Симметрии и точные решения в теории гетеротической струны тема автореферата и диссертации по физике, 01.04.02 ВАК РФ

ei^d-i/ôoï-ï

Joint Institute for Nuclear Research

Laboratory of Computing Technique and Automation

Зрр^ра - Агиллр Альфре-до

СимметЫ тоЦйые решения 6 теории

гетеротмЦес&ои. струны

Symmetries and Exact Solutions of Heterotic String Theory

Alfredo Herrera Aguilar

Т. НАУК

û-f.ûkoZ

A Dissertation Submitted to the Bogoliubov Laboratory of Theoretical Physics in Candidacy for the Degree of Doctor of Philosophy

Dubna 1999

j\

/i/Is

То ту mother

It is difficult for me to express in words the real feelings of gratitude that I owe to many people who, in one form or another, have played such a significant role in my studies. The point is that when you are studying abroad, it is not only your studies but your whole that is abroad. This means that I do not only have contact with teachers, professors and scientists, but also interact with a whole set of people, who have greatly supported me in my studies and life in Russia. However, there are several persons who played a crucial role in this stage of my life and I want to make some special mentions.

First of all I would like to thank my advisor and friend Dr. O. V. Kechkin for introducing me to my present field of research, for his great patience to me during my learning and for sharing with me not only his time in hours and hours of work, but also his valuable friendship.

I would like also to thank N. Makhaldiani for his help during the preparation of my candidacy exams, for inspiring me a general interest in science and for his helpful discussions on my work.

I am really grateful to Profr. V.G. Kadyshevsky for giving me the opportunity of carrying out my Ph.D. training at JINR and supporting my application for a fellowship. I really appreciated the unique atmosphere of the Institute and its ability to inspire research students from around the world.

It is a pleasure to thank Profr. B.S. Ishkhanov and Profr. I. V. Puzynin for creating such favorable working conditions both at DEPNI, NPI-MSU and LCTA, JINR; for stimulating the collaboration between both Institutes and for encouraging me during my studies. Special thanks are due to all my former teachers at Russian Peoples' Friendship University and, in particular, to Profr. G.N. Shikin, my first advisor, and Profr. Yu.P. Rybakov for their support.

Particularly warm thanks go to those people with whom I shared part of my life outside the physical community, I mean my family and friends. Perhaps I received the biggest support from them and for this reason, I would like to say: thanks a lot to everybody!

Finally I acknowledge the financial support from JINR, CONACYT and SEP.

