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 = a a 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 /• i- , f,2 n * --fd2z/7gabd x^a.x^- Jdn(S) IlSgJI2 = j d 2z j g (gacgbd + C g V ) 8gab8gcd (1.10) to give ll8gabH2 = (1+ 2C)Jd2zyi( 89+VcdVc)2 + \ d \ f g ¥byb (1.11) where cp = log(&2(z)) and Yab = Va(dVb) + Vb(dVa) - gabVc(dV°) (M2) From this Polyakov derives the measure for the integration over the metrics to be dn(gab) = D VDetL = exp - ^LJd2z [^L (d ^)2+ k e9] (1.15) where k 2 is a renormalization constant. The dependence of Det L on the conformal factor is an anomaly of the quantum theory which must be 1 3 cancelled by the conformal anomaly arising from the embedding integration. In a quantum theory with a conformal anomaly, it is found that the conformal anomaly manifests itself in the development of a non-zero trace for the stress-energy tensor. Using this knowledge, Polyakov derives the anomalous dependence of the embedding integration on the conformal factor by using the trace anomaly relation, so if exp(-F) = J d X(z) exp-{-ijd2z./g g^dX^d^ + xjd\/g } (1.16) then the trace anomaly relation is ^ 5g = Sab< Tab > = 2^ [R(Z) + C0I1St] (L17) where R is the scalar curvature. After integrating by parts in the first term in the exponential of (1.16), one can write formally exp (-F) = [ yjDet A ] exp-^J*d2z y g (1.18) where A is given by A = da(Vg gab3b). In the conformal gauge, R = —^— 32(log fl2(z)) (1.19) Q (z) Polyakov then regulates (1.18) to give F = - ^ J d 2z [ l 0 a Z = Jd The integral in the exponent is the Liouville action, and it is clear that in d = 26, the conformal anomaly would exactly cancel the metric anomaly, and one would not have to worry about quantizing the Liouville theory. Numerous researchers have followed up on Polyakov’s work, and have made significant contributions to clarifying the theory.[1-5] - [1-11] Since the material is so extensive, only certain results will be cited here. The paper by Alvarez introduced the concept of the Teichmuller space into string calculations. This is essentially a global result that states that even after factoring out the invariances from the theory, there remains a finite dimensional space over which one must integrate. This means that the choice of the conformal gauge does not fix all of the gauge degrees of freedom. The domain space in the path integral is the space RiemMet(W) x Emb(W) where RiemMet(W) denotes the space of Riemannian metrics on W, and Emb(W) denotes the space of embeddings of W; however, to find the true configuration space, one must factor out the gauge group ConftW) © DifftW). Generally this space is simplified by factoring only by the diffeomorphisms connected to the identity, Diff0(W). Doing this leads to the quotient space RiemMet(W)/ [ConftW) © Diff0(W)], which is the finite-dimensional Teichmuller space mentioned at the beginning of this paragraph. This is discussed in appendices A and B of Alvarez' paper [1-6], the paper by Moore and Nelson [1-7], and in the appendix on Riemann surfaces in this thesis. The work by Alvarez [1-6] followed closely upon that of Polyakov, but in a much more general form, extending Polyakov's work to surfaces with boundary. The boundary terms will not be pertinent here, so no discussion of them will be included. As the discussion in the previous paragraph indicated, Alvarez included the integral over the Teichmuller 1 5 parameters, which made it necessary to work in a gauge which reflected the fact that the conformal gauge did not work globally. Thus the metrics were written with reference to a fiducial metric which was a function of the Teichmuller parameters g = I t Q2(z) = i te2° (1.22) where the index t represents the dependence on the Teichmuller parameters. Including the Teichmuller parameters, Alvarez finds for the Polyakov path integral Z = V^jf[da][dx][Teich](Det'P^jPj) exp -S[X] (1.23) where the determinant comes from gauge fixing in the integration over metrics (which was based on a decomposition similar to (1.9)), and V^f is the volume of the diffeomorphisms perpendicular to the gauge orbits. S[X] is the action used by Polyakov with the addition of a topologically invariant piece equal to the Euler number of the Riemann surface S[X] = i j d 2z1/i g abax ^ bx^ + xjd2z1/i' + £ [ y dy j R (1.24) where the last term in brackets is the Euler number of the Riemann surface. Since this is a topological invariant, it does not enter into the dynamics of the theory. After an integration by parts in the action, the X integration is performed as a simple product of d gaussians, introducing into (1.23) a term [Det A]‘d/2. The regulating procedures for the determinants are based on the diffusion operator o o oo InDetD^ = -Jy Tr'[exp-(tD(n+))] = - jy ^ exp'^ V (L25) e £ where the are the eigenvalues of the operator, and the proper time 1 6 cutoff e introduces a length scale to the theory, ruining conformal invariance and giving rise to anomalies. However, if the different anomalies can be made to cancel, a conformally invariant theory can be recovered. After regulating the determinants, Alvarez finds 1 Z = f ( H - S ^ ) [ d o ] v Diff[detH(P^][det I^P^] 2exp- { SQ+ Sconf[a,g[]} [equation (1.26)] where the integral over the tu represents the integral over the Teichmuller parameters, S0 is independent of the conformal factor, and the determinants arise from the gauge fixing. Sconf is the Liouville action Sconf[0 ’5.] = K2jd 2z V f e 2° + l l ^ J d i f V 3!,0 + M {X21) There is a problem with (1.26) because it does not allow all of the anomalies to cancel, and Moore and Nelson [1-7] point out that even after taking d=26 and renormalizing the cosmological term, one is left with the a dependent terms Volv ulDiff(W) DetH(P') (1.28) JM VoLConf(W)(s) Diff(W) and the determinant gives a non-local o dependence . This would ruin the chances of recovering the invariances of the theory even in 26 dimensions. Fortunately, Moore and Nelson are able to show that the a dependent term arises from a problem with the coordinates used for the gauge slicing in Alvarez' work. After going through a careful analysis of the gauge slice and the Teichmuller space, they are able to eliminate the non-local anomaly. For the one loop string calculation, the work by Polchinski [1-8] will provide a good point of comparison for the work in this thesis. He chooses to calculate the genus one path integral in the critical dimension 1 7 d=26. In this dimension, he performs the calculation without worrying about the conformal anomaly. The starting point of the calculation is the path integral Z = fv[dSab]r ] e*P-[fdy i ( J s 'V V + Xr + k2)] d'29) J VDiff(W)VConf(W) J where T is the string tension and X and k 2 are renormalization constants. The X integration is done in the manner described earlier, by integrating by parts in the X dependent term in the exponent, to give [Det'A] d/2, where the prime indicates that the zero modes of the laplacian have been removed. Polchinski accounts for the zero modes by putting the calculation in a d-dimensional box with sides of length L, so there is an extra contribution from the zero modes of the form d d - 3° ) Polchinski makes use of the fact that any metric on the torus can be put in the form ds2 = gabdzadzb = e (?\ dz1+ x dz212 where x = %l + i^ is a complex number with x2 > 0. The parameter x corresponds to the Teichmuller parameter discussed above. Here the conformal factor is related to the Q2(z) term in (1.2) by (p = log(£22(z)). This will be discussed more fully in the appendix on Riemann surfaces. The regulated form of the determinant is found to be Det'A = Y l . (47t2gabnanb) = (x^e 3 |f(e27m)|4 (1.31) nr n2 where oo 27iiT I T 27TinT\ f(e ) = H (l-e ) (1.32) n = 1 1 8 The final form of the path integral becomes 2n ix |—48 (1.33) H j 4ji(t2)2 Notice that because of the form of (1.32), it is difficult to envision any further simplification of this expression, and the Teichmuller integration probably cannot be performed. 2. THESIS INTRODUCTION Several comments can be made about the Polyakov approach which will provide some motivation for the line of research which will be followed in this thesis. The work in this thesis will depart in several significant ways from the standard prescription outlined by Polyakov. The primary difference is that the Polyakov prescription attempts to produce anomaly cancellation, while the object of this thesis will be to outline an approach where the anomalies may not arise. Polyakov chooses a non-conformally invariant measure for the integration over the space of metrics, which immediately gives rise to an anomaly, then adds a non-conformally invariant term to the action to produce a conformal anomaly which will cancel the metric anomaly. One aim of this thesis (to be developed for genus one surfaces in chapter 3) will be to investigate the possibility of finding a measure for the true configuration space (the quotient space RiemMetCW)/ [Conf^W) (s) Diff0(W)]) which will be group invariant and hence free from the metric anomaly. This eliminates the 1 9 need to add the non-conformally invariant piece to the action, and at no stage will such a term be added. The addition of non-conformally invariant terms to an action was contemplated in the 1970's in the context of conformal anomalies in other conformally invariant theories. In the paper by Duff [1-12] it is clearly stated that in a conformally invariant theory, it is only after the addition of counterterms to the action that it develops a non-zero trace and, further, that the counterterms should be conformally invariant. With respect to non-conformally invariant counterterms, these are considered to be ad-hoc additions to the action, with parameters which can be adjusted to cancel the true conformal anomaly.[see also 1-13] This is exactly what Polyakov did. By adding to the action the non-conformally invariant term X,jd2z 7g the theory developed a non-vanishing stress-energy tensor, making the dimension a free parameter which could be adjusted to cancel out the anomaly arising from the metric integration. For these reasons, it is of some value to attempt to formulate the theory without the ad-hoc addition of non-conformally invariant terms to the action, and the resulting theory will be developed in chapter 4, with calculations presented for genus zero and genus one surfaces. Before proceeding with the work in chapter 4, it is necessary to take a critical look at some of the assumptions and goals of string theory. In chapter 2, the idea that string theory is supposed to determine the structure and dimension of spacetime is considered carefully. It is shown that because of a connection between zero modes and sets of measure zero with respect to the gaussian measure, the avowed object of 20 finding a measure for the embedding integration which gives equal weight to surfaces of equal area is not achieved. In fact, in the standard Polyakov approach, it is shown that only surfaces with a linear spam of d dimensions (by which is meant that the surface cannot be enclosed in a box of the target space of fewer than d dimensions) contribute to the path integral. This is a consequence of the fact that the laplacian has zero modes, and is hence degenerate on constant functions. The zero modes correspond to surfaces that have a linear span of fewer than d dimensions, and are shown to lie in sets of measure zero. The result is that the measure chosen for the embedding integration is highly dimension specific. To avoid these difficulties, it is shown that a dimension non-specific measure can be found, which amounts to defining a variable dimension field theory. The work on the variable dimension field theory may be of independent interest outside the realm of string theory, and to the knowledge of the author, is the first attempt at such a formulation. The variable dimension theory is calculated non-perturbatively, and for a field theory based on a laplacian with translational and rotational zero modes, the infinite sum over dimension gives in the naive scaling limit the usual result [Det A]'1/2. It is particularly important that the infinite sum naturally compactifies the theory, so that it appears as a lower dimensional theory. The exact result represents a correction to the determinant factor. The third chapter takes up the question of strings in curved spaces. The standard works on string theory have all concentrated on euclidean target spaces, while much discussion has been made about compactifying the extra dimensions of string theory on some highly curved manifold at scales of the Planck length. With a general curved target space, the embedding integration based on a gaussian measure 21 breaks down completely, so one must reevaluate the theory. Chapter 3 considers the basic structures of string theory in curved space, beginning with a discussion of the existence of classical solutions to the equations of motion. The connection of strings with harmonic mapping theory and minimal surface theory is presented. Although this is standard material in the mathematical literature, its inclusion has many implications for the physical significance of string theory as a geometric theory, and gives an interpretation of the action as the energy functional of the embedding, and allows for the introduction of the notion of the tension field of an embedding. For genus zero and genus one surfaces, the mathematical aspects will be applied in a toy model of strings in S3, and an extension to cosmological membranes in S3 x R, based on background field expansions about a classical solution. To perform these calculations, a measure will be derived on the true configuration space which retains invariance under the gauge group. This will make use of a global parametrization for the Teichmuller space of genus one, so although the integral is "gauge-fixed", the global nature of the parametrization allows one to maintain gauge invariance. However, the primary purpose of chapter 3 is to provide motivation and background for the geometric approach to redefining string theory developed in the last chapter. In chapter 4, a purely geometric reformulation of string theory will be developed in a manner which is applicable to curved target spaces. The action will be interpreted as the global inner product between the intrinsic metric on the domain space Riemann surface and the pull-back metric induced on the surface by the embedding into the target space. This will stress the metric properties of the theory, and will obviate the need to treat the embedding functions X*1 as observables. The new observables will be the functions which represent the deformation of the 22 image surface away from the constant curvature surface. As discussed above, the redefined path integral will integrate over the true configuration space. The use of the true configuration space restricts the integral to be over conformally inequivalent Riemann surfaces, which is consistent with the desire to formulate a purely geometric theory. When considering the action as the energy functional described in chapter 3, this corresponds to treating the domain space metric as the comparison point for energy calculations, much as the definition of potential energy requires a comparison point. The resulting integral would be termed gauge fixed in the standard approach; however, it still retains invariance under transformations of the defining fundamental regions ( which correspond to gauge transformations), since this at most redefines the comparison point of the integral. A new measure for the integration over the space of embeddings will be presented, with the properties that it satisfies the dimension non-specific requirements outlined in Chapter 2, and is local in the sense that it is based on deformations of a given embedding. It will be shown that the measure naturally encodes a new regularization procedure for the infinities which arise in the functional integral by defining the regulated integral to be the integral over one cross-section of a bundle formed from pull-back metrics induced by the embeddings. The results are applicable to curved target spaces provided an auxilliary mathematical question can be answered. The regularization leads to a new concept called the Dynamics of Dimension, whereby mass terms or interactions may dynamically appear in the theory as a function of the target space dimension. As the target space dimension decreases, a natural compactification of the theory occurs, and it is clear how to formulate the theory for any target space dimension. The work may have 23 some implications for the philosophical debate about observability. An actual calculation will be presented using the new theory, for the genus zero case, and for the first non-trivial case of tori (genus 1). It will lead to well-defined and finite integrals. No conformal anomaly is present, and dimensional restrictions arise only from considering the compactification in the Dynamics of Dimension. There will be some concluding discussion of the aspects of general relativity which arise naturally from the consideration of the theory as a metric theory. Certain mathematical aspects will be required, and will be included in appendices. These include a brief review of Riemann surface theory, a section giving selected results from the mathematical theory of harmonic and minimal immersions, and a section on the mathematical theory of isometric imbeddings and immersions of Riemannian manifolds (here interest will focus mainly on surfaces) into higher dimensional Riemannian manifolds. This latter section will be necessary for the definition of the measure for the embedding integration. The final chapter will draw some conclusions based on the work in this thesis. It will then offer some speculation about the implications of the line of research presented in this thesis, and comment on future directions. 24 CH. 2 - POLYAKOV STRING THEORY IN VARIABLE DIMENSIONS 1. INTRODUCTION In the standard papers on Polyakov string theory, the integration over the space of embeddings is treated as a straightforward product of gaussian integrals. As with any gaussian field theory, the validity of this approach is based on an underlying gaussian measure, given by: N = JdX exp -(X,X) = JdXexp-CX2) (2.1) where X is the field variable, dX is the associated Lebesgue measure, N is a normalization constant, and (*,*) is the inner product. Of course, the existence of the gaussian measures is predicated on the ability to expand the field variable in a series over some complete orthonormal set: x=Xn a„f(n) where the fin) are the complete set and the an are the coefficients of the expansion. In the usual way , then, one can write: The inner product then reduces to a sum of the squares of the coefficients: (x .x )= X ai; n Equation (2.1) then becomes: n V 7 1 n where the normalization has been chosen so that N=l. The case where the target space has more than one dimension is 25 handled similarly. The only difference is that the field variables have d-components when the target space is d-dimensional, so that one now has fields X = (X^X2,...