Geometric Approach to Path Integration in String Theory

GEOMETRIC APPROACH TO PATH INTEGRATION IN STRING THEORY by Kevin Michael Short A thesis presented for the degree of Doctor of Philosophy of the University of London and the Diploma of Membership of Imperial College The Blackett Laboratory Imperial College Prince Consort Road London SW7 2BZ September 1988 2 ABSTRACT This thesis is concerned with developing a geometric approach to path integration, particularly with respect to Polyakov string theory. Introductory remarks are in chapter one. Chapter 2 develops a non-perturbative approach to Polyakov string theory in variable dimensions. This leads to a reinterpretation of the critical dimension and an effective compactification of the theory. The variable dimension theory is based on gaussian measures, and the dimension can be taken to infinity. The limit represents a correction to [DetA]'1^2, which arises as a naive limit. The analysis involves an interplay between zero modes and sets of measure zero with respect to the gaussian measures. There is a discussion of functional Haar measures, and source terms are considered. An appendix is added concerning the definition of gaussian measures based on cylinder set measures, and the sets of measure zero which arise. The third chapter will consider the extension of string theory to curved target spaces. It is shown that only trivial classical solutions exist in euclidean space, and then discusses the existence of classical solutions for curved target spaces. A toy model of strings in S3 will be developed, which will be extended to a simple model of cosmological membranes in S3 x R. In the fourth chapter, a new formulation of string theory which is applicable to curved target spaces will be developed. The observables in the new formulation represent the deformations of the image surface. The path integral is "gauge fixed", but it retains a global gauge invariance. A new measure for the embedding integration is developed which encodes a new geometric regularization procedure. The new I 3 formulation can be applied to target spaces of any dimension, and leads to the "dynamics of dimension", whereby mass terms or interactions dynamically appear in the theory. Calculations are presented for the genus zero and genus one surfaces. No conformal anomaly is present. There is some discussion of the connection with general relativity. Certain mathematical aspects arise and are included in appendices. The final chapter will draw some conclusions and speculate about the future direction of the research presented in this thesis. 4 PREFACE The work presented in this thesis was carried out in the Theoretical Physics Group at Imperial College under the supervision of Prof. C. J. Isham. The material presented is original, except where due reference is given, and has not been presented for any other degree in any other university. I would like to thank my supervisor, Prof. C. J. Isham, for his guidance and assistance at all stages of this research. I have also benefited from discussions with Dr. John R. Klauder concerning the work in chapter two, and with Dr. Graham Shore and Sunil Gandhi with regard to the work in chapter three. Finally, I acknowledge gratefully the financial support of the Marshall Aid Commemoration Commission during the years of my post-graduate study. TO MY FAMILY... with thanks for their unwavering support throughout the years. 6 CONTENTS PAGE Chapter 1 : Introduction 1. Background 9 2. Thesis Introduction 18 Chapter 2 : Polyakov String Theory In Variable Dimensions 1. Introduction 24 2. Zero Modes and Measure Zero Sets 28 3. Variable Dimension Field Theory 36 4. Functional Haar Measure 4L 4.A. Fixing the Translational and Rotational 46 Degrees of Freedom with Delta Functions 4.B. Fixing the Translational and Rotational 50 Degrees of Freedom with the Geometry of Function Space 5. Scaling limit as D—»©« 54 6. Complete Solution 55 7. Implementation in Polyakov String 56 8. Source Terms 58 9. Appendix - Sets of Measure Zero 60 7 Chapter 3 : Strings in Curved Space 1. Introduction 62 2. Existence of Classical Solutions 68 3. Applications 73 1. The Toy Model 74 2. Cosmological Membrane Model 85 4. Conclusions 92 Chapter 4 : New Formulation of String Theory 1. Introduction 94 2. Reinterpretation and Implications 96 3. Reformulation of the Polyakov Path Integral 99 A. Measure for the Riemann Surface Integration 102 B. Measure for the Embedding Integration 105 C. Reformulated Path Integral 113 4. Theory of Isometric Imbeddings and Immersions 114 5. The Dynamics of Dimension 120 A. Calculating the Fiber Volume for D > Dcrit 122 B. Compactification for D < Dcrit 124 6. Calculations 129 A. Genus Zero - The Sphere 129 B. Genus One - The Tori 132 7. General Relativistic Aspects 138 8 Chapter 5 : Summary 1. Conclusions 142 2. Speculation 147 ABB£_nflix.A: Riemann Surfaces 149 Appendix B : Harmonic and Minimal Maps 161 Appendix C : Theory of Isometric Imbeddings and Immersions 165 References 175 9 INTRODUCTION 1. BACKGROUND In a seminal paper appearing in Physics Letters in July 1981 [1-1], A.M. Polyakov proposed a model for the quantization of bosonic strings in euclideanized spacetime, based on an action previously considered by Brink, DiVeccia and Howe [1-2], and by Deser and Zumino [1-3], although the same action was considered by mathematicians, e.g. Eells and Sampson [1-4], in a more general form as far back as 1964. The action is given by s[x,g] = I p z /i (i.i) W where |i = l,...,d and a,b = 1,2. The different terms require some explanation. W represents the string world sheet, which in the euclideanized Polyakov model is just a closed Riemann surface, so it has real dimension 2. The intrinsic metric on W is given by gab. The functions X represent mappings of W into a d-dimensional euclidean target space, Rd, and are generically termed "embeddings". The coordinates on W are given by z = (z1, z2), while g = Det gab, so d2z Vg is the invariant volume element on W. The terms 9aX^3bX^ are just the components of the pull-back metric induced on W by the embedding into Rd, so the action can be viewed as the inner product between the intrinsic metric and the pull-back metric. The action (1.1) has several invariance properties which are important to consider when quantizing the classical theory. The action is invariant under difeomorphisms of W, since d2z Vg is the invariant volume element, and is also invariant under conformal transformations 10 of the intrinsic metric g -> n 2(z) g (1.2) where Q 2(z) is a positive function defined on W, which will be referred to as the conformal factor. These invariances combine in such a way that the gauge group is the semidirect product group Conf(W) © Diff(W), where ConfiW) is the group of conformal transformations on W, and DiffiW) is the group of diffeomorphisms on W. Further discussion about the gauge group will be found in the appendix on Riemann surfaces. The action has another trivial invariance under translations and rotations in Rd. Much of the work in string theory revolves around attempts to preserve these symmetries in the quantum theory, as will be discussed below. The classical equations of motion are derived by varying the action (1.1) with respect to the embedding X and the intrinsic metric g. This results in the equations AX^ = 3a(yiyb3b)X^ = 0 (1.3) and Tb = 3/3/ - = 0 0-4) The first equation (1.3) will be termed the laplacian equation, while (1.4) is the stress energy tensor equation. It should be noted that the stress energy tensor is traceless, and this condition holds without invoking the other equation of motion. This is simply a reflection of the fact that this is a two dimensional theory, since ga^gab= 2 in 2 dimensions. In Polyakov's original paper [1-1], he solves (1.4) to give the relation °abg = aa b (1.5) but it can be seen that this is just a particular solution. The most general solution of (1.4) is Sab = W d-6) which is evident once one realizes that the gabgC(* terms in (1.4) are conformally invariant. This implies that the classical solutions are conformal embeddings. In the quantum theory, Polyakov wants to sum over continuous surfaces in analogy with the sum over paths in path integrals. Consequently, the partition function developed by Polyakov is often called the Polyakov path integral, even though the paths are in fact 2-dimensional surfaces. Further, one would like to preserve the invariances of the theory, although this is not generally possible because of the presence of anomalies. In the path integral, Polyakov wants all surfaces of a given area to enter into the integral with the same weight, and tries to derive a measure which fulfills this requirement. For the purposes of renormalization, he adds an extra non-conformally invariant term to the action, of the form (1.7) For any functional of surfaces <j)[S], Polyakov gives the path integral as /• i- , f,2 n * --fd2z/7gabd x^a.x^- Jdn(S) <KS) = J[Dgab(z)]e' J ^J[D X (z)]e 2 <t>[X] (1.8) When Polyakov derives the measure for the integration over metrics, he finds that it is not conformally invariant. Consequently, there is a dependence on the conformal factor arising from the integration over metrics, and unless the metric anomaly can be cancelled by a conformal anomaly arising from the embedding integration, the quantum theory will not recover the full invariance group of the classical 12 theory. This is the primary reason for the addition of (1.7) to the action; it allows the desired anomaly cancellation to occur, in the manner described below.

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