Contents

Introduction 3

1 String Motivated Gravity Models 11

1.1 From String Theory to M-theory................................11

1.2 Effective Field Theory of Heterotic String ......................18

2 New Formulation of Heterotic String Theory 22

2.1 New Chiral Matrix................................................23

2.2 Matrix Ernst Potentials ..........................................27

2.3 Israel-Wilson-Perjes Solutions........................32

2.3.1 N=4, D=4 Supergravity..................................39

3 Symmetries of Heterotic String Theory 40

3.1 Charging Symmetries..............................................41

3.2 Linearizing Potentials..............................................46

3.3 Class of Invariant Fields under Charging Symmetries..........51

3.4 Solution Generating Technique................. 53

4 Truncated Models 56

4.1 Ernst Formulation of EKRD Theory ............................57

4.1.1 Dualization Procedure............... . . . 59

4.1.2 Axisymmetric Case.......................60

4.1.3 0(d+l,d+l)-symmetry in SL(2,R) Form................62

4.2 0(2,2)-Symmetric EKR Theory..................................64

4.2.1 Kramer-Neugebauer Maps................................66

4.2.2 Double Ernst Solution........................68

4.3 Ernst Formulation of EMDA Theory............................71

4.3.1 Charging symmetries.................. . 72

4.3.2 Linearizing potentials......................................74

Conclusions 79

Appendix A 81

Appendix B 84

Bibliography 85

Introduction

Superstring theory has been one of the most promising candidates over the past years for a theory that consistently unifies all fundamental forces present in nature; this implies having a consistent quantum theory of gravity. The elementary objects of this theory are one-dimensional strings whose vibration modes should correspond to the usual elementary particles. At distances large with respect to the size of the string, the low-energy excitations can be described by an effective field theory. Thus, contact can be established with quantum field theory, which turned out to be successful in describing the dynamics of the real world at low energy. The evidence that superstring theories could be unified theories is provided by the presence in their mass-less spectrum of enough particles to account for those present at low energies, including the graviton. It is precisely in this low-energy limit that Einstein gravity arises supplemented by some matter fields. These string gravity models preserve the long-distance behaviour of the mysterious quantum gravity and in special cases (BPS-saturated) exactly reproduces it.

It is well known that there exist five consistent superstring theories which all could pretend to be the unique one, namely:

• Type I: open-closed strings with gauge group 50(32).

• Type IIA: N = 2 closed strings (non-chiral).

• Type IIB: N = 2 closed strings (chiral).

• Heterotic: N = 1 closed strings with gauge group 50(32).

• Heterotic: N = 1 closed strings with gauge group Es x Eg.

In fact, this is not a good feature of superstrings since one would like to have an unique theory to describe the physics of our world. Some of these theories are more realistic than others, but it seems that there is no theoretical reason that could help in reducing their number. However, there is a hopeful attempt to unify all these theories by a large web of non-perturbative dualities in the framework of the so called M-theory:

Figure 1: The conjectured M-theory whose various limits in its 'moduli' space produce the weakly coupled ten-dimensional superstring theories.

It is important to stress that the unification of superstrings theories cannot take place at perturbative level, because it is precisely the perturbative analysis which singles out the five different string theories. It is by going beyond this perturbative limit and taking into account all non-perturbative effects that the five superstring theories turn out to be five different descriptions of the same physics, giving raise to M-theory. Recently, the structure of M-theory has begun to be uncovered, with the essential tool provided by su-persymmetry. Its most striking characteristic is that it indicates that spacetime should be eleven-dimensional since eleven-dimensional supergravity is assumed to be the low-energy effective theory of M-theory. This is appealing since this theory is totally constrained and no supergravity in more than eleven dimensions can be constructed. However, the current research is very far from understanding how the conjectured unification concretely occurs in nature due to the fact that the dualities which relate the known string theories are mostly intrinsically non-perturbative.

Now when we have a general picture of what M-theory pretends to be, we shall concentrate our attention on one of the five string theories, namely, on the SO(32) heterotic string theory, more exactly, in its low-energy effective field theory. Thus, the low-energy effective action of TV = 1 heterotic string

theory with gauge group SO(32) (its bosonic sector) is the subject of the present work. This action is a string gravity model which can be studied on its own. It turns out that we can draw a parallel between the effective action of this theory, when compactified to three dimensions on a torus, and the stationary action of the Einstein-Maxwell (EM) theory based on a striking similarity of the group structures of both theories. This fact enables us to construct a matrix formalism which in turn, permits us to make a simple and detailed analysis of the symmetries of the three-dimensional string gravity model under consideration and some of its truncations. Moreover, such a formalism allows us to directly extract (solving the equations of motion) some classes of classical stationary solutions for them.

Thus, the aim of Chapter 1 is to give the framework in which the string gravity model corresponding to the low-energy effective field theory of the heterotic string with gauge group SO(32) arises. In Section 1.1 we begin presenting a brief historical perspective on string theory and its further understanding as M-theory due to the large web of dualities that relates the different string theories and the eleven-dimensional supergravity. Then, starting from the low-energy effective action of the heterotic string, in Section 1.2 we review the three-dimensional cr-model that emerges after a toroidal compactification. In particular, the chiral formulation and main properties of the theory are presented.

The chapters which follow contain material which is based on original works.