^). Each component is expanded over the complete orthogonal set as above: XV= I < f(n) n and the resulting integral is a product of d such gaussian integrals: '-jriirS «p-E <*i>! v=l n v 7C v,n The importance of the underlying gaussian measure is that one can introduce a non-degenerate operator into the inner product to properly define a new functional integral. It is at this point that the connection with the action for the field theory becomes evident. If (*,*) is the inner product in the function space associated with the complete orthonormal set (f(n)}, and A is a positive definite (non-degenerate) operator on the function space which produces the field theory action, S[X], when incorporated into the inner product, i.e. (X,X) -> S[X] = (X,AX) then the path integral for the (free) field theory can be computed: fdX exp-(S[X]) = fdX exp-(X.AX) = 1 J J /DetA A typical example for the above would be the case where the field variables are taken to be elements of a Hilbert space, the inner product would be the usual Hilbert space inner product, A is an operator on the Hilbert space, and Det A is the product of the eigenvalues of A. Since the Hilbert space is usually ©o-dimensional, there may arise the need to regulate the infinite product of the eigenvalues, but that is of no concern at the moment. 26 In Polyakov string theory, techniques of gaussian path integration play a crucial role because the integration over the space of embeddings is computed as a straightforward gaussian integral. The field variables are the embedding functions, mapping a 2-dimensional riemann surface into a d-dimensional euclidean target space. By simply considering the embedding functions as field variables living in a d-dimensional euclidean space, the underlying integration measure is taken to be a d-fold product of gaussian measures. The string action is quadratic in the embedding variable, X, and the integral is performed. To look at this more closely, the Polyakov path integral will be written symbolically in a way that separates the embedding integration from the integration over riemannian metrics: Z = [[Metrics] e'S[sl f[Emb] e'SIX'8] = [[Metrics] e'S[sl Z (2.2) J J J Emb As in the first chapter S[g] and S[X,g] have the form: S[g] = A.Jd2z y j + Pjd2z R and S[X,g] where g = det ga^, and the repeated index p indicates a summation between |i=l and (i=d, the dimension of the target space. The object of study will be ZEmb . Integrating S[X,g] by parts converts it to a useful form incorporating the laplacian on the Riemann surface (the boundary term is zero for closed Riemann surfaces). This gives: S[X,g] = - i j d 2z X^AX^ where A= d ^ g \ ) The embedding variable X has d-components, each of which is expanded over a complete orthonormal set as before. The laplacian operates on the components of the field variable expansion in the usual way, pulling out the eigenvalue of fin). For a given component of X, call it Xv, the result is: 27 A X V = A a > ) 'X aX f(n> where on is the eigenvalue, and the sign is chosen so that an > 0. For the same fixed component of X, the integration in the action reduces to a summation of the coefficients squared times the appropriate eigenvalues: i f d 2z X vA X v = - X (a£>2oa The path integral for the v-th component becomes simply a product of gaussian Riemann integrals: |dXv exp -[XVAXV] = (XTexP -[X (aV/°J =tn oj 2 = [Det A] 2 J J n V7C n n Now when the product over all d-components of X is taken, the standard result is obtained, for p = 1 to d: _d JdX1...dXd exp -fjjVzX^AXa = [Det A] 2 The discussion above appears to be well defined, but a problem arises because the laplacian is a degenerate operator whenever the mode expansion of X includes constant modes. Consider again the mode expansion of X and let a0 represent the constant term in the expansion (f(0)=l), along with the action of the laplacian: oo oo xv = X a> ) AXV = X aX f(n> n=0 n=l Now when the laplacian acts on the expansion, the a0 term drops out (or can be considered as having a zero eigenvalue), so the term in the functional integral corresponding to da0 becomes infinite. This leads to the problem of zero modes, which arise whenever the space of functions over which the integration is performed includes constant functions. The 28 standard solution to this problem is to remove the zero modes "by hand" from the domain of integration, so that the determinant has all zero eigenvalues removed. However, since the treatment of zero modes will be one of the most powerful tools in developing the variable dimension field theory in this chapter, a separate section will be devoted to further exposition of the problem and an explanation of the connection with sets of measure zero. 2. ZERO MODES & MEASURE ZERO SETS In the first section it was indicated that the underlying gaussian measure for the path integral involves an integration over all of the coefficients in the expansion of the field variable, including the constant components. Also, it was stated that this underlying gaussian measure induced a well-defined measure when a non-degenerate operator was incorporated into the inner product. It is just this requirement of non-degeneracy which the laplacian fails to satisfy, since it acts on constant components with zero eigenvalue. To ensure consistency, the calculation should first be done using the underlying gaussian measure: (2.3) where only the v-th component has been shown; the term Vol(a0) is infinite and the determinant has only non-zero eigenvalues. However, the generally accepted definition of this integral is: (2.4) It is evident that this involves normalizing the infinite constant to 1. 29 Equivalently, this second definition can be viewed as removing a translational degree of freedom in the target space (since the constant term serves only to translate the field configuration away from the origin) by implicitly incorporating a 5-function into the integrand. Zero modes are generally considered to be constant field configurations (i.e. when all terms in the field expansion vanish except for the n=0 term) and are thought to give rise to the infinite constant in the definition of the integral (2.3). However, the discussion above indicates that the so-called infinities arising from zero modes are inherently tied up with the choice of the definition of the gaussian integral, and the infinities are not only associated with constant field configurations So, even at the simplist level of a one-dimensional field theory, the problem of dealing with constant modes arises, and it is a matter of convention whether or not the infinite constant is included. Now that the choice of definition of the gaussian integrals has been indicated, it is important to consider what contribution to the integral is made by constant field configurations, or field configurations in which some of the components are constant. If one considers either of the definitions (2.3) or (2.4), it can be seen that the set of constant functions lie in a set of measure zero with respect to the gaussian measure. First of all, the integrand is positive definite, so if the domain of integration is taken to be two disjoint sets, the contribution to the integral should be additive. Now the function space of field configurations includes constant functions and non-constant functions, where it is probably beneficial to keep in mind that a point in this function space is represented by a particular expansion over the complete set (f(n)}. Divide up the integration in (2.3) and (2.4) into an integration over the constant functions, C, and the non-constant functions, S. The result 3 0 is shown below for both the unnormalized measure (2.3) and the normalized measure (2.4): [dXv exp -(XV,AXV) + |*dXv exp -(XV,AXV) = [dXv exp - (XV,AXV) + (2.3') e s c -J Det A or JdXv exp -(XV,AXV) + JdXv exp -(XV,AXV) = fdXv exp - (XVAX V) + — - ■■ (2.4’) e s c yDetA Since the second term on the right-hand side of equations (2.3') and (2.4') is equal to the result in (2.3) and (2.4) respectively, the contribution to the path integral from the integration over the constant functions is zero, and since the integrand is positive definite, the constant functions must lie in a set of measure zero with respect to the measure. Note that this is a function of the degeneracy of the laplacian and exists irrespective of which definition of the gaussian integral is chosen. The discussion here has been largely heuristic; for a more technical explanation based on the definition of gaussian integrals through cylinder set measures, see section 9 of this chapter. This understanding of the relation between the zero modes and the sets of measure zero is crucial when one attempts to understand the further complications which arise when the dimension is increased and the product gaussian measure is imposed on the path integration. The starting point for the discussion will be a consideration of embedding configurations in which some of the components are constant, so for a d-dimensional target space and embedding variable X = (X^X2,...^), some of the X^ are constant functions. When the d-fold product gaussian measure is used in the path integral, the integral over any set of embedding configurations with any fixed component X^ constant would 31 be zero. This is because the constant components lie in sets of measure zero with respect to the measures corresponding to the v1*1 components. This can be illustrated by a simple analogy with Riemann integrals. Consider an embedding function, X, mapping the domain space, W, into a d-dimensional euclidean target space X:W-»Rd, and consider the case where some of the components of X are constant. It suffices to consider one constant component, say Xd = A. Then X = (X^X2,...^*1^) can be viewed as an extension of an embedding function with d-1 non-constant components Y:W—>Rd_1 by trivially embedding Rd_1 in Rd. The field configuration is determined by the non-constant components, whereas different values for A serve only to translate the configuration along the axis in the target space corresponding to that component. For ease of visualization, restrict the number of dimensions so that W has one dimension parametrized by t, and in the target space d=2. Let Y:t->(t,f(t)) [figure 1] and X:t—»(t,fit),A) [figure 2], as pictured below: 32 It is clear from figure 2 that if the measure dxdy is chosen, then one finds ffxo)dxdy = 0 but if the dimension is restricted jY(t) dx = Jf(odt* 0. Consequently, if the measure dxdy is imposed on the integration, all functions of the form X:t—»(t,fit),A) would be contained in a set of measure zero, so would not contribute to the integral. Of course, one might claim that by translation invariance the function X can be extended to all values of A, as below; however, here jj X(t) dxdy = °o, which is no better. This restriction persists when the dimensions of W and the target space are increased. None of this is at all surprising when applied to Riemann integrals, but when working with functional integrals the choice of measure is more 3 3 obscure, and it is important to realize that the analogy holds. Whenever an embedding configuration has some constant components, it will lie in a set of measure zero with respect to the d-fold product gaussian measure. The specification of constant components is somewhat of an oversimplification which was used for clarity. Obviously, coordinates can be chosen to describe a plane in 3-space in such a way that there are no constant components; however, by a reparametrization (rotation) of the 3-space basis vectors, one of the components can be made constant. There is no essential difference for the case of the embedding configurations. The more accurate statement is that the linear manifold spanned by the embedded configuration is of a given dimension. The dimension spanned by a configuration is the smallest number of dimensions necessary for a linear manifold to contain the configuration (it can be thought of as a euclidean space containing the configuration, with the proviso that the origin of the target space need not coincide with or be contained in the spanning linear manifold). This means that if X = (X1,...^ ) has a linear span of p < d dimensions, then there is a linear dependence between the Xv. Hence p components of the embedding function are independent, and the others may be expressed as linear combinations of these. Thus if the embedded configuration has a linear span of p dimensions when viewed as a submanifold of the target space Rd, then the embedded configuration can be contained in a p-dimensional linear subspace of Rd, but it cannot be contained in any q-dimensional linear subspace of Rd where q < p. The end result is that whenever an embedding configuration spans fewer than the maximum number of dimensions, it must lie in a set of measure zero with respect to the product gaussian measure, since by expressing the embedding configuration in terms of the p independent 34 functions, the remaining d-p functions can be eliminated (up to a constant). There are many aspects of string theory that make it very important to consider the role of the zero modes, the sets of measure zero, and the linear spans of the embedding configurations. Since the domain space for the embedding variables is a 2-dimensional Riemann surface, the embedding configurations will generally be referred to as embedded surfaces. The primary reason one must worry about zero modes and sets of measure zero is that string theory ostensibly claims to predict the dimension of spacetime. From the discussion above it should be evident that selecting a particular value for d in the functional integral automatically forces to zero the contributions from any embedded surfaces that span fewer than d dimensions. While the standard approach to Polyakov string theory determines the value of d from the requirement of anomaly cancellation, this value is directly dependent on the regulation scheme used, and further, if string dynamics were later found to favor some other dimension, the standard approach would never reveal this, because the contributions from these lower-dimensional interactions would lie in sets of measure zero. It is of some value to consider the elements of the theory to determine if it is reasonable to expect the embedded surfaces to span a particular number of dimensions. The action for the string represents the global inner product between the intrinsic metric on the Riemann surface and the pull-back metric induced on the surface by the embedding into the target space, where the pullback is given by 3aX^3bX^. Since there is a summation over \i from 1 to d, it is evident that the pullback is well-defined even if some of the components of the embedding function are constant, although there would be no contribution to the 3 5 action from the constant functions since 3(constant) = 0. Clearly, nothing in the action seems to specify that strings should span d dimensions, although one might suspect that some form of entropy would cause them to span the maximum number of available dimensions. Two considerations make this assumption questionable. Firstly, if there exists a ground state or minimal configuration for the embedded surface, it is possible that excitations above the ground state would deform the surface so that it would span a greater number of dimensions. Then, since the ground state and excited states have differing spans, one or the other of them would lie in sets of measure zero with respect to the d-fold product gaussian measure. Secondly, since the action is additive in p from 1 to d, the higher the dimension of the span of the surface, the higher the number of dimensions through which the surface must be deformed, and naively one would expect a greater contribution to the action. The action enters the functional integral as a negative exponential, so any increase in the action decreases the contribution to the integral. With respect to the concept of strings vibrating or deforming under tension, these considerations have been described as the analog of equilibrium versus entropy relations.[2-l] While there is no assurance that these considerations play a role in string theory, the only way to determine this is to adopt an alternative viewpoint on the correct measure to use in the calculation of the functional integral. This measure will be constructed so that the embedded surfaces may span fewer than the maximum number of target space dimensions, d. When the configurations are concentrated on a linear submanifold of Rd, the support of the measure will be concentrated on that submanifold. The dimension is thus treated as a variable, and will be allowed to approach infinity. Defining this measure can be 3 6 quantified by first returning to the basic question of defining a field theory in Rd. 3. VARIABLE DIMENSION FIELD THEORY At its most fundamental level, the field theory path integral is designed to sum up the contributions from all inequivalent field configurations. The use of the euclidean target space provides for great simplification of the problem because of its intrinsic vector space nature, the ability to define a Hilbert space, etc. However, there is the inherent understanding that because of the trivial invariance of Rd under rotations and translations (i.e. the semi-direct product group SO(d) ® Td), the path integral written down in Rd overcounts equivalent field configurations, so at some point one must formally divide by the volume of this group. In this context one can postulate the existence of an abstract space of Inequivalent Field Configurations, call it IFC, where each point in the space is a unique (inequivalent) field configuration. Reconsider the (integrated) measure for a functional integral based on a product gaussian measure in Rd, written symbolically as: J [Field VarCR^] = Jdx\..dXd exp-(X;X) = J[ffC]Jd)i(SO(d) © Td) (2.5) where the second term is as previously defined, the third simply expresses the overcounting of equivalent field configurations because of the trivial invariance in Rd, and dp(..) just represents the Haar measure on the group. In the standard approach to field theory, particularly when the value of d is known from the outset, or when the linear span of the field variables is expected to have some generic value, one need go no further than the above (bearing in mind the considerations of the 3 7 previous section and the fact that configurations that have linear span < d lie in sets of measure zero). This is acceptable because whenever the integrand is invariant under the action of the group, the integral over dp becomes just the volume of the Haar measure, which is then normalized to 1. However, as the discussion will show in a later section, this normalization of the Haar measure is by no means trivial, and it may play a role in the eventual regularization of the final results. In order to develop, the variable dimension theory, one must find some way to transform the abstract measure in (2.5) into a form which gives support to field variables which span fewer than d-dimensions. In other words, the task is to develop a graded, dimension non-specific measure, and this will require careful consideration of the abstract space, IFC, and the Haar measure. In order to learn more about the abstract space of Inequivalent Field Configurations, and with an eye towards the application to Polyakov string theory, it is useful to introduce some notation. Let Emb(W,Rd) be the space of functions mapping the domain W into Rd (where W will be identified with the world sheet Riemann surface). Emb(W, Rd) will be used in the sense of Polyakov and others, and should be viewed as containing C°° functions. It should be noted that this space overcounts equivalent configurations because of the SO(d)(s)Td invariance of Rd. Let the space of Inequivalent Field Configurations be denoted by EMB(W), where every point in this space represents a unique and hence inequivalent configuration. Now one can fiber Emb (W, Rd) over EMB(W), where the fibers are just the equivalent configurations related by SO(d)®Td. The fibration is represented by, Emb(W, Rd) <------{X ~ X iff g X = X for some g e SO(d)® T 6} EMB(W) 3 8 In attempting to write a path integral which is not dimension specific, one can try to write the integral as an integral along the base space EMB(W) and then along the fibers, as in the last expression in (2.5). The measure on the abstract space EMB(W) is not known; however, since it is the space of inequivalent field configurations, one can take advantage of the uniqueness of its elements. Each element of EMB(W) can be characterized by its linear span when viewed as a submanifold of Rd , so EMB(W) can be decomposed into a union of disjoint sets according to the linear span. Thus each element of EMB(W) is endowed with an intrinsic dimension which is equal to the minimum dimension of the linear submanifold of Rd which can contain the configuration. The decomposition can be represented by the disjoint union: EMB(W) = u EMBp(W) where a function f e EMBp(W) has intrinsic dimension p and hence spans a p-dimensional linear submanifold of Rd. The Haar measure on the group can also be decomposed by using the coset decomposition of the group. Here the cosets for the group SO(d) are given by the relation SO(d) ~ [SO(d)/SO(d-l)j SO(d-l) ~ Sd_1 SO(d-l), where S dA is the d-1 sphere. Likewise, the translation group decomposes into products. Using these cosets, one can write the Haar measure on the group SO(d) © T ^as [2-2]: d|i[S0(d) © Td] = dn[S0(d)]dn[Td] = dp[sO(d)/SO(d-l)]«»dp[ SO(p+l)/SO(p)]dn[sO(p)]dn[Tdp]d|x[Tp] = dn[sdl]»« dp[sp]d|i[SO(p)]dp[Td'p]d(i[Tp] 39 Using the decompositions of the space EMB(W) and of the Haar measure, along with the additivity of an integral with respect to disjoint sets, the measure in (2.5) can be rewritten (Eq. 2.6): J[Emb(W,Rd)] = ^ J[E M B p(W)] Jd|J.