless annihilated by Ln. These states involve oscillator excitations along both V and the other spatial directions. The solution to the Virasoro constraint in this case complicates matters considerably. At high energies we expect the emission amplitude to be independent of polarization. Thus we will calculate it only for the first 23 polarizations and assume that the last polarization will not significantly change our conclusions. Chapter 4. Inhomogeneous D-brane Decay 90 4.7.3 Emission Amplitude So now I have to calculate (V) = j dudveiquueiqvv (j9**} (Vtp). (4.20) In both the U and V directions the state is only a plane wave, so the amplitude should be symmetric under U «->• V. The fact that our gauge choice does not treat U and V symmetrically does not change the physics of this. The required amplitude was first calculated by Sen [6] and found to be (% = / («) = — i - . (4.21) 1 + n\eu Thus also (eiqv^ = f(y). As pointed out in [69], the amplitude in the spatial directions gives only a phase. As physical quantities always depend on the complex square of the amplitude, there is no need to calculate the phase. Thus (V) ~ j dudveiquueiqvvf(u)f(v) (4.22) As we wish to interpret the result in real time, we first rotate back to the usual coordinates: u = (t + iy)/y/2 and v = (i — iy)/y/2. Here t is the zero-mode of X°, corresponding to time in target space, while y is the zero-mode of X1, the spatial direction along which the inhomogeneity lies. Rather than do both integrations in (4.22), we will Fourier transform from c/i to y to get V(E;y). We see that the amplitude (4.22) diverges at finite time at certain real physical locations. Away from these divergences, which are understood to represent the formation of lower-dimensional D-branes, the Chapter 4. Inhomogeneous D-brane Decay 91 amplitude is finite and can be integrated over all time to arrive at a function of energy. The remaining integral is (V(E;y)) = [ dteim ^ — (4.23) J (1 + ae73)(l + a*e^) where a = irXe^. Away from y = \/2irn I can do a partial-fraction decomposition of this, to get /AEt p AEt dt - + A* / dt j- (4.24) 1 + ae^ J l + a*ev/2 where A=(l- e " ^ ) - 1 . The poles in the i-plane, for fixed y away from y/2im, are simple poles at t = - V ^ l n ( 7 r A ) - » y + M r v ^ ( 2 n + l ) (4.25) where n is any integer. Thus [dt fEtr r = 2 n r e ^ ( - h ( ^ - i ) ) = 2me-iV2Eh1^ £ e-^B{2n+1-^].{4.26) UHP The poles are in the upper half-plane (UHP) when 2n + 1 — y/(\/27r) > 0. We can restrict y to the range — y/2n < y < V^TT without loss of generality; the integral is periodic under y —> y + 2y/2ir as expected from the form of the interaction (4.17). Thus the sum is ^ sinh( v / 27r£;) The sum only converges for e~s^2l'E < 1 but this can be continued to all E. Chapter 4. Inhomogeneous D-brane Decay 92 Plugging this result into the integral (4.26) yields / eiEt e-iV2Eln(n\)+yE dt ^ — F F = 4m F . (4.28) 1 + n\{*v/>/2et/V2 smh(V2irE) The emission amplitude is now /piEt r JEt dt ^ + A* / dt j -= ~ 8 7 r e (eiy/V2£yE _e-iy/Vle-yE\ sinh(v^7r£?) sin ^ V / for - V ^ T T < y < V ^ T T , and V(y) = V(y + 2y/2ir). At the maxima of the potential, for example at y = 0, the amplitude needs to be recalculated. At y = 0 there is a double pole in the integral, so we get <V(ig;y = 0 ) ) ~ (4-30) sinh(y27r.E) which is finite. This can also be seen by taking a limit of the amplitude found above. Thus the amplitude is continuous at y = y/2~Tr(2n). On the other hand at the minima of the potential, y = ^/2ir(2n + 1), the amplitude really does diverge. In [8] the stress-tensor was also found to diverge at these points. These divergences are interpreted as corresponding to the formation of codimension-1 D-branes. 4.8 Analysis At large E and fixed y, the squared amplitude behaves like Chapter 4. Inhomogeneous D-brane Decay 93 Thus the amplitude depends on position, unlike in the homogeneous case. As a result the exponentially growing density of states is not exactly cancelled, and we can expect the total radiated energy to diverge for some region of y. The number of emitted particles (4.16) is f=E^I<V.>| 2 (4-32) and similarly the emitted energy density is s The large-energy states are closely spaced and contribute most to the sum over states s, so we send Yls ~~I dEp(E) where p(E) is the density of states. Since we are not counting all polarizations here, the correct density of 24 /— 23 /— states to use is not p ~ e"^ "7 "^ but rather p ~ e"8" 7 r V n. Using the mass-shell condition we can also write the energy in terms of level number as E ~ 2y/n. Up to polynomial prefactors, the integrands of the total energy (per unit length) and number are then e ( M 7 r _ 4 v ^ 7 r + 4 | s , | ) > _ (434) Far away from the D-branes, where y is small, the integrand is dominated by the exponential decay of the emission amplitude. However, close to the D-branes the decay is much softer; in fact the integrand diverges for large E due to the large density of states. Although the amplitude is finite everywhere between the D-branes, the emitted radiation density diverges in finite regions close to each D-brane. Chapter 4. Inhomogeneous D-brane Decay 94 There are finite regions between the D-branes where the integrated ra-diation density appears finite. It is possible, but seems unlikely, that the missing polarization will contribute substantially in these regions. A more likely scenario is that those regions will disappear when one includes further inhomogeneities. Since the radiation density diverges on finite intervals close to the D-branes, the total emitted radiation must be infinite. Thus we do find that an infinite amount of energy is emitted for the space-filling brane undergoing this inhomogeneous decay. This lends support to the conjecture of [69] and others that D-branes decay completely, despite the fact that the space-filling D-brane does not decay completely via the homogeneous decay. Our result is robust in the sense that it does not depend on the finely tuned cancellation of exponentials. This means that all lower-dimensional branes should also decay completely if they have a sinusoidal spacelike inhomogeneity. It is known that a space-filling D-brane undergoing homogeneous decay does not decay completely [69]. We find the largest decay rate close to the new lower-dimensional branes. We believe this clumping of radiation implies that formation of lower-dimensional objects is a more efficient decay process than the homogeneous decay. Thus we expect a typical decay to involve a cascade of branes decaying to lower-dimensional branes. Note that the array of lower dimensional D-branes has less energy than the initial space-filling brane [7]. 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