Chapter 2 is devoted to the presentation of an alternative matrix formulation of the effective field theory of the heterotic string toroidally compactified down to three dimensions. In this Chapter we also obtain a class of stationary classical solutions that illustrates how our formalism works. Thus, in Section 2.1 we construct a new chiral matrix [1] which parametrizes the 'matter' sector of the string gravity model. Further, in Section 2.2 we reformulate the theory in terms of two matrix Ernst potentials [1]—[2] which are connected with the coset chiral matrix of the model. This new formulation enables us to establish a map between the effective field theory of the heterotic string toroidally compactified down to three dimensions and the stationary Einstein-Maxwell theory. We show as well that our matrix formalism includes the stationary Einstein-Maxwell theory as a special case. In Section 2.3 we go on presenting a simple algorithm to obtain extremal sta-

tionary solutions that generalize the Israel-Wilson-Perjes (IWP) class of the Einstein-Maxwell theory for the theory under consideration [3]—[4]. It turns out that the above mentioned map allows us to solve directly the equations of motion making use of the introduced matrix Ernst potentials. We thus obtain a rotating axisymmetric asymptotically Taub-NUT dyonic solution in terms of a single (d + 1) x {d + l)-matrix harmonic function and 2n real constants [5]-[6]. We define the physical charges of this field system and show that they saturate the Bogomol'nyi-Prasad-Sommerfield (BPS) bound due to the extremality condition. Then we show that the gyromagnetic ratios of the corresponding rotating field configurations appear to have arbitrary values. A subclass of rotating dyonic black hole-type solutions arises when the NUT charges are set to zero. Furthermore we consider the particular case d = 1, n = 6, which corresponds to the bosonic sector of the action of N = 4, D = 4 supergravity, and obtain explicitly a rotating dyonic solution which reproduces (in the Einstein frame) a previously known supersymmetric dyon.

In Chapter 3 we study the U-duality group of transformations of the three-dimensional string gravity model under consideration and derive a pair of matrix potentials that transform linearly under the action of the subgroup of finite symmetries that preserve their asymptotics [7]. Thus, in the framework of this symmetry approach in Section 3.1 we establish the action of the full symmetry group of the theory on the matrix Ernst potentials. Then we construct all the finite symmetry transformations which preserve their asymptotics: first we split the U-duality group into gauge and non-gauge sectors, then we fix the gauge (with trivial field asymptotics) and finally we construct a representation of the theory which linearizes the non-gauge sector. It turns out that the transformations of this sector form a charging symmetry subgroup (we call them in this way because they generate charged solutions from neutral ones). After that, in Section 3.2 we introduce a pair of new matrix variables that transform linearly under the action of all charging symmetry transformations. First we derive these linearizing potentials for the Einstein-Maxwell theory in its Ernst potential formulation (details can be found in the Appendix A) and then we directly generalize the result to the heterotic string theory case. Further we arrange these two potentials into a single (d 4-1) x (d + n + 1)-dimensional matrix potential which transforms linearly under the action of charging symmetries in a transparent SO(2,d - 1) x SO(2,d + n - 1) way. This allows us to establish the

structure of the charging symmetry group (the non-trivial structure of its so(2,d — 1 + n) subalgebra is studied in the Appendix B). Then we construct on the basis of this single linearizing potential one charging symmetry invariant which vanishes for BPS-saturated field configurations. In Section 3.3 we continue by performing an investigation of heterotic string fields with a linear dependence between the linearizing potentials. We show that such fields are invariant under the action of charging symmetries. It means that this class of fields cannot be generalized using charging symmetry transformations. By setting the charging symmetry invariant of the theory to zero we single out a subclass of field configurations which coincides with the general Israel-Wilson-Perjes class of solutions of the effective three-dimensional heterotic string theory. Finally, in Section 3.4 we formulate the most general technique for generating new solutions using charging symmetries. We show how starting from the pure Kaluza-Klein theory one can generate all the massless fields of the bosonic string theory and, further, of the full heterotic string theory sector.