[sd1] — dp[sp]dp[SO(p)]dp[Tdp]dp[Tp] p Recalling the earlier discussion where it was shown that when the underlying gaussian measure on a space of functions mapping into RP is altered by incorporating the laplacian into the inner product for the gaussian measure, the support of the measure is concentrated on functions with linear span p. All other embedding functions with a linear span * p He in sets of measure zero. From a physical point of view, it would overcount equivalent configurations because of the invariance group SO(p) © TP. Thus, it corresponds to the integral over functions in EMB(W) of intrinsic dimension p, and one can give support (in the measure) to functions of intrinsic dimension p by identifying: Using this identification, (2.6) becomes: J[Emb(W,Rd)] Vol[Sd4] • • • Vol[sp] Vol[Td p] Jd X 1..,dXp exp-(Xp,AXp) (2.7) p Since the desired result is the integral over EMB(W), one must divide (2.7) by the volume of SO(d) (s) Td. Again using the coset decomposition of the groups, the factors of the volumes of the spheres can be made to cancel, leaving: 1 JdX 1...dXp exp - (X^,AXP) (2.8) P Vol[SO(p)] Vol[Tp] This is the desired dimension non-specific measure which will be used to construct the variable dimension field theory. From (2.8) one can write 40 the general expression for ZEm^: ip 1 (2.8') P Vol[SO(p)] Vol[Tp] / Det A Notice that it gives support to embedding configurations with any intrinsic dimension within the range selected for p. Also note that a given configuration of intrinsic dimension q will lie in a set of measure zero with respect to (2.8) for every value of p except when p = q. The problem of zero modes has been removed as well, because for a function of intrinsic dimension q in EMB 4(W), it would have a linear span of q-dimensions, so there would be no constant components and hence no zero eigenvalues. The volume factors in the front compensate for the overcounting in the gaussian integral resulting from the rotational and translational invariance in the target space. The net effect is a sequence of appropriately weighted field theories concentrated on field configurations of intrinsic dimension ranging up to the dimension of the target space Rd. Here the maximum dimension d will be treated as a variable, and later will be allowed to approach infinity. It is now appropriate to recall the previous discussion on the possible definitions (2.3) and (2.4) of the gaussian measure incorporating the laplacian. In (2.3) the translational degrees of freedom were not removed which led to the infinite factor Vol(a0), while in (2.4) the volume factor was removed by eliminating the translational freedom. The translational freedom talked about in (2.3) and (2.4) is the same translational freedom in (2.8), so if the definition (2.3) is selected, there would be a factor [Vol(a0)]P which will cancel the factor Vo1[Tp] in (2.8). Likewise, if definition (2.4) had been selected, the translational degrees of 41 freedom would already have been removed, so the factor Vo1[Tp] would never have appeared. Thus, by maintaining consistency in the definition of the gaussian integral, it is a simple matter to handle the infinite volume of the translation group. Handling the rotation group, however, is no simple matter. The manipulations on the Haar measure were very formal, and how to determine the volume factor for SO(p) has not been defined previously. The primary difficulty is that the SO(p) invariance induces an action in the function space of embeddings, so the object that must be calculated is really a functional Haar measure. The next section will be devoted to further discussion of the functional Haar measure and the development of an approach which will calculate the volume factors. 4. THE FUNCTIONAL HAAR MEASURE A Haar measure for a group is defined as the measure on the group manifold which is invariant under the action of the group. It will simplify the discussion if one distinguishes between the topological space of the group and the object space on which the group acts. In this case, the topological space and the object space are the same, since the group is acting on itself. A Haar measure is generally associated with defining an invariant integral over the group manifold. Then the group integral is required to satisfy: J Jdn(X) exp - i j d 2z y j = Jdp(X) exp - l 0 0 0 • x 1' x 1 • sin 0. • cos 0. • • 1 1 • P4 X t •• • l • • • 1 (2. 11) • • • -cos 0. i • sin 0.i • ,d d X X ••• • l where X represents an equivalent configuration, R{ is a generic generator for SO(d) and the bold 1 represents a unit matrix which fills the matrix so that the overall dimension is d x d. Since each component of X has an expansion over a complete orthonormal set, it is evident that R acts simultaneously on all terms for a given Xv. In general, the complete orthonormal set over which the embedding functions are expanded will form an infinite dimensional Hilbert space, and the properties of Hilbert spaces will be used in what follows. The function space of the embeddings can be considered as the 44 direct sum of d Hilbert spaces, for the case when the target space is Rd. The terms "function space of embeddings", "space of emedding configurations", and "d-fold sum space" will be used interchangeably. This d-fold sum of Hilbert spaces is a euclidean space since the orthogonality of the basis elements in each Hilbert space makes them separately euclidean, and the direct sum makes them all mutually orthogonal. If X = (X1,...^ ) and each of the X^ has the expansion: Xv = a* f(i) where i ranges from 0 to and v corresponds to the component of X and can range from 1 to d. Then a point in the d-fold sum space can be represented by an (infinite) vector: ,1 2 d 1 2 d 1 . V X ^ ,...,2-j ,3-2>&2’ *’**^2* *^3* ) Then the generators of induced SO(d) transformations in the d-fold sum space are of the form: 0 0 0 0 • • 0 M 0 0 0 • • (2. 12) 0 0 • 0 0 • • 0 0 0 • 0 • • 0 0 0 0 • • • where X is the infinite vector in the d-fold sum space, and the elements along the diagonal [R.] are the d x d generators of SO(d) appearing in (2.11). The matrix is infinite in extent in all directions, and the only non-zero elements are the matrices [Rj] along the diagonal. It is now possible to verify equation (2.9) in a straightforward manner. First consider the action. Since a rotation in euclidean space is isometric, the action is invariant, with the result: 45 n r i k -»w n n ^ n=l ^.=1 n=l |j.=i where the factors of k have been dropped, and [J] is the Jacobian of the transformation. However, the Jacobian is 1 since it is just the determinant of the matrix in (2.12) which reduces to Det [Rj] and this equals 1 for any element of SO(d). Thus, the conditions of equation (2.9) are satisfied. It is possible to learn more about the equivalent embedding configurations (when considering the function space as the d-fold sum of Hilbert spaces) by endowing the d-fold sum space with the inner product inherited from the separate Hilbert spaces. If (X,X) is the inner product in the d-fold sum space, and (XV,XV) is the inner product in the v-th Hilbert space, then: (X,X) = (x '.x 1) + (X2,x2) + . • • + (Xd,xd) = X <®i>2 + X d (XX) = X X«): = X fri>: i=l p=1 i=l Since each of the SO(d) matrix elements along the diagonal in (2.12) acts isometrically on a group of coefficients ranging from a1 to ad for a particular value of i above, the magnitude of the r. is preserved. This means that there are the further constraints: d X(af)2 = rf fori=l,...,~ (2.15) So in the d-dimensional subspace associated with a particular value of i in (2.15), the equivalent points are again constrained to lie on a sphere, this time of radius In order to calculate the contribution to the path integral from just the inequivalent states, it is easiest to work with the initial path integral including the overcounting, and then try to factor out of the integral the contribution from the equivalent states. In this way the volume of the Functional Haar measure will be determined and automatically factored out of the result. Two approaches to this problem will be considered. §4A FIXING THE TRANSLATIONAL AND ROTATIONAL DEGREES OF FREEDOM WITH DELTA FUNCTIONS 47 The most obvious approach to the problem is to try to incorporate some fixing parameters into the path integral, so that the rotational and translational degrees of freedom in the target space are removed. The object is to determine a set of delta functions for target spaces of any given dimension that will achieve this fixing of the configuration. For ease of exposition, consider functions embedding a one dimensional domain into Rd. Let each component of the embedding function X = (X1,...^ ) be expanded in a Fourier series: oo ©o X v(t) = X bnvcos(m) + X an sin X(°) = X bn = 0 ^ 8 X b n n=0 n=0 For this case (2.16) the action of the laplacian pulls out a factor of -n2, so the action for the path integral reduces to: (X,AX) = - X n2(b / + - X n\? (2-17) n=l n=l Now the full path integral with the delta function included becomes: o o rr^Vdb t —r da q. exp - (2.18) HI-pIT-pS X b„ X ^ + bn) n=0 V K n= l V K n=0 n=l 7^7 Det A where the determinant has all the zero eigenvalues removed because of the delta function in the integrand. This should not be at all surprising, 48 since the removal of translation invariance is exactly what led to the definition (2.4) of the measure. The extra factor in front comes from the constant in the measure. If the dimension of the target space is increased so that d = 2, one must fix the two degrees of translational freedom plus one degree of rotational freedom. The translations can be fixed using delta functions as above, but another condition is needed to fix the rotations. For this one can specify that the tangent vector at the origin of the domain space must push forward to lie on a particular line in the target space, which will be taken to be the positive part of the X1 axis, so X'(0) = (A,0) for any A>0. This leads to the following: HFx ‘ ( t ) U (2.19) i x 2(t)lM) = 0= £ n c (2.20) where the expansion coefficients of X1 will be a and b, and those for X2 will be c and d. The path integral will have integrations over b and d just like in (2.18) which just bring down the eigenvalues and the factor in front, so consider the a and c integrals: oo X 2 2 m a - A (2.21) m m am = i2 A TT1 i m m = l VK m=l m=l m=l oo X . 2 2 1 1 k c, lim —= (2.22) k S = m k=l V 71 k=l k=i V i t k=l k— -J k The b and d integrations give a contribution of the form: o o (2.23) 49 The final result for the path integral in d=2 is the product of (2.21),(2.22), and two terms like (2.23). The paired products of the eigenvalues give the standard 1/VDet A factor, so the final result is: (2.24) The important thing to realize is that this is the answer to the path integral for the inequivalent states, and the last expression in brackets results from factoring out the Functional Haar measure. There are two very important implications of this result; first, the idea of normalizing the Functional Haar measure to 1 involves an infinite factor and, second, when working with a target space of dimension greater than one, the last expression in brackets should be incorporated into any regularization of the determinant. While these rudimentary results are encouraging, they do not generalize to higher values of d in an obvious way. The limitation is that one is constrained to derive translational and rotational fixing conditions at one point. Attempts to derive higher derivative conditions to fix the translational and rotational degrees of freedom are unsatisfactory for several reasons. Primarily, the tangent vector fixing condition for d = 2 has the well-defined geometric interpretation of the push forward of a tangent vector, whereas for higher derivatives this geometric picture is more obscure. Even if one blindly posited conditions to fix the translational and rotational degrees of freedom based on the idea that higher derivative vectors could be made to lie in some plane, 3-space, etc., the calculations which arise are too unwieldy. Instead another approach has been developed which is more geometric, and makes use of the structures and constraints derived previously for the embedding space viewed as a d-fold direct sum of Hilbert spaces. §4.B FIXING THE TRANSLATIONAL AND ROTATIONAL DEGREES OF FREEDOM WITH THE GEOMETRY OF FUNCTION SPACE The fundamental idea in this second approach is that it should in principle be possible to use the constraints and the generators of equivalent configurations in the function space of embeddings to determine the volume of equivalent configurations within the d-fold sum of Hilbert spaces. Earlier it was shown that for a given configuration XQ (i.e. a point in the function space of embeddings) of magnitude (XQ,X0) = R2, any equivalent configuration must lie on a sphere of radius R in the function space of embeddings (the translational freedom has already been excluded). It is also known how to go from the point XQ to an equivalent point by transforming as in (2.12). This information should be sufficient to determine the volume of equivalent configurations on a particular sphere of radius R in the function space of embeddings. Consider the case where the target space has dimension 2. The generator of rotations in the function space (2.12) takes on a particularly simple form, with the matrices along the diagonal given by: sin 0 cos 0 (2.25) -cos 0 sin 0 Now let the reference point X0 be given the expansion: (2.26) n n Then in the function space of embeddings [with the topology of the d-fold sum of Hilbert spaces], X0 can be represented by an (infinite) vector: 51 In terms of the reference components, the coefficients of any equivalent point can be found: a.(0) = a° sin 0 + b° cos 0 and b.(0) = -a.° cos 0 + b° sin 0 (2.27) Since the relations (2.27) parametrize the path (for higher target space dimension it would be a volume) of equivalent states on the sphere of radius R, the total length can be calculated: 2tc i _ _o _ 2* d b;(0)’ V ( 0 ) = d e (2.28) d0 However, the integrand reduces to: -.2 i2 da.(0) db.(0) + = X [ (ai)2+(b°)2] = r2 (2.29) d0 d0 So finally, V(0) = 2tiR (2.30) This means that on each sphere in the function space of radius R, one must factor out V(0). By converting the measure for the path integral to a set of (infinite) spherical coordinates, this calculation can be done. The conversion to spherical coordinates presents some notational difficulties, but since the function space of embeddings is a euclidean space, the conversion is conceptually simple. In the following discussion, the integration over the constant term in the expansion will be left out, since the earlier discussion showed that the infinite factors arising from translation invariance cancel out for both definition (2.3) and (2.4) of the gaussian measure. There will, however, be some differences in the factors of k when compared to the results using delta functions because of the Vrc in the denominator of the dbQ term. For the target space of dimension d = 2, the gaussian measure will be written: 52 d|i(X) = lim (2.31) X— rn=l i r i=l tV ^ TC p -I. n,i f s d J When this is converted to spherical coordinates (and noting that as it is written there are 2X variables) it becomes: dr d n sinm0 d0 (2.33) f n ^ ) which causes all intermediate terms to cancel out, leaving only the numerator of the first term (=1) and the denominator of the integration over the last angular variable. This gives: 2 i/jt d n = rot) (2.34) where dfl just stands for the angular integrations. The radial integral becomes: dr r2X-1 (2.35) r exp-r-(r2> = —p r [ir^ 4)] j J n 2jtr 2 ( J n f L2 2 J Putting together both terms and taking the limit as X —»°° gives: 1 ™ - 7 > 1 1 lim lim — = (2.36) 2tt 271 X— rot) ^->°° V X The second term in (2.36) was found using the relation: 53 m - i ) r(n+a) - a l o g n lim —> l i m ------(2.37) n — > ° ° r(n) T(X) When this result is used in the path integral incorporating the laplacian, the result is the same as (2.24), except for the factor of k which arose from the different definitions: 1 1 1 1 d x 'd X 2 exp -(X^,AX^) = li m — = (2.38) 2 K Vol(0) Det A v X The beauty of this second approach is that it provides a well-defined algorithm for finding the functional Haar measure as the dimension of the target space is increased, which is in principle calculable. Given a point in the space of embeddings, X0, equations (2.25) and (2.26) generalize to allow one to write any equivalent configuration as a function of XQ and the parameters of the SO(d) transformation, call them oc,p,y,... This relation will be written explicitly as X = (a,p,y,...,X0). Then the volume of equivalent configurations can be found by calculating: Vol(a,p,y,...) = f [Det 3;X(a,p.....Xo)9.X(a,P,...,Xo) ]2 docdpdy... (2.39) where i and j run over the parameters a,p,y,... and one might notice that the term in brackets is just the pull-back metric from the space of embeddings onto the parameter space of the SO(d) transformations. While the actual calculation of (2.39) appears quite difficult, it is clear from considering the constraint equations and dimensional requirements that for a d-dimensional target space one would expect to factor out the volume of a (d-l)-sphere from the sphere of radius R in the function space of embeddings. Since the interesting results lie in the radial integrations, those will be considered first, to derive a scaling limit 54 as d —> oo. 5. SCALING LIMIT AS d^oo To determine the scaling as the dimension of the target space is increased, the volume of a (d-1) dimensional sphere of radius r must be factored out of the integral. For the purposes of deriving a scaling limit as d— the radial integral must have a factor of l/r4*1 included in the integrand to account for the power of r associated with the (d-l)-sphere. Two integrals will be useful: 00 oo « 2 i i* 2 i i 1 dr r2"'1 e‘r = - T(n) and I dr r e'r = - T(n + -) (2.40) 0 0 Using (2.40) gives: jctf — r exp-Cr) = - 1 rt"t [ —dA, - - d + I 1 ]-\ (2.41) 0 r To find the scaling from the angular integrations, reconsider equations (2.31) and (2.32) for a target space of d dimensions. This would give dX variables, and the last angular integration is over 0dx_2. The scaling from the angular integrations is then: dO —^ 1 (2.42) j a F L 2 J When (2.41) is multiplied by (2.42), and using the expression (2.37), the factor which compensates for the overcounting of equivalent states has the form: ■tdi l i m (2.43) k—»«> 55 This result can now be used in conjunction with (2.8') to determine the scaling limit of IP-1 1 P 1- Jx ^Em b ■"* Jim - i - v - * »Z P= i | y x ■^Det A A— DetA - 1 1 (2.44) ■J Det A The limit was obtained as a simple geometric series. It gives the result that by summing over contributions from all dimensions, the answer (2.44) is what would be expected for a standard field theory in one dimension. It should be noted that the calculation did not involve any switch of the summation and the limit. Since the effective dimension of the variable dimension field theory in the naive scaling limit is 1, it appears that an effective compactification is built into the theory. With this in mind, the complete solution will be considered in the next section. 