In Chapter 4 we consider some truncated string gravity models that arise in the framework of the effective field theory of the heterotic string, namely the Einstein-Kalb-Ramond-Dilaton (EKRD), the Einstein-Kalb-Ramond (EKR) and the Einstein-Maxwell-Dilaton-Axion (EMDA) theories. The first field system corresponds to the case when the matrix Ernst Potential A is dropped, being completely defined by the potential X which describes the gravitational, dilaton and antisymmetric tensor fields. In the second model we consider the subsequent vanishing of the dilaton field in the five-dimensional theory. The third model describes four-dimensional gravity coupled to dilaton, axion and one vector field. All these models, when toroidally compactified to three dimensions, admit classical procedures well known for the stationary Einstein gravity since their field variables can be mapped onto the Einstein ones. Thus, in Section 4.1 we start by considering the EKRD theory. We show that in terms of the matrix Ernst potential X, the target space duality group of the field system becomes the SL(2, R) matrix-valued one. Its three subgroups are identified as the matrix generalizations of the gauge shift, rescaling and Ehlers transformations of the stationary vacuum Einstein theory. Furthermore, we construct the matrix scalar-vector Lagrange representation of the model. We discuss later the properties of this theory reduced to two dimensions. We introduce a new Hermitian Ernst-

like matrix potential in the nontarget space formulation and map it onto the Ernst-like matrix potential of the real target space using a complex transformation which generalizes the Kramer-Neugebauer map for the stationary axisymmetric Einstein theory. Further we define a non-coset chiral matrix, which possesses the same properties that the Belinskii-Zakharov one for pure vacuum, on the basis of the real and imaginary parts of the new Ernst-like potential. Finally, the global 0(d+ l,d+ 1) symmetry transformations are written in the SL(2, R) matrix-valued form using both Ernst-like matrix potentials. The results presented in this Section were obtained in [8].

We go on in Section 4.2 studying the particular five-dimensional EKR theory admitting two commuting Killing vectors. It turns out that this system allows a Kahler representation which is formally defined by two vacuum Ernst potentials. A discrete transformation between these potentials gives rise to an alternative 2 x 2-matrix formulation of the model. Then we impose axial symmetry and show that in this case, there are three pairs of descriptions of the theory; each pair contains a target space (nondualized) representation and a nontarget space (dualized) one. Further we establish the corresponding Kramer-Neugebauer-like maps between the dualized and nondualized variables. Later on we present the five-dimensional line element explicitly depending on the Ernst potentials and construct a class of five-dimensional black hole solutions; among them we identify EKR dipole configurations which are hidden inside the event horizon. This Section is based on results obtained in [9].

Section 4.3 is devoted to the study of the EMDA theory which describes four-dimensional gravity coupled to a single vector field and non-trivial dila-ton and axion fields. This system can be regarded as a simplified model of N = 4, D = 4 supergravity. A complete formal analogy between vacuum Einstein and EMDA theories in three dimensions allows us to construct the Kahler formulation of the latter field system in the stationary and stationary axisymmetric cases. Thus, in the context of this theory we derive a set of complex potentials which transforms linearly under the action of the charging symmetry subgroup [10]. This potentials provide a useful description of asymptotically flat field configurations and allow us to establish a general invariant of the charging symmetry subgroup that vanishes for BPS-saturated configurations.

Finally, in Conclusions we list the main results obtained in this thesis; Appendix A contains the derivation of the linearizing potentials of the charging symmetry subgroup of the Einstein-Maxwell theory, and in Appendix B we display the commutation relations of the charging symmetry algebra of the heterotic string theory.

Chapter 1

String Motivated Gravity Models

In this Chapter we begin by presenting a quick and sketchy introduction to string theory (M-theory) in order to give the reader the context in which the string gravity models that we study arise. We did not intend to be exhaustive in the referencing. The classical manual for perturbative string theory is Green, Schwarz and Witten [11], whereas a useful review which contains non-perturbative dualities is [12] and the references therein. Then, in Section 1.2 we focus our attention on one of the five consistent superstring theories: in the SO(32) heterotic string theory, more exactly, in the bosonic sector of its low-energy effective field theory. It is precisely this field system that gives rise, after its toroidal compactification down to three dimensions, the string motivated gravity models that are the subject of our investigations.

1.1 From String Theory to M—theory

As it has been mentioned-above, nowadays, string theory is the leading candidate for a theory that consistently unifies all fundamental forces of nature, including gravity. However string theory was born as a theory to describe the interactions of hadrons. In 1968, in the framework of dual models for hadrons, Veneziano proposed his famous amplitude [13]. During the sixties there was a large amount of experimental data relevant to the strong interaction. The behaviour of lots of particles or 'resonances' with increasing mass and spin was very complicated and could not be explained in the context of the model for electro-weak interactions. However, there were two regularities observed: a big number of these resonances obey the Regge behaviour (by plotting the mass versus the spin we obtain almost straight lines) and a conjectured duality between the s- and ¿-channels for the hadronic amplitudes.

These facts led Veneziano to propose his amplitude realizing both the duality and the Regge behaviour. Another important feature of these amplitude is that it implies an infinity of poles of increasing mass and spin in both of the dual channels. This means that the underlying theory, whatever it is, must have an infinite spectrum of massive excitations (i.e. particles). From here we can deduce that the theory which produces such an amplitude is not a conventional quantum field theory.

In 1970 it was realized by Nambu [14] and Goto [15] that theories of quantized strings reproduce such amplitudes, and thus the infinite particle spectrum. One may therefore consider the infinite number of particles as an indication for abandoning locality, and the string as the most simple object which gives rise to an infinity of vibrations of higher and higher energy. Moreover, this spectrum of particles is completely fixed by the quantization procedure. However, such theories had several shortcomings in explaining the dynamics of strong interactions: all of them seem to predict a tachyon (a particle with imaginary mass); several of them seemed to contain a spin 2 particle that was impossible to get rid of; all these theories required a space-time dimension D — 26 in order not to break Lorentz invariance at the quantum level; and finally, they contained only bosons.

In 1971 Ramond [16], and Neveu and Schwarz [17] formulated the fermionic string in order to have a model that describes space-time fermions as well. This implied the introduction of fermions on the world-sheet of the string that had to be related to the world-sheet bosons (the embedding coordinates) by supersymmetry. Thus, these strings were called superstrings. In order to consistently quantize them, the dimensionality of space-time must be D = 10. Moreover, it turned out that one can consistently discard the tachyon in the framework of these superstring theories.

It was at the beginning of the seventies that experimental data obtained in the region of high-energy scattering at fixed angle showed that at even higher energies hadrons have a point-like structure; this fact opened the way for quantum chromodynamics (QCD) as the theory that correctly describes strong interactions.

In the context of the string theory attempt to describe strong interactions there was a total lack of experimental evidence of a massless spin 2 particle. Nevertheless, in 1974 Scherk and Schwarz [18] proposed that by rescaling the string parameter ameasuring the inverse string tension, from the strong

interaction scale to the Planck scale, i.e. making it much smaller, the massless spin 2 particle can be interpreted as the graviton. At that moment, string theory ceased to be a candidate for a theory of hadrons, becoming instead a candidate for a theory describing, and unifying, all known interactions, including gravity. In fact, if we take the limit a' —>• 0 (the zero-slope limit), all the massive string modes decouple and we are left with the massless sector. The latter includes the graviton and many other particles with lower spin, which could account (at least in some of the string theories) for all the known 'light' particles, matter and gauge bosons. Then, it is possible to write a field theory of these 'low-energy' modes and in this context the Einstein gravity is correctly recovered.

In 1976 Gliozzi, Scherk and Olive [19]—[20] introduced the so-called GSO projection which simultaneously removed the tachyon and imposed spacetime supersymmetry. This last property of superstring theories will turn out to be very important for the present-day developments: not only the world-sheet fields, but also the string spectrum in 10 dimensions obeys a symmetry relating bosons to fermions. Contemporarily to the research in string theory, an extensive research on supersymmetry, and on its local generalization: supergravity, has been done in the seventies. It is important to mention that space-time (extended) supersymmetries severely constrain the form that an action can take. Moreover, while all superstring theories are defined in D = 10 space-time dimensions, a supergravity theory can be formulated at most in an eleven-dimensional space-time.