6. COMPLETE SOLUTION To derive the complete solution requires only that when calculating the compensating factor which accounts for the overcounting of equivalent states, the most general form for the volume of the (d-l)-sphere which corresponds to Vol(a,p,y,...) be used in the gaussian measure. The appropriate gaussian measure is just the higher dimensional generalization of (2.32). The volume of the (d-l)-sphere is given by: £ VolCS11'1) = Id 1 dr[f] r[f -f 4] = lim (2.47) X—>oo7T 2 n f ] This result can now be used in conjunction with (2.8') to derive the complete result for Z gm^: y r t rrtf] r[f-f,i] Z = lim (2.48) Emb 1— « l^J 2r[f] The extra factor of T[d/2] in the numerator does not grow quickly enough to stop the convergence of the sum, so the limit exists and represents a non-perturbative correction to [Det a ]-1/2, which arises as the naive scaling limit (2.44). 7. IMPLEMENTATION IN POLYAKOV STRING The developments of the preceeding sections must now be placed in the broader perspective of the full Polyakov prescription. This involves the regularization of the determinant and the addition of the integration 57 over metrics. Polyakov worked in the conformal gauge g = p(z)8ab, and used this in determining the form of the regulated determinant. This was found using the trace anomaly relation, and gave the form of the regulated determinant as: Z£[p(z)] = exP - fd2z[i@ log p(z))2 + |i2p(z)] (2.49) 48tcJ 2 Applying the same scheme for regularization of the determinant to the results of section 5, gives for the scaling limit: 2^[p(z)] = exp - -J—|d2z[i(3 log p(z))2 + p2p(z)] [Scaling] (2.50) 48itJ 2 The result for the complete solution of section 6 is: 2£[p(z)]- lim£. exp - — fd2z[i(9 log p(z))2 + p2p(z)] “K—**> d=l 48t:j 2 n —]2 [Complete] (2.51) Polyakov then proceeds to gauge fix the metric integration, and regulates the Faddeev-Popov determinant, with the final result: Z Re®= Jd 0 (z)exp - — [d2z[i(9 CO Z** = U z) limY.rtflrty-f+j] exp - jd2z[i@ <|)(z))2 + p2e°] n — ] 4871' 2 2 J [Complete] (2.54) 58 From these results it is apparent that by reinterpreting the measure for the embedding integration, correlating the zero modes and sets of measure zero, and factoring out the Functional Haar measure, the Polyalov string prescription is radically altered. It becomes impossible to select a critical dimension in which the Liouville action disappears, as can be seen from (2.53) and (2.54), so quantization of the Liouville theory must be undertaken. In the scaling limit the infinite sum over dimension has been performed, so the Liouville theory is clearly defined. For the complete solution, the limit of the infinite sum exists, and would lead to a fixed effective dimension, call it deff. Since it represents a corrrection to the scaling limit, which has effective dimension 1, deff would presumably be a small number. The factor multiplying the Liouville action would be [26-deff]/487t, so the quantization of the Liouville theory would be necessary. It should be noted that in subsequent papers, other authors recognized that the initial paper by Polyakov did not include certain technical difficulties arising from considerations of moduli on Riemann surfaces.[2-4,5,6] However, these problems arise at the level of the integration over the metrics, and this will not affect the calculation of Z Emb- Zero modes were also ignored in the Polyakov paper and require special treatment, whereas in the approach of this chapter they have been accounted for in a natural way. As a final application of the variable dimension measure, it will be shown that source terms can be included. 8. SOURCE TERMS The inclusion of source terms in the prescription of this chapter is straightforward since every step is based on simple gaussian integrals. 59 Consider the generic case of functions with linear span p. As usual, X will represent the embedding variable, and let J be a source. J maps the domain space W into the dual RP*, which can be identified with RP. The functions X and J will be given appropriate expansions, x* = X a*f(n) and JV = X *#(") (2.55) n n and they will be coupled through the inner product, i.e. (J,X) = fd2z X V = a£b£ (2.56) n=l n= l Adding the source term to the action, the path integral is still gaussian and can be computed (where only the integration over the v-th component ofXis shown): n=l V ^ n=l n=l = Y[—t= ; exp - — = — , exp - _L = - exp-(J,A_1J) (2.57) n=1J a n a n VDetA an VDetA where the index v has been dropped for ease of notation. The result would be the same for each component of X, and when incorporating it into the variable dimension theory, the factors determined by the functional Haar measure would be included. Note that the exponential in (2.55) converges ifZ[(bn)Van] converges. Since J is some functional expansion in L^(W), it must satisfy 2(bn)2 < ©o. If all but a finite number of the a are >1, then X^< where the prime indicates that the sum includes only terms for which a > 6 0 1, and the remaining terms are finite. For all cases of interest the on will satisfy these conditions. 9. APPENDIX - SETS OF MEASURE ZERO Gaussian measures are defined by means of non-degenerate scalar products in a linear topological space, which is taken to be an infinite dimensional Hilbert space for the case at hand. The discussion will follow closely the presentation in Gel'fand and Vilenkin, Generalized Functions Vol.IV.[2-7] Let d> denote the linear topological space and B( dH^X) = Jdy exp-iB(\|/,\|/) (2.58) Xe4< where ye'F and d\|/is the Lebesgue measure corresponding to B( The preceeding chapters have all been concerned with Polyakov string theory under the restriction that the target space is Rd. This restriction has many serious implications, not least of which is that compactification schemes imply that the extra dimensions of string theory must be curved up at a very small scale, so it is difficult to see how the Polyakov prescription can be extended to a compactified string theory. This chapter will deal with the generalization of string theory to general curved target spaces, beginning with the action and the equations of motion. The first section will consider the existence of classical solutions to the equations of motion by showing the connection between strings and the mathematical theory of harmonic mappings and minimal surfaces. By considering strings in this context, it will provide an interpretation of the action as the energy functional, and a description of the second equation of motion in terms of a tension field defined on the embedded string surface. The mathematical aspects will be applied to a toy model for genus zero and genus one surfaces in an S3 target space, based on background field expansions about a classical solution. Since the integral is conformally invariant, the calculations will make use of a global parametrization for the Teichmuller space of genus one to derive a new measure for the Teichmuller integrations. This approach corresponds to developing a "gauge-fixed" path integral; however, the global parametrization and measure are chosen in such a way that the integral is invariant under the gauge transformations between the fundamental regions described in appendix A. The toy model will then be 6 3 extended to develop a simple (time sliced) model for cosmological membranes in S3 x R. The gauge choice corresponding to time slicing will allow for the specification of the intrinsic metrics on the membrane. In order to maintain a consistent presentation, some notation will be set out. As before, the domain space, W, for the Polyakov string theory will be closed Riemann surfaces, endowed with an intrinsic metric, g. The target space will be a general manifold, M, with a fixed background metric, G. The metric G will be arbitrary, but will not be considered as a variable of the theory. The notation X:W—>M will again represent an embedding function taking W into M. The action is generalized to take into account the presence of the target space metric, G, but it is still just the global inner product between the intrinsic metric and the pull-back metric, written: (3.1) where the pull-back metric is: [x*G]ab = 3aX^bXv G^(X(z)) (3.2) In the above, the explicit dependence of G on X(z) is shown, which essentially means that since G is arbitrary, it can vary from point to point in the target space; consequently, the pull-back metric depends on the position in M where the image of W resides, i.e. X(z). This will be important when deriving the equations of motion. The equations of motion are derived by varying with respect to X and g, but not G since this is not a variable of the theory. This produces a slightly altered form of the equations: tab = a ax ^ Kxv b (3.3) (3.4) 64 In the above, the convention has been introduced that latin indices refer to the coordinates on W, and greek indices refer to coordinates in (a coordinate patch of) the target space. The extra term in (3.4) results from the variation of G with respect to X(z). When strings are allowed to embed into a general curved target space, the connection of string theory with the mathematical theory of harmonic mappings and minimal surfaces becomes evident. The standard reference for the mathematical work on harmonic maps is by Eells and Sampson.[3-l] The general framework of [3-1] is the embedding of compact Riemann surfaces into general Riemannian target spaces, where the action is the same as (3.1) and is identified as the energy functional of the embedding. The Euler-Lagrange equations resulting from the variation of the embedding lead to the definition of the tension field of an embedding, which will be shown to be equivalent to (3.4). The tension field for an embedding, X, is defined to be: Wv = a x ’1' + gab Mn a(5 axVxPa b (3.5) where ab a2 ab Wj_c 3 A = g «-«. 3**. b- g ab -v c (3.6) oz dz dz In (3.5) and (3.6), WT and Mr represent the Christoffel symbols on W,M respectively, and in (3.6) the derivatives on W have been written explicitly to avoid confusion. The connection with (3.4) can be established by expanding the derivative in the second term, and after some manipulations, identifying: xv g yY s it i gab0a3bxv)G a D o = v yBiTi v gab(—-—--va-xb ) ov dz dz 3a(7 5 ^ i g ab)3bXvG av = - 7 5 ^ 1 glbV b3cX’G ro (3.7) 6 5 VDetggabd.Xvd G - -i/Det g A x M G = .VDetg gab3 X^, Xv Mn G Adding up the terms on the right hand side of the equations (3.7) shows the equality of (3.4) and (3.5). Embeddings for which the tension field is zero, i.e. they satisfy the equation of motion, are called harmonic maps. There is a clear physical interpretation that can be given to the tension field and harmonic maps, which gives some perspective on the terminology and motivation of the mathematical investigations. This interpretation of a harmonic map X:W—>M comes from Eells and Lemaire [3-2]: "Imagine that W is made of rubber and that M is made of 'marble': the map X constrains W to lie on M. Then with each point p e W we have a vector Tx(p) = div(dX(p)) at the point X(p) e M representing the tension in the 'rubber' at that point. Thus X is harmonic if and only if X constrains W to fie on M in a position of elastic equilibrium." It will be seen that this interpretation of the string equation of motion (3.4) as the tension field will prove useful when trying to learn more about the classical aspects of strings in curved space. The discussion about harmonic maps only made use of the second string equation of motion. Employing the first equation of motion imposes the further restriction that the pull-back metric must be conformally equivalent to the intrinsic metric, since (3.3) can be solved to yield gab ™ where Q (z) is a positive function on the domain Riemann surface which will be referred to as the conformal factor. Any embedding whose pull-back metric is conformally related to the intrinsic metric is called a conformal embedding, so (3.7) and (3.8) combine to impose the restriction 6 6 that any embedding that satisfies the classical equations of motion (3.3) and (3.4) must be a conformal harmonic mapping of W. At this point it is probably worthwhile to mention that the term "embedding" used in string theory is only loosely defined. It is generally used in such a way that it should be considered to be a general smooth function mapping W to M, which can be a many-to-one map. However, for the purposes of the mathematical discussions which relate to the pull-back metric, it is best to restrict to the spaces of immersions, Imm(W,M), and imbeddings, Imb(W,M), since for these spaces the image of W in the target space is such that the pull-back metric induced on the surface is non-degenerate at all points of the image. These restrictions are reasonable for several reasons, and at any rate they probably correspond to the mental image that is meant to be implied by the term embedding. First of all, the spaces Imm(W,M) and Imb(W,M) are open, dense subspaces of C°°(W,M), so a general map can be approximated to any degree of accuracy by imbeddings and immersions, so their use should not significantly alter any results which may be derived. Further, since the intrinsic metrics are not allowed to be degenerate or singular, and since the equation of motion (3.3) requires the pull-back to be conformally related to the intrinsic metric, it is reasonable to require the pull-back metric to be non-singular. The distinction between Imb(W,M) and Imm(W,M) is that an imbedding is required to be a one-to-one map onto its image, whereas an immersion is allowed to have self-intersections so long as the push forwards of the tangent vectors at the points of intersection are independents 3-3, p.45] The fact that classical solutions to the equations of motion must be conformal harmonic maps can be used to relate the solutions to minimal immersions. A minimal immersion is a map which minimizes the area 6 7 integral formed from the pull-back metric: A(X) = Jd2z J d s i 3aX^3bXvG v (3.9) w This is equal to the energy functional S[X] whenever the embedding is conformal and, further, one has the proposition [3-1 p.126]: If dim W = 2, then for any smooth function from W to M, A(X) < S[X], and equality holds only when X is conformal. Since (3.3) requires that X be conformal with respect to g, the proposition shows that for any classical solutions, A(X) = S[X], and results from the theory of minimal immersions are pertinent. This relationship is stated in the theorem [3-4 p.641, Th. 1.1]: If X:W^M is a conformal (branched) immersion with respect to the conformal structure g on W, then X is harmonic with respect to g if and only if X is a (branched) minimal immersion. In the above, the term "branched" refers to maps with self-intersections, generally associated with uniformization of Riemann surfaces as multi-sheeted coverings of 2-manifolds. For the cases considered in this chapter, branch points will not occur. The converse of the previous theorem is stated in the corollary [3-1 p.126]: If a map X:W—»M minimizes A(X) and is conformal, then X minimizes S[X]. These results are not surprising, since for a conformal immersion X of the form (3.8), one can compute 6 8 iJd2zV5irig V 3bx\ = y I gab^ ) g ab w w w ww w (3.10) w w Thus it has been shown that classical solutions to the string equations of motion must be conformed, harmonic, minimal immersions. 2. EXISTENCE OF CLASSICAL SOLUTIONS The existence of solutions to equation (3.5) is quite problematic. Eells and Sampson point out that the equation is suggestive of the simple equation Au + (j)(u)grad2u = 0, but the presence of curvature adds severe complications. For this reason, the proof of the existence of solutions for specific cases or under certain curvature restrictions has been the focus of research in this area. In this section, some of the results of such work will be considered to give an overview of the area. The next sections will make use of certain specific cases where constructive solutions are known, then will present sample calculations applying the results. The first specific case to consider is when M is flat. Then its Riemannian curvature is zero at every point, and one can select a smooth covering by coordinate charts so that Mr = 0 in each chart. Equation (3.5) then becomes T(X)^ = AXY, where A is given by (3.6), and in these local coordinates the equation T(X) = 0 is linear. This implies that when the target space is Rn, any harmonic map X:W—»Rn must be constant (i.e. must map all of W to a point in M). This can be seen by employing the maximum principle; this states that the solution to the laplacian 6 9 equation on a compact region must be constant or attain its maximum and minimum on the boundary, and since W is compact and without boundary, this implies that X is constant.[see also 3-3, p.119] Of course, if all of W is mapped to one point of M, the pull-back metric is decidedly \ degenerate, i.e. zero. Since the first equation of motion (3.3) requires that the pull-back be conformally related to the intrinsic metric, it is obvious that embeddings of strings into a euclidean target space can never satisfy the classical equations of motion, unless one is willing to admit solutions where the conformal factor is zero. This drawback of the euclidean approach to string theory effectively rules out the application of techniques like background field expansions to the calculation of the string path integral, and if the observable physical states are expected to satisfy the classical equations of motion, the flat space formulation raises questions about what would be observable in string theory. Since the observable states are generally believed to be solutions to the classical equations of motion, the lack of such solutions in euclidean space implies that string theory is an unobservable theory. Even if one admitted the constant maps as classical solutions with conformal factor equal to zero, then the observable theory would be a theory of point-like particles, not strings, since constant functions map the entire domain into a point. This provides initial motivation for investigating the implications of strings in curved target spaces, and this requires careful consideration of the non-linear equation (3.5) for non-flat M. When considering solutions to (3.5) in the presence of curvature, the problem is no longer trivial. The initial paper by Eells and Sampson established the existence of solutions by inventing gradient lines of S[X] in the function space of maps from W to M, the trajectories of which lead to critical points of S[X].