Thus, at the beginning of the eighties there were three superstring theories: two theories of closed strings with extended N = 2 space-time supersymmetry (one chiral and one non-chiral), and one theory including open strings, with N = 1 supersymmetry. All these theories were shown to be finite at one-loop, and it is expected that the finiteness will remain at all loops. This is because all the massive string modes regulate in such a way the effective theories coming from the massless sector, that the ultra-violet behaviour is much softer than in ordinary field theories. This success was really stimulated by the explicitly space-time supersymmetric formulation of the superstrings given by Green and Schwarz [21]—[24]. Moreover, after a general analysis of gravitational and gauge anomalies 1 made by Alvarez-Gaume and Witten [25], it was realized that anomaly-free theories in higher dimensions are very

1 An anomaly is the breakdown of a classical symmetry in the quantum theory.

restricted. Green and Schwarz showed in [26] that open strings in ten dimensions are anomaly-free if the gauge group is 50(32) (or E% x Eg, but this group could not appear in the framework of the open string theory).

A big impetus for string theory came during 1985, when Gross, Harvey, Martinec and Rohm [27]—[28] formulated two other kinds of superstrings called the heterotic strings. These theories contain closed strings and are built asymmetrically combining features of the toroidally compactified bosonic string and superstrings. Heterotic strings have N = 1 supersymmetry which is enough to preserve all the remarkable features of the superstrings, while the bosonic sector allows the spectrum to fall into representations of £0(32) or Eg x E$. The massless sector contains all the vectors in the adjoint representation of the group, which correctly translates into a low-energy effective theory consisting of N = 1 supergravity coupled to a super Yang-Mills (SYM) theory.

The heterotic strings aroused a big interest because they appear to be promising candidates to incorporate the Standard Model of strong and weak interactions, thus unifying most simply all the known interactions. In the framework of the previously known closed superstring theories it was impossible to find, at the perturbative level, a non-abelian group, in the low-energy sector. Later it was understood that duality will predict, and evidence has now been found, a gauge symmetry enhancement in these theories by non-perturbative effects. Nevertheless, at that time there was little hope to get, even after compactification, a viable non-abelian gauge group from the N = 2 superstrings. It turned out that in order to fit the SU(3) x SU(2) x U( 1) gauge group of the Standard Model, the Eg x Eg group of one of the heterotic strings was the most suitable. Then people focused their attention on heterotic strings theories more than on the others.

In order to make contact with phenomenology one has first to compactify from ten to four space-time dimensions, and then reduce the supersymmetry from the unrealistic N = 4 (in four dimensions) to the more workable N = 1 set up, for example. The compactifications on Calabi-Yau manifolds [29] and orbifolds [30]—[31] appeared as promising tools to solve these problems. So, at the end of the eighties an enormous amount of work on phenomenologic aspects of string theory has been done and people looked at string theory as the 'Theory Of Everything'.

Although at this stage it was clear that string theory provides a finite

theory of quantum gravity (at least in perturbative theory) and that it seems to have all the right properties for Grand Unification (it produces and unifies with gravity not only gauge couplings but also Yukawa couplings), to date, its main shortcomings are its numerous different vacua: the fact that there are five consistent superstring theories that look different (in this line of thought, eleven-dimensional supergravity has the same uncertain status); the huge quantity of ways of compactification (this means that in the attempt to recover the Standard Model physics, the number of parameters one has to tune is much more than the number of parameters of the Standard Model itself); and most importantly, supersymmetry breaking. At the same time, however, it was realized that perturbative T-duality (for a review see [32] and references therein) relates some of the string theories. This duality acts on the string theories when they are compactified on a generic d-dimensional torus, for example.

A new wave of enthusiasm towards string theory came with the famous paper of Witten [33] where it was shown that most of the strong coupling limits of all the known string theories can be reformulated through dualities in terms of the weak coupling limits of some other string theory, or in terms of the eleven-dimensional supergravity. Thus dualities came into play and M-theory was born. It is worth noticing that the concept of duality that Witten used was already introduced and utilized in string theory since the beginning of the nineties by some authors, in particular Sen [34]—[36], based on the the original conjecture formulated in [37], studied and provided evidence for the 5-duality (this duality relates the strong and weak coupling regimes) of the heterotic string theory in