[3-1, p.115] To do this they let C°°(W,M) be the space 7 0 of smooth maps from W to M. Then for every Xe C°°(W,M), the set of smooth maps u:W—>T(M) from W into the tangent bundle on M, such that tc - u = X, forms a vector space TXC°°(W,M), where n is the projection operator from T(M) onto M, and TXC°°(W,M) is the tangent space to the function space C°°(W,M) at the function space "point" X. The map u is called a vector field along X. The space TXC°°(W,M) has a natural inner product defined by AvS[X] = l [ S [ X (]] t=0 where X ((P) = expx(p)(t v(P)) (3.12) Xt(P) is the geodesic segment in M starting at X(P) and determined in length and direction by the vector tv(P) e M(X(P)). Then the directional derivative can be related to the tension field by AvS[X] = - ix= t T (X t) (t„ 4 s [X t]= T(Xt)> dvw (3.15) dt w then using _D 3X, a2x “ M 9x* 9x ' ____+ Mp a ______1 (3.16) 3z* 9t 3z3t dz1 3t they find, ld2S[Xt] f ii D /9Xt. d /dX \ fiiM _ _9X|19X'' = g,J<— (^-1) , ii(_ i)> d V w - gu MR „ X“.Xf. ' ' dVw J & 3 3t J 3t J w r a ^ v ti tj 0t 0t w Equation (3.17) One can see that if the sectional curvatures of M are non-positive, then (3.17) is greater than or equal to zero. From these results it is evident that if M has non-positive sectional curvature, and a solution ^ satisfying (3.14) for all t > tQ exists, then d S [X l —dt------>0 as t °o so as t approaches the solution converges to a harmonic map. After a long and elaborate proof, Eells and Sampson are able to establish the existence of harmonic solutions, although certain restrictions on the curvature and global behavior of M were required, as well as some assumptions about the t sequences for t > tQ. Later work by Hartman [3-5] removed some of these restrictions, so his generalization of the theorem [see 3-1, p.158.] by Eells and Sampson is cited here; where it should be noted that Hartman is considering W compact and M complete with non-positive sectional curvatures [see 3-5, p.674]: 72 (A) If X0:W—>M is a C1 map, then the initial value problem (3.14) has a solution X(z,t) for all t > 0. (B) If X(z,t) is a solution of (3.14) for all t > 0, and its range is in a compact subset of M, then X J z ) = lim X(z,t) t — exists in Ck(W,M) for every k and is a harmonic map X^W—>M homo topic to X0. Notice that these results are not as helpful as one might hope, since if X 0 is homotopic to the null vector, the harmonic map for that homotopy class would be the degenerate case of a constant map. This leads to further specification about the geometry of the target spaces which admit non-degenerate harmonic maps. A great deal of research on harmonic and minimal immersions has followed the early work. Below, some of the interesting results are summarized; the exact statements of the theorems are collected in Appendix B. 1) If W has non-negative Ricci curvature, and M has non-positive Riemannian curvature, every continuous map X:W—>M is homotopic to a totally geodesic map, and harmonic maps are either constant or map onto closed geodesics.[3-1] 2) Homotopic harmonic maps are homotopic through harmonic maps.[3-5] 3) If X^ is a non-trivial harmonic map onto a closed geodesic in M (where the sectional curvatures are < 0), then any harmonic map homotopic to X^ is just obtained by rotation along the geodesic, so non-trivial harmonic maps are essentially unique.[3-5,3-6] 7 3 4) When there is a one-to-one map between the fundamental groups of W and M induced by X:W-»M, then there exists a minimal immersion homotopic to X (and hence a harmonic map).[3-6,3-7,3-8] These results show that in many cases there exist harmonic maps or minimal immersions, so for target spaces which fulfill the necessary restrictions, there exist classical solutions to the string equations of motion. 3. APPLICATIONS The previous results were purely existence theorems, and hence, not amenable to performing physical calculations. However, it is possible in some instances to choose a particular target space and find an actual embedding which satisfies the equations of motion. For these purposes, it is probably easiest to make use of certain minimal imbeddings into S3. By a theorem of Lawson [3-9], one is assured that every compact, orientable surface can be minimally imbedded in S3. Thus it appears that this would be a useful manifold to consider since it would allow for classical solutions for every genus. Certainly considering S 3 as the target space for string theory can only be a toy model; however, by considering more complicated target space topologies with an S 3 component, the results for the toy model can be carried over. The calculations for the toy model path integral are then performed as background field expansions around the classical solution for genus zero and genus one. One extension of the toy model is the development of cosmological membranes. The topology of spacetime can be postulated to be S 3 x R, and 74 the calculation can be performed as a time sliced path integral. The classical solutions in the S3 toy model would be carried over to the product target space by adding a time component, so if Xcl is the classical solution in S3, then the classical solution in S3 x R is (Xc|,t). This extension of the toy model will be calculated in a background field expansion (up to second order) for the cases of genus zero and genus one. Other applications which will not be investigated here include taking a time sliced spacetime to be R3 x R and attaching to each point of the spacelike hypersurface a non-physical S 3 vacuum. Recall that for flat target spaces the classical solutions do not exist because the only harmonic maps are constant. By attaching the S 3 vacua, the classical solutions would sit in S 3 in a non-trivial state which would be identified with the ground state. One might then discuss questions of tunneling from the vacuum at one point to the vacuum at a neighboring point, construction of a model of spacetime foam, etc. This approach is essentially the inversion of the problem of compactification, since one first picks a target spacetime (in this case R3 x R) and the compactifying manifold (in this case S3) and then expands around non-trivial classical solutions. §3.1 THE TOY MODEL In the toy model, the target space will be taken to be S3. The string path integral will be calculated in a background field expansion around a classical solution, Xcj. The calculation will be performed to second order in the fluctuation,O. The technique is standard, but will be outlined here for completeness. The action is expanded as 7 5 8X°(z')8Xp(z") where the functional derivatives are evaluated at Xcl. The fluctuation terms have the form [equation 3.18A] and The path integral is then performed over the fluctuations O and the intrinsic metrics on the Riemann surfaces, giving (3.19) Now one can consider the calculation for genus zero and genus one. § 3.LA. GENUS ZERO - THE SPHERE If one considers the terms in the expansion of the action about Xc], it is apparent that they still retain conformal invariance in the intrinsic metric terms. Thus the integral depends only on the conformal equivalence class of metrics, and hence, the integral should be taken over conformally inequivalent Riemann surfaces. These matters are discussed more fully in the appendix on Riemann surfaces. For the genus zero case, there is only one conformal equivalence class, that of the standard sphere. Consequently, the domain space will be the Riemann 7 6 sphere with the usual metric induced on the unit sphere in R3. Since there is only one conformally inequivalent genus zero surface, there is no integration over metrics on inequivalent surfaces. The domain sphere S 2 is given the usual spherical coordinates the "longitude" and 0 < 0 < k measures the polar angle or "co-latitude". The metric on S2 is given by g n=1 gi 2 = g 2i= ° g 22 = sin20 (3.20) The target space S3 is also given the usual spherical coordinates (j^Gj ,02, where 0 < W d aXV.G cl ^ = < 0 ’,0 a X cl. > = 0 for a = 6,0 T’ This implies that (D1 = O2 = 0, so written as a vector, O = (0,0, n 2n S[Xd] = ±[d2z1/i'gaba = J J sin9 #4n d9 = (3.23) 00 and 77 = 0 (3.24) as one would expect for a classical solution. However, for the second variation, one finds (3.25) x=x cl which one would not have expected. This implies that the classical solution is exceptionally stable, and there are no quantum corrections up to second order. The path integral up to second order is just (3.26) where N is an infinite constant, and one is free to choose the normalization of (3.26). § 3.I.B. GENUS ONE - THE TORI The first serious problems with the background field approach arise when considering genus one surfaces. Primarily, it becomes impossible to pick one classical solution which will produce a zero first variation of the action for all of the conformally inequivalent tori. This is simply a reflection of the requirements of the equations of motion; since the first equation of motion requires the imbedding to be conformal, and since there is a one (complex) dimensional space of conformally inequivalent tori, it is apparent that any classical solution can only be a solution for one equivalence class of tori. Thus, there will be first order 7 8 terms in the expansion of S[Xcl+0]. These will be handled by making a field redefinition to complete the square in the exponential. The tori will be defined over the unit square in the manner discussed in the appendix on Riemann surfaces. It is shown in the appendix that all conformally inequivalent tori can be generated by defining the metric on the unit square to be of the form 'l u &ab 2 2 (3.27) U U + V where u,v take their values in a fundamental region of the upper half plane, which will be defined by the relations - 1/2 < u < 1/2, Vl-u2 < v < «>. As discussed in the appendix, any fundamental region is suitable, but the one selected here is the standard choice. The coordinates on the unit square will be x,y, although a compact notation z = (x,y) and d2z = dxdy will generally be used. For the calculations to follow, it is easiest to choose a different set of coordinates on S 3, related to the Euler angles. This coordinate system is derived from the coordinates on R4 according to the relations a i x. + ix~ = r cos-2. exp^-(\j/ + x3 + ix4 = r sin-2- expi(\j/ - X ,(x,y) = -^(cosx.sinx^osy.siny) (3.29) J l It is evident that this surface lies in an S3 sub-manifold of R4, and in the coordinates on S 3 it is given by X cl(x,y) = (j, x+y, x-y) (3.30) The pullback metric is simply X*G = 3 aXM;a.Xv1G cl b cl HV I 8 ab (3 .3 1 ) so this imbedding is conformal with respect to the flat metric on the torus. A simple calculation shows that the imbedding is harmonic, so this is a true classical solution with respect to the flat metric on the unit square. It may be interesting to note that this classical solution is unique because (even locally) it is characterized as the only non-totally geodesic minimal surface of constant curvature in S3.[3-9, p.340] At this point it is appropriate to mention that any translation of the solution Xcj along the 0-direction is again a classical solution. The fluctuations must be chosen so that they are perpendicular to the classical solution. As before this can be achieved by requiring that there be zero inner product betweenthe fluctuation and the tangent vectors to the surface at every point, i.e. 0^0 aXV.G cl = < O, 0 aX cl> = 0 for a = x,y This implies that s p y = y ~ jd 2z g \ d bO l = Ad)1) (3.35) and the laplacian can be written out explicitly as A = -i[9 2 + 2u9x3y + (u2+ v2)92] (3.36) It is evident that if O1 included constant modes in its functional expansion, then A would act on those modes with zero eigenvalue, which would lead to a poorly defined functional integral. Fortunately, as was mentioned in the last paragraph, the constant mode in O1 represents a translation to a different classical solution, so must be factored out anyway. The string path integral (3.19) can be calculated once some choice has been made for the measure for the integration over the inequivalent tori. All of the information necessary for the specification of the 8 1 conformally inequivalent tori was included in Appendix A. This measure must vary over all values of u,v in the fundamental region, but must also satisfy the group invariance property under a transformation to any other fundamental region. The tori will be specified by defining period parallelograms endowed with the flat metric inherited from the plane. All inequivalent tori will be generated by choosing one side of the parallelogram to be the unit vector on the real axis. The other side of the parallelogram will be chosen in one of the fundamental regions of the upper half of the plane. Letting the real axis be denoted by u and the imaginary axis by v, the choice of the fundamental region will be the standard region defined by -—1 < U < 1— U2 + V2 > 1 2 2 which is the region above the unit circle and between + 1/2 and -I/2 depicted in Figure 7. By choosing all possible vectors in the fundamental region, all inequivalent tori will be generated. The measure for these integrations must be chosen so that it respects the invariance of the theory under different choices of the fundamental region. This requires a measure that transforms invariantly under the generators of the elliptic modular group z->z+l and z—>—l/z, since this is the group which transforms one fundamental region into another. The measure which satisfies these properties is given by du dv (3.37) ~2U + V2 The invariant transformation properties can be seen quite easily. Under the translation z—»z+l, the path integral is invariant after making a simple change of variables. However, under the transformation z->-Vz 82 letting w(z) = x + iy = -V 7 = [u2 + v2] ^-u + iv] the transformation becomes du dv —> ^ L - |j| with J = 2 2 2 2 U + V x + y [X2 + y2r [x2 + y 2]2 This measure satisfies the desired group transformation properties, and will be used for the integration over the Riemann surfaces. So far as the author knows, this is a new approach to defining the measure for the Teichmuller integrations. The string path integral (3.19) becomes 1 2 2 -__(l+4v u +v ) f_ --(cD,A AzG(z,z') = 5(z,z') (3.41) VS Using the eigenfunctions for a fourier expansion of a field over the unit square, the Green function is seen to be <*> -l G(z,z') = ^ [m2+2umn + (u2+v2)n2] exp i[m(x - x') + n(y - y')] (3.42) m,n= -oo In (3.42) it is apparent that the Green function is undefined for m=n=0, which results from the zero eigenvalue of the laplacian on the constant mode. Thus it will be necessary to employ the Faddeev-Popov procedure to remove the zero modes. To do this, insert into (3.38) a term of the form 1 = jda S(jd2z c ®(a))Det(j-Jd2z cO(a)) (3.43) where c is some constant and a represents the zero mode, which is a translation in this case, so one can write O(a) = a + O' (3.44) where O' can be taken to have no constant modes. Since the integral of O' over the unit square is zero (since it has no constant mode in its fourier expansion), (3.43) reduces to 1 = Ida 5(ac) Det (c) (3.45) This is inserted into (3.38) to give [equation (3.46)] Z =M " 4 ^ t J[DO] 8(°tc)Det(c) exp-{S[Xd]+I(0, A O )~fd 2zd2z'A2G(z,z')} Now the O integration can be done since the delta function effectively restricts the integration to be over the non-constant modes, giving [equation (3.47)] Z =fdaf"d"?Z- — 1— -exp-{ l ( u 2+ v2+ 1)-^[^-(u2+ v2- 1 ) ] jd2zd2z'G'(z.z')} u + v yj Det 'A 84 where the prime on the determinant indicates that the zero eigenvalues have been removed, the prime on G indicates that the m=n=0 term in (3.43) has been removed, the A2 term has been pulled out of the integral since it is independent of z,z', and the constant c has been set equal to 1 . Now since the m=n=0 term in (3.43) has been removed, it is evident that Id2 z d2z' G(z,z') = 0 so only the first term in the exponential arising from S[XC|] remains, leaving .If. 2 2\ . 1 /, 2 2 \ — ( 1 + u + v ) ( l + u + v ; du dv 4 v du dv 4 v = n |. z-UJ 2 2 2 2 (3.48) U + V VDet'A U + V VDet'A where the infinite constant N results from the zero modes. Since the eigenvalues of A are of the form = —[m 2+ 2umn + (u2+ v2)n2] (3.49) it is apparent that the determinant cannot be taken outside the integral. Consequently, to perform the u,v integral in (3.48), one must choose some regularization scheme for the determinant which retains its dependence on u and v. If such a regularization could be found, then it would be possible to consider integrating over u and v. There may again be problems with divergences in the u,v integral, however, as long as the regularized determinant had powers of v > -2, and there were no problems at u = 0, the integral would be convergent. There might then be an interpretation of (3.48) as the analog of a probability integral for the intrinsic metric. The calculation of the path integral (3.38) could have been as easily performed by simply removing the constant modes by hand, and defining the integral to be an integral over the non-constant modes. This would 8 5 correspond to choosing the measure for the integration to be of the form T—r da U - p m,n =1 y ^ where the a^ are the expansion coefficients, as discussed in the first chapter. The term quadratic in O would become 2 - Y x mnv (a rrm7 )' m ,n=l and the linear term in O would be zero since the constant modes in O have been removed by hand, and the integral of any non-constant mode over the unit square is zero. The result would be the same as (3.41) with N=l. §3.2 COSMOLOGICAL MEMBRANE MODEL One application of the toy model is to develop a time sliced theory of cosmological membranes for a universe of the topology S 3 x R. In the spirit of a time sliced approach, the R coordinate will be identified with time. The classical solutions used in the toy model for genus zero and genus one become classical solutions in the product topology by simply adding a time component. The domain space for the calculations thus becomes W x R, where W is a closed Riemann surface as before, so the membranes which will be discussed are spatially closed surfaces sweeping out a 3-dimensional manifold in time. The choice of the time sliced gauge is different from the usual "physical gauge" choice made in bosonic membranes [3-10], and removes only one degree of reparametrization invariance for the membrane, while maintaining 2-dimensional reparametrization invariance on W. This will allow for the specification of the form of the intrinsic metric on the domain space. To distinguish between the components in the metric tensors the notation will be adopted that the intrinsic metric is given by Tab • g = (3.50) • -1 where Ya^ will be related to the metric on W from the toy model, and in the target space G = (3.51) • -1 where is the metric on S3 from the toy model. The action is extended to give sm = dtV‘Deti gJ9ix°ajx p (3.52) and the convention has been adopted that i j indices run from 1 to 3, indices a,b run from 1 to 2, indices G,p run from 1 to 4, and indices |i,v run from 1 to 3. It is interesting to note that no cosmological term has been added to the action, and the theory which will be derived will not be trivial. This is an indication that these cosmological membranes in the time-sliced gauge in curved space are quite different from the usual bosonic membranes, since the usual membranes are found to require a non-zero cosmological term.[3-10, p.7][3-ll, 12,13] The action cam be expanded as S[X] = Jd2z dt ^ D e t gab { g V V G F - gab3 x \ x 4 - a,x^a,xvG, x43, +9x4} (3.53) The time slicing can be achieved by the identification of X 4 with t, which is effected by removing one reparametrization degree of freedom from the membrane. This corresponds to including in the functional integral a 87 term of the form 1 = j’[D § 3 3 = 8 8 which was made to maintain a consistent time slicing with the target spacetime. Together these requirements restrict metrics on W x R to be of the form ^2(x>y) gab • (3.57) • -1 where ga^ in (3.57) is the metric carried over from the toy model, which corresponds to (3.20) for genus zero and (3.27) for genus one. In the genus zero case the integration over the metrics in the path integral will involve only an integration over Q. For the genus one case, it will involve an integration over both Q and the u,v parameters. §3.2 A GENUS ZERO The S3 x R target space is given the coordinates ($,6^62,0 where the angular coordinates (j),01,02 are defined as in the genus zero toy model. The metric G^v on the spacial part, S3, is given by (3.21). The classical solution for the imbedding of W x R in S 3 x R reflecting the gauge choice is given by Xd«j),0,t) = ($, 0, 0, t) (3.58) It is a simple exercise to check that (3.58) satisfies both equations of motion. The requirement that the fluctuations are perpendicular to the surface implies that O = (0, 0, 5S[X] Oa(z',t') = 0 (3.60) 5XCT(z',t') x cl and 5S[X] Oa(z',t')d>p(z",t") = 0 (3.61) 8Xa(z',05XP(z",0 x cl which is not much of a surprise after the result of the toy model. The path integral is simply -4nT Z = J[DQ]J[DCD] exp - spy = e ^ T/[DQ]J[D § 3.2.B GENUS ONE The S3 x R target space is given the coordinates (0,\|/, S[XcI] = i [ l + U2+ V2] + J vjdx dy n 2(x,y) (3.65) where T is the range of the time interval. The first variation is 8S[X] jd 2z'dt' ®°(z',t') = A [ u2+ v 2-l]fd x d y dt (3.66) 8X®(z',tO xcl The second variation is 8S[X] jdVdt'dV'dt" 0°(z',t')0P(z",t") = |d2z dt 1 f-2 « ,2 2. -n2 \ V ^2^2 a = --L(a2+ 2u d d + (U2+ v2) a2) + ^ n 2a2 (3.68) 4v x x* y v ' y'y 'A 4 t The path integral to second order in the fluctuations becomes Z = J[D£2]|^i^J[DO]exp-{S[Xd] - ijd 2z dt AO - ijd 2z dt ®AO } (3.69) U + V where A = (2v)_1[u2+ v2 - 1]. As the discussion in the section on the genus one toy model indicated, it is necessary to remove the spatial translational degree of freedom in the 0 direction, since this represents just a translation to an equivalent classical solution. Since the formal manipulations to remove the zero mode changed only the normalization of the integral, it was shown that it was equivalent to remove the 0-translational modes by hand, as will be done here. The [equation(3.70)] 2 1 2 rt2 2 21'1 i[m (u-uO+n(v-vO] ip(t-t') G (z,t; z',t') = j* ~ - ^ 4[m2+ 2umn+ (u2+ v2) n2- Q2v2p2 ] m , n = - ° ° where the prime on the summation indicates that the m=n=0 term has been removed. Then making the field redefinition 1 (\2 , , . O —» O = O + iJdVdt' G(z,t;z',t') A(z',tO (3.71) the path integral becomes [equation (3.72)] Z = J [D n ]J ^ J [M ]e x p -{ S[Xd] - 1(®, AO)-IJd2zdtJd2z'dt'A2G(.,.)} U + V The O integration can be done (formally), leaving Z = f [ D f l ] f d2UdV2 — L = r exp - { - ^ (1+ u2+ v2)^- ~ v Id 2 z £2 o 2 u + v J Det'A ------— [u2+ v2-1 ] fd2z dt fd2zrdt' G (z,t; zx,t")} (3.73) (4v) J J where the prime on the determinant indicates that the translation zero modes have been removed. It is evident that (3.73) is gaussian in Q, and the integrand is exponentially decaying in the u,v parameters, so there is some hope that regularization of the determinant and evaluation of the Green function would lead to a well-defined theory. However, since the laplacian and the Green function contain Q, the functional integrals over Q and O do not separate, and the determinant is explicitly a function of Q. Consequently, the manipulations above are purely formal, and before a clear result could be found it would be necessary to consider more carefully the intermixing between Q and O. It might be worth mentioning that the inclusion of a cosmological constant or other Q dependent terms in the action may provide a mechanism which would remove some of the complications discussed 92 above. Since the theory is not conformally invariant, the addition of such terms would not break a classical symmetry of the theory. For example, by adding a cosmological constant of the form - j j d 2z dt Detg = - j vJd2z £22 the theory would be spatially conformally invariant with respect to conformal transformations on W, with the only dependence on G in A and G. 4. CONCLUSIONS The results of the work in this chapter serve two purposes. The primary role of the theoretical aspects and existence theorems is to provide motivation for a study of string theory in curved spaces. In curved spaces it becomes possible to find classical solutions of the equations of motion, and if one expects the classical solutions to play the role of "on-shell" particles, the existence of an observable theory would seem to require a curved space formulation. If one allows for speculation about the development of a "string detector", it appears that such a detector would have to trap strings in a "curved space", which might effectively be produced by a strong gravitational field or a field with which the strings interact. The second benefit of these investigations is that for a particular curved target space it may be possible to find non-trivial classical solutions for surfaces of a given genus and conformal class, and then following the background field approach outlined here, derive actual results after regularization. Such work would then allow one to consider problems like cosmological membranes or an inversion of the problem of 9 3 compactification. Finally, if one could find classical solutions for more than one equivalence class for surfaces of a given genus, it would be possible to consider questions of tunnefing from one classical solution to another. To put the work of this chapter in the broader context of this thesis, the background field approach contrasts with the standard flat space approaches to string theory discussed in the second chapter. First of all, by making a choice of target space and background field, one is effectively choosing the span of the surfaces which contribute to the integral. In the case of the toy model, the target space was three dimensional and the classical surface spanned two of the dimensions, with fluctuations into the third. This reduced the problem to an integration over one fluctuation field rather than the three which are allowed in the standard approach; consequently, the determinant factor Det'A enters to the power -V 2 rather than the of the standard approach. If the codimension was larger, say p>l, one would expect that there would be integrations over p fluctuation fields, but the discussion of the zero modes (which can include rotations and translations, etc. in higher codimension) suggests that the determinant factor Det'A would not necessarily scale as -P/2. This adds further impetus for an investigation of string theory which places no restriction on the span of the embedded surface, and suggests that a formulation which is applicable to curved space should be sought. The next chapter will develop such an approach. 94 CH. 4 - NEW FORMULATION OF STRING THEORY 1. INTRODUCTION After the developments presented in the second and third chapters, it is now appropriate to attempt to reformulate the Polyakov string theory in a way that allows for embeddings into a general curved target space. The motivations for this work follow naturally from the preceeding chapters, but it is probably of some value to collect them here. The second chapter illustrated the problem with choosing the product gaussian measure for the embedding integration, with the implication that the standard approach only allowed contributions to the path integral from embedded surfaces with a linear span equal to the number of target space dimensions. Since string theory is supposed to be a theory of geometrical objects, which are Riemann surfaces in the Polyakov formulation, it is important that the formulation of the theory include contributions to the path integral whenever the geometrical object is well-defined. The linear span restriction discussed in the second chapter fails to satisfy this criterion, because the embedded Riemann surfaces can be well-defined as a surface with a linear span of varying dimensions. The third chapter stressed that the string equations of motion in a general curved target space are in general a non-linear set of differential equations. The solutions to the classical equations of motion were shown to correspond to conformal, harmonic, minimal immersions. Such solutions do not exist for mappings into flat target spaces. This seems to suggest that there is no observable classical theory for strings in flat 9 5 space. Finally, one could put forward the argument that the extra flat dimensions in string theory are not meant to be flat at all, because compactification schemes are expected to "curve up" the extra dimensions at an extraordinarily small scale. However, this will be shown to have severe implications for the validity of the definition of the flat space path integral. With this motivation, an approach to a reformulation of Polyakov string theory in a general curved target space will be developed. The Polyakov path integral for strings has been written down earlier, but for the sake of a self-contained presentation, it will be copied here; Z = J[Dg] e ^ J lD x ] e~SM = J[Dg] e ^ 1 Z £mb (4.1) Again the terms S[g] and S[X,g] have the form S[g] = \ [ d 2z V i + pjd2z VJ r and S[X,g] = ± [d 2z g’^ / S / The terms in the purely metrical action, S[g], are extra terms which are added for the purpose of renormalization. The original paper by Polyakov only made mention of the first term in S[g], the second was added in subsequent papers. As discussed in the first chapter, the viewpoint will be adopted that the extra renormalization terms, S[g], should not be added at the outset, and the reformulated theory in curved space should attempt to derive a consistent theory based solely on the action S[X,g]. The generalization of the Polyakov string theory to curved target spaces can be effected by simply incorporating the target space metric, G, into the string action, as in equation (3.1) of the preceeding chapter. The target space metric is again treated as arbitrary, but fixed, so it will not 9 6 enter into the theory as a field variable. The generalized Polyakov path integral becomes: Z = J[Dg]J[DX]e-S[X’8'Gl=j[Dg] Z Emb (4.2) where S[X,g,G] has the form s[x,g,G] = ijd2z,/Dttg gabax ^ bXvG^v (4.3) W For the sake of clarity, the embedding maps will be written X:W—>M, and one should understand that the intrinsic metric on W and the target space metric on M may or may not be related by the embedding. The remainder of this chapter will be concerned with the implications of this generalization of the Polyakov path integral and, ultimately, with a reformulation of the integral which preserves the content of the theory without the restrictions and drawbacks of the flat space formulation. 2. REINTERPRETATION & IMPLICATION The curved space string action (4.3) represents the global inner product between the intrinsic metric on the domain space Riemann surface and the pull-back metric induced on the surface by virtue of its embedding into the target space. This raises the possibility that instead of treating the embedding variables, X, as the observables, one could treat the pull-back metric as a single geometric object, which plays the role of the observable. Notice that this treatment of the metrics as the fields in the theory corresponds quite closely to the role of the metric in general relativity. Such an approach will be explored in the reformulation which will be developed. The pull-back will be denoted by L[x *g J] ab =a x^a.xvG a b (4.4) 97 and the action will be written using a bracket notation < . , . > which indicates that the tensors inside the brackets are fully contracted s[x,g,G] = AJdVDetg gaba,X*l3bXVG(iv = iJdVDetg After the preliminary discussions of the preceeding sections, it is now possible to begin reformulating the Polyakov path integral. At every stage, the geometrical properties will be emphasized rather than the field theory aspects, and both the geometrical structures on Riemann surfaces outlined in appendix A and reinterpretations of section 2 will be exploited. The starting point will be the initial definition of the Polyakov path integral for closed bosonic strings: Z = J[Dg]J[DX]e-S[X'8’G! = J[Dg] ZEmb (4.2) where S[X,g,G] is given by 1 0 0 S[x,g,G] = ijd^Detg gab3X^bXvG^ (4.3) w The Polyakov prescription for strings requires summing the contributions to the path integral over all inequivalent Riemann surfaces and all embeddings of the Riemann surfaces into the target space. As it is written, the path integral is purely symbolic, and it will be the primary object of the remainder of this chapter to specify further all of the elements of the path integral so that it will lead to a well-defined and calculable reformulation of the Polyakov path integral with a clear procedure for application to curved target spaces. As indicated in section 2, the statement that the string path integral should add up the contributions from all conformally inequivalent Riemann surfaces will be taken literally. In most of the standard works on string theory, there is an attempt to derive a measure on the space of Riemannian metrics, which must then be gauge fixed by the analog of a Faddeev-Popov procedure to reduce to the degrees of freedom associated with the space of Riemann surfaces. The resulting decomposition of the "measure" associated with the integration over the space of Riemannian metrics and the resulting gauge fixing procedures are extremely complicated, while the integration over the embeddings is considered to be a trivial gaussian integral. The viewpoint that will be adopted in this chapter is quite the opposite. Instead, one should consider the gauge-fixed path integral from the outset and develop a way to integrate over the conformally inequivalent Riemann surfaces, rather than Riemannian metrics. This corresponds to the interpretation of the metric on the Riemann surface as a comparison point for the measurement of the string energy which arises from the stretching and contorting of the string when it is embedded in the target space. Thus, 1 0 1 the interesting aspects of the calculation are those that arise by virtue of the embedding of the string in the target space, which is a significant shift from the standard view. So, following the discussion in section 2, the integral will be developed so that it integrates over conformally inequivalent Riemann surfaces directly, without going through the mechanism of the gauge fixing procedure. It will be evident that both approaches should be equivalent, since the gauge fixing procedure should restrict the integral to just the conformally inequivalent Riemann surfaces, but it will be seen that the reformulated approach affords a great simplification of the calculation. The first thing to consider is the measure for the integration over the space of inequivalent surfaces. This will replace the measure [dg] and its associated gauge fixing terms in (4.2), and it represents a new approach to performing the Teichmuller integrations. At this stage, the work will still be purely symbolic, but will be specified later. Recall that a conformal structure on a Riemann surface was an equivalence class of metrics, [y], where the brackets denote that the object under consideration is the equivalence class itself. Riemann surfaces were then defined by a topological manifold (of a certain genus), M, and a conformal structure, written (M,[y]). The bracket notation will be extended now to allow for the specification of an equivalence class of conformally equivalent surfaces (i.e. equivalence in the sense of surfaces). This will be written [(M,[y])]. Symbolically, then, the measure for the integration over the space of conformally inequivalent surfaces will be written: D[(M,[y])]. Later, some specification will be given to the symbolic measure so that it will be useful in calculations. The measure for the integration over the space of embeddings into a general curved target space will require special attention because of the 1 0 2 difficulties discussed in section 2. It will not be possible to base it on a gaussian measure or a product measure because the function space of embeddings has no vector space structure for a general target space. The only specification that will be made now is the imposition of the proviso in section 2 defining the generic term embedding to be immersions and imbeddings (i.e. they induce non-degenerate pull-back maps). The symbolic measure [dX] will be written as D[Emb(W,M)] to reflect this specification. In accord with earlier remarks, the action will be viewed as the global inner product between the intrinsic metric and the pull-back metric, and the bracket notation < . , . > will be used as in (4.5). The Polyakov path integral can then be written symbolically as Z = Jd [(m,[y ]) ]J°[Bmb(W,M)] exp —ijd 2z -/Det g < g, X*G> (4.6) w In order for this to have meaning, some specification must be given before the calculations can be performed. §3A. MEASURE FOR RIEMANN SURFACE INTEGRATION Following the discussion outlined in appendix A, the integration over the conformally inequivalent Riemann surfaces will be achieved by integrating over the fundamental polygons in the constant curvature covering spaces. The polygons will be taken so that no two polygons represent the same Riemann surface. This can be achieved by integrating over only those polygons whose sides are generated by vectors or non-euclidean segments in the fundamental region of the covering space for surfaces of a given genus. In the case of genus one tori, this corresponds to choosing the 1 0 3 period parallelogram so that one side lies at the point (1,0) on the real axis, and the other side, x, is a vector extending into the defining fundamental region, i.e. the cross-hatched region depicted in figure 7. By choosing all possible vectors x in the fundamental region, all possible inequivalent tori are generated. Since these tori all are endowed with the flat metric, and the intrinsic metric enters the path integral only through the conformally invariant terms VDet g ga^, this specification of the flat metric on the conformally inequivalent tori gives all of the information needed for the calculation of any objects in the path integral. Denoting the real part of the vector x by u, and the imaginary part by v, one need only integrate over a fundamental region in the plane where u,v take on all their values. However, since any fundamental region region can be taken into any other by a discrete automorphism, the measure must be normalized by imposing the requirement that it be invariant under the action of the appropriate automorphism group. This form of the measure was constructed explicitly for genus one surfaces in chapter 3, section 3.1 .B. For the higher genus Riemann surfaces the object would be to choose canonical non-euclidean polygons in either the Poincare disk or the upper half plane corresponding to all conformally inequivalent Riemann surfaces of a given genus. The specification of 2g (independent) non-euclidean segments, xp ..., x2g for the 4g sided polygon would completely specify the polygon ( the other sides, Xj-1,..., x2g_1, would be unambiguously defined by the symmetry requirements imposed by the elements of the defining group of non-euclidean motions). The determination of the fundamental region for the genus g surface would then allow one to generate all conformally inequivalent surfaces of genus g by choosing all possible combinations of non-euclidean segments in the 104 fundamental region. Since the canonical polygons are endowed with the constant curvature metric of the covering space, this specification would again provide all of the information needed for the calculation of the path integral. Denoting the real part of xi by Uj and the imaginary part by v^, one need only integrate over a region in the covering space where Uj, vi (for all i) take on all of their values. The measure for the integration over the space of conformally inequivalent surfaces of genus g would then have to be normalized so that it would be invariant under any transformation to an equivalent fundamental region. It must be noted, however, that the determination of the fundamental regions for higher genus surfaces is still an open question in mathematics. However, that is no bar to proceeding with the formalism. The measure for the integration over the space of inequivalent Riemann surfaces is thus a function of the parameters ui5 vi and the normalization factor, N(u,v), which insures the group invariance property under a transformation to another equivalent fundamental region. This will be denoted by d|i(...) and one can identify D[(M,[y])] ~ dn(Uj,v.,N) This transformation property of the measure is the equivalent of a gauge transformation under the group Conf(W) (s) DifKW), since by transforming to another fundamental region, one is effectively going to a set of equivalent surfaces, with inequivalent metric conformal structures. So even though it appeared from the outset that this prescription was the equivalent of a gauge fixed approach to the path integral, it still retains the invariance associated with the gauge transformations. This will be illustrated in section 6 for the genus one calculation. 1 0 5 §3B. MEASURE FOR EMBEDDING INTEGRATION Turning now to the question of defining the measure for the integration over the embeddings, it will be necessary to construct this measure in a new and unique manner since the presence of curvature in the target space invalidates the gaussian-based approach. The starting point will be an insistence that the pull-back metric should be the object of study, rather than letting the field variables be represented by the X^. The pull-back metric X*G is the metric induced on the Riemann surface, so it must conform to all of the requirements of a metric on a surface. Notably, it must lie in some equivalence class of metrics, i.e. a conformal structure. As such, if the pull-back metric is designated by X*G = gj, and if gj lies in the equivalence class denoted by [gj°], then gj must satisfy the relation gj = £22(z)gj°,where gj° represents a fiducial metric in the equivalence class. By adopting the viewpoint that the pull-back metric should be the field variable in string theory, it becomes apparent that the variables serve only in the capacity of producing the pull-back metric, and the path integral measure must be constructed in such a way as to exploit this property. The operative technique which will be employed to construct the integration over the embeddings will be a quasi-fibration of the space of embeddings, Emb(W,M), over the space of Riemannian metrics on W, RiemMet(W), i.e. every embedding will be mapped down to the Riemannian metric it induces on the embedded surface through the pull-back map. [The term quasi-fibration is used because, in general, the decomposition is not a fibration in the strict mathematical sense of the word]. Let X:W—>M be an embedding of W into M. Then the fibers are determined by the equivalence relation: X ~ X' if and only if X*G = X'*G, so the fiber F can be written 1 0 6 F = { X g Emb(W,M) : X*G = ge RiemMet(W)} (4.7) Define 7CG as the map to the base space; then the quasi-fibration can be expressed diagramatically as -»Emb(W,M) 7Zf'G i RiemMet(W) By viewing the space of embeddings in this way, it is possible that the in teg ration uv ei uixe cuiucuuiiigo can he accomplished by integrating across the base space and then along the fibers. This also encodes the desire to treat the pull-back metric as the field variable in the theory, since the base space is then the space of pull-back metrics. Further, since every element in a fiber has the same pull-back metric, the value of the action is constant along a fiber, i.e. since S[X,g,G] = ijd 2zVDetg < g, X*G > w and X*G is constant along a given fiber, the action is constant for a fixed intrinsic metric g. This then represents a quasi-fibration according to equi-energy embeddings, and if they exist, any global cross-sections of the quasi-fibration would contribute equally to the path integral. Since the quasi-fibration conforms with the interpretation that the field variables should be the pull-back metrics, it is now possible to proceed with the construction which will allow for the action to be considered as a true global inner product. Attaching to each fiber an index j to designate different fibers, one can say that any embedding Xj e Fj has a corresponding pull-back gj. Then gj must lie in some conformal equivalence class of metrics [gj°], and so it is represented as gj = £22(z)jgj° 1 0 7 for some positive function Q 2( z )j defined on the Riemann surface and some fiducial element of the equivalence class gj°. However, the earlier discussion on Riemann surfaces indicated a good way to choose the fiducial element of the equivalence class; that is, to select the constant curvature metric on the fundamental polygon. Then the conformal factor Q2(z)j is just a positive function defined on the fundamental polygon. To integrate over all pull-back metrics, then, one need only integrate over all conformal structures [gj°], and all conformal factors £22(z)jdefined on the fundamental polygons with fiducial metric gj°. Thus, the action for the integration takes on the form S[X,g,G] = JdV D etg< g. X*G > = Jc^z^DeTg < g, G2(z)jgj° > (4.8) w w Recall that the specification of the Riemann surface W follows along similar lines. One selects a fundamental region in the constant curvature covering space endowed with the metric inherited from the covering space. Evidently, the metrics g and gj° are both constant curvature metrics on the fundamental polygons, so indices will be attached to g (i.e. g = gj0) to distinguish the constant curvature metric of the domain Riemann surface W from the constant curvature part of the induced metric, gj°. However, it has not yet been established whether the choice of the gj0 can be restricted to the fundamental region of the covering surface, since the fundamental region arose after factoring by the group DifffW). If this proves to be the case, then the choice of the gj0 and the gj° can be made in the exact same way. Before proving this, one must consider how the calculation of the global inner product in the action will proceed. The global inner product in the action is calculated on the domain space Riemann surface W. The specification of W has been accomplished 1 0 8 by choosing a fundamental polygon in the covering space, which will be denoted by Bj. The metric on the polygon is the constant curvature metric gj0. However, the pull-back metric corresponds to a Riemann surface defined by some other fundamental polygon Bj, with the metric £22(z)jgj°, where the function H2(z)j is a positive function defined on Bj, and gj° is the constant curvature metric on 13j. In order to calculate the global inner product, there must be a common region over which to compare the metrics, which will be denoted by fiQ. Then to take the inner product, both (fi^gj0) and (Bj,Q2(z)jgj°) must be mapped isometricallv onto the comparison region B0. This means that if fiBj—»B0 and h:Bj—»B0 are the maps to the comparison region, then the metric on Bj is pulled back onto B0 by (f-1)* and the metric on Bj is pulled back onto B0 by (h-1)*. Since the maps are isometric, they do not affect the calculation in any way, i.e. lengths, angles, and inner product are preserved. It does facilitate the calculation of the global inner product, so the action can be written now as a calculable inner product S[X,g,G] = Jd% /Det (f'Vgf < (fVg° (Q2 • h'1)(z)j(h'1)*gj° > (4.9) Bo Although the above form looks a bit complicated, it is still simply the inner product of the metrics, now shifted to the comparison region, where the conformal factor £22 is defined by composition with the inverse map from the comparison region. In this form it will be possible to determine whether the gj° can be found from just the fundamental polygons in the fundamental region, and if this restriction holds then the gj0 and gj° can be specified in the same way. Recall that the Polyakov path integral prescription requires that one integrate over all embeddings of all conformally inequivalent 1 0 9 Riemann surfaces. This is all that is necessary to show that the gj0 and gj° can be chosen in the same way. Let [gj°] and [gk°] be two conformal structures on the topological surface W, where [gj°] ^ [gk0]. Thus in terms of metrics the conformal structures are inequivalent. Now let the surfaces formed with these conformal structures be equivalent in the sense of surfaces, so (W,[gj0]) ~ (W,[gk0]) and [(W,[gj0])] = [(W,[gk0])]. Since the surfaces are equivalent, there exists a diffeomorphism l:(W,[gj°])->(W,[gk0]) such that l*gk° = o2(z)gj°. Now let fij and fik be the respective fundamental polygons defining these two surfaces. Without loss of generality, one of the polygons, say fij can be chosen so that it is generated by vectors or non-euclidean segments in the fundamental region of the constant curvature covering space. By definition, then, fik cannot be so generated, and can be found by acting on the fundamental polygon Bj with an automorphism of the covering space. Consider the action for an embedding X such that X*G = gk°, where there is no loss of generality in the choice of Q = 1. Further let hj:/3j—>J30 and hk:Bk—>B0. Compare fik with some fixed domain Riemann surface Bi via the map f:Bi— »fi0: S[X,g,G] = Jd\yD et(f’Vgj < (fV g°, (hkVg° > (4.10) «o Now since is equivalent to fij, the function hk can be decomposed into hj o l"1, so (4.10) becomes S[X,g,G] = JdV D et(f Vg° < (f'Vgj, (1 o h:‘)*g“ > (4.11) Bo However, the second term in the bracket can be simplified as (1 O h.Vg” = hj1* ^ ) = h'1*(o2(z)g]>) (4.12) So the action reduces to a form that is entirely equivalent to that of an embedding that pulls back into the equivalence class [g:°] S[X,g,G] = jd2z^/Det(f ')*gj < (f'Vg“, (o» . h/'Xz) h^g” > (4.13) If this is compared with the form (4.9) it is evident that this is just the action that would result from an embedding with pull-back metric given by a2(z)gj°. Since the Polyakov prescription requires that the integration be performed over all embeddings of inequivalent surfaces, it is apparent that there must exist some X' such that X'*G = o2(z)gj°. If the integration included both X and X', then it would overcount the embeddings of inequivalent Riemann surfaces. Hence the constant curvature part of the pull-back metric must be restricted to those metrics arising from polygons formed from vectors or non-euclidean segments in the fundamental region of the covering space. This is the same restriction which applies to the integration over the domain Riemann surfaces, so they can be specified in the same manner. This results in a double Teichmuller integral in the reformulated path integral. This restriction factors the base space RiemMet(W) (in the quasi-fibration of the space of embeddings) by the group Diff(W), and the quasi-fibration passes to the quotient: Fg. * Imm(W,M) 71q DifKW) » RiemMet(W) RiemMet(W)/Di®W) Thus when integrating over the constant curvature part of all pull-back metrics, one need only integrate over the metrics determined by the polygons in the fundamental region and the conformal factors, which still remain to be considered. The conformal factors Q2(z)j are simply positive functions defined on the Riemann surface, which is specified by its fundamental polygon in the constant curvature covering space, flj. The physical significance of n2(z)j is that it represents the way in which the Riemann surface is deformed when it is embedded into the target space. Since the Riemann surfaces are being specified by their constant curvature metric representatives, Q2(z)j represents the deformation away from the constant curvature case. Considered this way, it is even more apparent why the intrinsic metric should be treated as a comparison point for (the energy functional interpretation of) the action, and as one would expect intuitively, the physics arises from the deformations of the Riemann surface which result from the embedding. Since the path integral prescription demands that the integral be taken over all possible configurations, it requires that all possible functions £22(z)j be considered. Since these functions are defined on the fundamental polygon (or the sphere for the genus 0 case), they can be written as the square of a functional expansion over a complete orthonormal set associated with the covering space and the polygon. In the genus zero case of the sphere, the functional expansion is in spherical harmonics, and for the tori the expansion will be over a double fourier series. Higher genus analogs exist, but will not be used here. The requirement of all possible functions can be achieved by allowing the coefficients of the expansion to range over all values. Finally, when the fundamental polygon flj is mapped to the comparison region I30, the functional expansion of f}2(z)j is mapped in such a way that it becomes a functional expansion over the polygon B0. This functional expansion is completely independent of j, and it is apparent that the deformations completely decouple from the constant curvature metrics gj°. Thus the index j on Q2(z) can be dropped. These results will be clearly illustrated in section 6 when the calculation for the tori is carried out. The final question in the integration over the space of embeddings is how to calculate the volume of the fibers in the embedding space quasi-fibration. This is a highly non-trivial question. The volume of the fibers can be very much dependent on the metric in the target space, the number of dimensions in the target space, the deformation of the surface arising from the embedding (and hence the pull-back metric), and possibly even the conformal class of the surface. Implicit in the work of most string theorists has been the assumption that there exist embeddings for the domain space Riemann surfaces, but even this is not true in general. Because of the depth of these questions, the next section will be devoted to this topic. For the moment, however, consider the fiber volumes as the transformation that takes the infinite dimensional space of embeddings to the infinite dimensional space RiemMet(W)/ Viewed in this way, it becomes reasonable to call the volume factor the Jacobian of the transformation, and it can be a function of G, Q2(z), and gj°. It will be written J(G,Q,gj°). Putting together the results about the specification of the constant curvature part of the pull-back metric, the conformal factor, and the Jacobian, the measure for the integration over the embeddings can be expressed: D[Emb(W,M)] = d|i(u.,v.,N.) D[Q(z)] J(G,Q,gj°) where d|i(U j,Vj,N j) has the exact same form as that of the integration over the domain Riemann surface, except for different indices which express the requirement that the two be varied independently. §3C. REFORMULATED PATH INTEGRAL The results of the previous two subsections can be employed to write the reformulated path integral in a form which will be directly applicable to calculation once the particular genus has been specified. The original form (4.6) becomes = Jdn(ui,vi>N.)Jd|l(uj,v.,N.)D[n]j(G,Q,g. °) exp -S[g° Q.gf] (4.14) where S has the form s[g°,n,g°] = i[d 2z>/Det(h:1)*g? < h:‘*g° n2m'}*g° > (4 .15 ) and and hj represent the maps of Bj and Bj onto B0 respectively. Notice in (4.14) that there is a double Teichmuller integral, one coming from the integration over the conformally inequivalent Riemann surfaces, and the other coming from the the integration over the constant curvature part of the pull-back metric. In (4.15) there is no index j on £12, representing the decoupling of the conformal factor under the mapping onto B0. The integral is gaussian in Q(z), but the important point is that it is a one (functional) dimension gaussian, and the discussions of chapter 2 indicate that this reduction is necessary for proper definition of the integral. The inner product between the intrinsic metric and the pull-back metric is a completely contracted tensor, so the calculations can proceed in a straightforward manner. Finally, as the next section will show, all questions of dimensional requirements and curvature requirements are contained in the Jacobian factor. Before complete answers to these questions can be given, it is necessary to consider the existence of embeddings of Riemannian manifolds into higher dimensional Riemannian spaces, the topic of the next section. 4. THEORY OF ISOMETRIC IMBEDDINGS AND IMMERSIONS The primary concern of this section will be to consider the problem of isometrically imbedding or immersing a given Riemannian manifold into a higher dimensional Riemannian manifold. Because of its applicability to the problem of immersing strings, concentration will be on the two dimensional case, but most of the theorems are of a general nature. One can see that the problem of isometrically immersing or imbedding Riemann surfaces is the converse to the problem of fibering the space of embeddings over the space of Riemannian metrics. That is, if a given immersion (imbedding) induces a pull-back metric g on W, then that immersion can be viewed as an isometric immersion (imbedding) of (W,g). Hence, by studying the existence of isometric imbeddings and immersions, one can determine a great deal about the nature of the fibers in the quasi-fibration of the space of embeddings, particularly its dependence on the number of target space dimensions and the degree of deformation induced by the embedding. The study of such questions about the isometric embeddability of Riemann surfaces arose naturally once the modem abstract definition of a surface as a 2 (real) dimensional domain with a specified metric structure was adopted, since prior to that, the classical theory assumed that the metric structure on a Riemann surface was always induced by an immersion or imbedding into a higher dimensional euclidean space. Since it is easier to specify an abstract two-dimensional metric than it is to specify a physical surface in euclidean space which carries that metric, the study of isometic immersions and imbeddings was necessary to unify the classical view of surfaces with the modem abstract view. The classic paper on the subject is the paper by Nash [4-5]. An extremely good review on the subject which summarizes and outlines nearly 100 years of development is the paper by Gromov and Rokhlin [4-1]. Rather than digress into extensive mathematical proofs, only a summary of the mathematical theory will be given here, with the technical statements and proofs of the theorems grouped into Appendix C. The next section on the dynamics of dimension will require manipulations based on the mathematical results, so it is important to have a clear statement of the problem. The work will be discussed for the case of isometrically imbedding and immersing into a higher dimensional euclidean space, since the analysis is easier to formulate in that framework. Later, some comments will be made about the extension of the results to other Riemannian manifolds. To state the problem clearly, the isometric immersion of an n-dimensional Riemannian manifold (W,g) into Rq will be considered, where g is the metric on W, and since the desired immersion is to be isometric, the pull-back metric induced by the immersion must be equal to g. This is equivalent to finding q functions ... ,<|)q mapping W->R such that g = (d^)2 + (d<])2)2 + . . . + (d 3 < t> . where d(J). = 2 —“dxp (4.16') 1 p=i 3xp Notice that this is just the standard form of a pull-back metric, but this notation will prove to be useful. If such functions can be found, then the desired isometric immersion is given by X:(W,g)— , <{)q). The functions Theorem 2.9.4 [p.36] - A compact C°° (Ca) Reimannian manifold can be isometrically C°° (Ca) imbedded in an euclidean space of sufficiently high dimension. Theorem 2.9.5 [p.37] - For every free C°° (Ca) imbedding fQ of an n-dimensional compact C°° (Ca) manifold V in RQ with q > sn + 4n + 5 and any C°° (Ca) function These theorems will allow for the determination of many properties of the fibers in the quasi-fibration of the space of embeddings. The consideration of these properties leads to a new concept called the Dynamics of Dimension, which will be introduced in the next section. Before going on to consider the dynamics of dimension, it is perhaps worthwhile to consider isometric imbeddings into target spaces which are non-euclidean. It is certainly true that if the non-euclidean target space is flat and of dimension > sn + 4n + 5, then all Riemannian manifolds W of dimension n can be isometrically embedded. This follows almost trivially from the recognition that the non-euclidean target space has itself an isometric imbedding into a higher dimensional euclidean space, where its image is flat. Let M denote the target space, and let dim(M) = sn + 4n + 5 = q. Then M can be isometrically imbedded in Rp for p > sq + 4q + 5 by some map f:M—>Rp, and its image Im^M) is flat and 1 2 0 must span a q-dimensional submanifold in Rp. Now by Nash's theorem, W can be isometrically imbedded into any small volume of Rq, so choose a q-dimensional neighborhood U in Rp which is completely contained in Im^M) and isometrically imbed W into this neighborhood by h:W—»U. Then by composing h with the inverse isometric map f 1:Im^(M)->M, one finds the desired isometric imbedding f'1» h:W—»M. For target spaces M which are more general Riemannian manifolds, i.e. non-flat, there is still the assurance of an isometric map of M into a euclidean space of sufficiently high dimension, so if the image of M contains a flat q-dimensional neighborhood (for q, W as above), then W can be isometrically imbedded into M in the same way as above. For arbitrary target space metrics, however, it would be necessary to examine the problem more closely to determine if the target space admitted isometric imbeddings. 5. THE DYNAMICS OF DIMENSION The reformulation of the Polyakov path integral in section 4 was explicitly dependent on the quasi-fibration of the space of embeddings. It was developed in such a way that the integration would be performed on families of cross-sections of the quasi-fibration, each of which was presumed to be isomorphic to the base manifold RiemMet(W)/DifflW). There are several qualifications that must be made. First of all, there must exist isometric embeddings of Riemann surfaces reflecting every possible Riemannian metric on W. This was assured for (euclidean) target spaces of dimension > 10 by the discussion in the previous section reflecting the work of Gromov and Rokhlin, in fact, the results held for imbeddings, not just immersions. However, this does not in any way 1 2 1 ensure that the volume of each fiber of the quasi-fibration is equal, and since there is no obvious measure on the fibers, the concept of volume has little meaning anyway. Ostensibly, one would like to prove that the quasi-fibration can be decomposed into a canonical family of cross-sections, with each cross-section isomorphic to the base manifold, and no element of the total space on more than one cross-section. The end result would be that one could regularize the functional integral by integrating over only one cross-section. This represents a fundamentally new approach to regulating functional integrals, and in the discussion later, it is shown that this regularization introduces a new concept of locality which is appropriate for curved spaces. One must note, however, that there may not exist cross-sections isomorphic to the base manifold, which introduces some complications. In fact, the non-imbeddability theorems prove that if the target space dimension falls below a critical value (which must be > 4) then certain Riemann surfaces with a particular metric structure cannot be isometrically imbedded into the target space. This would mean that for a metric which cannot be isometrically imbedded, the fiber over that point in the space RiemMet(W)/Diff(W) would be empty. This is the essence of the Dynamics of Dimension; as the dimension of the target space changes, the structure of the quasi-fibration of the space of embeddings must change, altering the theory. And, even though the stated results were for flat target spaces in particular, it is true in general that as the target space dimension decreases, it becomes more difficult to embed surfaces isometrically; hence, the term Dynamics of Dimension is appropriate for all target spaces. The factor J(G,Q,gj°) in (4.14) is designed expressly to take into account these changes in the quasi-fibration as a function of G and the dimension, which is implicitly included in G. In the following 1 2 2 discussion, attempts will be made to determine the possible form of J, and will indicate that the reformulation of Polyakov string theory should lead to an integrable, non-perturbative theory. §5A. CALCULATING THE FIBER VOLUME FOR D > DCRrr The object of this section is to show that for flat target spaces of sufficiently high dimension, the quasi-fibration of the space of embeddings can be decomposed into a canonical family of cross-sections. This essentially means that for any given isometric imbedding of a Riemann surface with a given metric, that imbedding can be extended to an isometric imbedding for any other metric in RiemMet(W)/Difi(W). This represents a deformation of a given fiducial image surface in the target space to produce a cross-section of the embedding space quasi-fibration. Consequently, the quasi-fibration of the space of embeddings has a natural family of cross-sections, with each cross-section associated with the deformations of a particular isometric imbedding. Since the reformulated path integral involves an integral across the base space, which has the same value for any cross-section, the regulated value for the functional integral will be defined to be the integral over one cross-section. Providing that the remaining integral is finite, this represents a new regularization procedure for path integrals in general, and the reformulated path integral (4.14) in particular. The proof of this claim will depend upon the theorems in the last section and Appendix C, and is essentially contained in them. Actually, it should be pointed out that the theorems refer to imbeddings (i.e. without self-intersections), so the following development is really for less singular surfaces than the immersions which have self-intersections. 123 To show that the theorems of the last section lead to a unique family of cross-sections for the embedding space quasi-fibration, one needs to consider two distinct imbeddings which belong to the same fiber (distinct in the sense that they are not just euclidean translations or rotations of each other). Call the two imbeddings fQ and f0', and let W be a C°° (Ca) Riemannian manifold, and then f0, f0':W—>Rq induce the same metric on W since they are in the same fiber over RiemMet(W)/Diff(W), so that gives g(f0) = g(fo0 (where g(f) is the pull-back metric, using the notation introduced in the previous section). By Th. 2.9.5 in Appendix C, if q > sn + 4n + 5, there exist extensions f and F of f0, f0' respectively which satisfy g(f) = g(f0) + (d<»2 and g(F) = g(f0') + (d<(>)2 with g(f) = g(fi) and <|):W-»Rq a C°° (Ca) fimction on W. That f and F are distinct functions can be seen in the proof of Th.2.9.5, since the functions are given by f = ft»f2 and F= fx"o £z' where f2 = f^' since it is independent of the imbedding into Rq and is the identity on W in the first component. However fx ^ f / since fo = fi | WxO: W x D2(a)->Rq and f0' = f/1 Wx0:W x Int D2(a)->Rq are distinct by assumption. Hence, f * F, i.e. they are distinct. As in the proof of Th. 2.9.6, this can be extended to all Riemannian metrics. The end result is that any element of a fiber in the embedding space quasi-fibration can be used to generate a cross-section, providing the target space dimension is > sn + 4n + 5. This generates the family of cross sections and shows that the elements of the fibers can be put in one-to-one correspondence, so the Jacobian factor in (4.14) is an infinite constant. Thus by integrating over only one cross section in the family of cross sections, one can regularize the path integral. There are several implications of these results. First, it sets an upper limit on the critical dimension, above which the embedding space quasi-fibration can be decomposed into the canonical family of 124 cross-sections, which gives dcrit < sn + 4n + 5 = 16 for n = 2. Recall also that the value sn + 4n + 5 is the general value for isometric imbeddings of Riemannian manifolds of any dimension, and specialized techniques allowed this value to be reduced to 10 for the n=2 case [see 4-1, p.46], so one might hope that the application of the same specialized techniques would reduce dcrit to be < 10 for n = 2. Secondly, the cross-sections all arise as deformations of a given embedding, which indicates that the procedure gives a path integral based on local deformations of a surface. This amounts to a definition of locality for curved space path integral calculations. The calculation essentially compares a given fiducial embedding configuration with all of those configurations into which the fiducial surface can be deformed. In a general curved space, this is as good a notion of locality as one can hope to find, since configurations that are separated in space cannot be related unless there is some regular geometrical structure which can be exploited. In this sense, the regulated path integral represents a local field theory. §5B. COMPACTIFICATION FOR D < DCRrr Whenever the target space dimension falls below the critical dimension, the Jacobian will become non-trivial. Similarly, if the target space was non-flat, one would expect that the Jacobian would play a fundamental role in the theory. The object of this sub-section is to consider the structure of the reformulated theory in the presence of a non-trivial Jacobian, a process which will be termed compactification because of the obvious implications for flat target spaces. The starting point for this investigation is a consideration of the possible forms the Jacobian may take. The basic purpose of the Jacobian 125 is to provide a relative measure of the volume of the fiber over a given point in the base manifold RiemMet(W)/Diff(W). As was discussed earlier, a point in the base manifold can be characterized by its conformal equivalence class of metrics [gj0] and its conformal factor G2(z), so the Jacobian can certainly depend explicitly on these. Similarly, for a given target space, its metric structure and dimension can provide obstructions to the embeddability of surfaces, which would thus affect the volume of the fibers. Thus, the Jacobian can depend implicitly on the target space metric G and the dimension. The explicit dependence of the Jacobian on gf and £l2(z) bears further consideration. Since the gj° all represent constant curvature metrics on the surface, one would intuitively expect that the isometric embedding of a genus p Riemann surface with constant curvature metric gj0 would naturally extend to an embedding of any other surface of genus p with a constant curvature metric gj° for any j * i. This result has not yet been proven explicitly, so far as the author knows. However, there are several existence proofs by Garsia and Rodemich [4-15] and Garsia [4-16,4-17] which indicate that this may in fact be the case. Garsia proves that if the target space is R3, there exist C°° surfaces in R3 which are conformally equivalent to every equivalence class of compact Riemann surfaces for all genus > 1. [The case of genus 0 is satisfied trivially because the standard sphere in R3 is conformally equivalent to any other genus zero surface]. This means that every conformal equivalence class of Riemann surface has some representative among the surfaces of 3-space; however, the imbedding is not isometric with respect to the constant curvature metrics on the defining fundamental polygons. Consequently, all of the constant curvature surfaces must undergo some deformation to fit into 3-space. The fact that they can all be so deformed is 126 an indication that the particular constant curvature metric g^° does not play a fundamental role in determining the embeddability of a Riemann surface, and that the volume of the fibers over the base space may depend only on the conformal factors Q?(z). However, in the absence of a proof, this can only be a plausible conjecture. The fact that the proofs by Garsia were only existence proofs and not constructive proofs makes it impossible to check this explicitly. In defense of the plausibility argument, it should be said that the constructions in Garsia's paper [4-17] were all based on deformations of a standard torus, so at least the method leading to the conformal imbedding was consistent for all gj°. The precise statement of this conjecture is that if the fundamental polygon Bi corresponds to the Riemann surface (W^0) and if XifWjgj0)—>R^ is an imbedding which induces the pull-back metric X*8=Q2(z)igi° on fy, then there exists for all j * i a conformal embedding Y:(W,gj0)—»Rd with pull-back metric Y*8 = G2(z')jgj° such that the conformal factors H2(z)i and Q2(z')j are equivalent. Conformal factors are said to be equivalent if composition with the affine transformation between the fundamental polygons makes the functions equal, i.e. if Vji:(B.,g°)-»(B.,g”) then (n.2 • V..)(zO = a 2(z) If a proof were to be found of this statement, it would imply that any conformal imbedding (with respect to an intrinsic constant curvature metric, g° say) could be extended to a conformal imbedding with respect to any other intrinsic constant curvature metric with equivalent conformal factors under the pull-back map. This in turn would imply that the volume of the fibers in the embedding space quasi-fibration would depend only on the conformal factors Q2(z). Assuming for the moment that the above conjecture proves to be 127 true, it is then possible to show that the reformulated path integral leads to an integrable theory and can provide a natural mechanism for the development of massive terms in the action for d < dcrit (or whenever the Jacobian is non-trivial), which could represent an alternative method for spontaneous symmetry breaking. First of all, since the fibers would depend only on Q2(z), the Jacobian would depend only on £22(z). As discussed earlier, the conformal factors must be positive functions, which can be expressed as the square of a general functional expansion, so J can be written as a bilinear functional J(Q ,Q). The path integral would still be formally integrable using the relation (4.17) where of course it may be necessary to regulate the trace of J since it is infinite dimensional. The ability to generate massive terms in the action follows from (4.17) since powers of J would also be formally integrable, so by inserting into the path integral a term of the form t p njn(n,Q) = exp-j If the conjecture proves to be false, then it will be necessary to incorporate some dependence on the conformal equivalence class of the surface into the Jacobian. If the dependence on the conformal factor separates from the dependence on gj° = gj°(uj,Vj), then the Jacobian can be written J(G,£2,gj°) = A(Q,£2)F(gj°) for A(Q,£2) a bilinear functional formed from the conformal factor, and for some function F which depends on the conformal equivalence class of the surface. As in the preceeding paragraph, the [D£2] integral is defined through equation (4.17). For the Teichmuller integrals over gj0, gj°, since the integrand of the path integral is a negative exponential depending on gj°,gj°, the integral should be defined provided that F grows only like a polynomial. The observables in the theory are functionals of the pull-back metrics, and for simplicity, the functionals will be assumed to separate into a part dependent on the conformal factors and a part dependent on the constant curvature metrics. The part dependent on the conformal factor can be converted to a bilinear functional B(Q,G), and the part dependent on the constant curvature metrics will be denoted G(gj°(Uj,Vj)). Then the correlation functions can be expressed as r -s[g°,n,g°] Jdn(u.,v.,Ni)dn(u.,vj,Nj)D[n]j(G,n,gj°)B(n,n)G(g“)e 1 J (4.19) As before, this should be an integrable theory when the Jacobian has one of the forms described above, and for functions G(gj°) which grow only like a polynomial, and bilinear functionals B(Q,£2) which have a trace which can be regularized. Since the conformal factors represent the deformation of the embedded surface away from the constant curvature surface, the theory can be interpreted as a field theory of the deformations, with an extra integral over the constant curvature metrics which acts as an averaging integral over the different comparison points 129 of the energy functional. The above discussions indicate that the reformulated theory provides a natural environment for considering string theory in a target space of any dimension, with any metric structure. The problematic part of the prescription is to determine the structure of the embedding space quasi-fibration, and once this has been determined, the theory should be calculable. This can be considered as an adjunct mathematical problem for string theory, the results of which would have immediate application. 6. CALCULATIONS §6A. GENUS ZERO - THE SPHERE For the genus zero case of Riemannian spheres, the application of the reformulated approach reduces the calculation to an almost trivial case. Any complications which arise must result from irregularities in the target space which are reflected in a non-trivial Jacobian factor in (4.14). As discussed in the previous section, if the target space was a euclidean space of sufficiently high dimension, then the regulated path integral would be defined as the integral over one cross-section in the canonical family of cross sections, which amounts to treating the Jacobian as an infinite constant. The great simplification in the genus 0 case is that there is only one conformal equivalence class of spheres, so any metric on the sphere is conformally equivalent to the usual metric on the sphere induced by the inclusion map in R3. This eliminates the integration over dji(...) in (4.14), and one can simply let g0 represent the usual metric on the sphere. Thus any pull-back metric induced by an embedding X:S2-»M must be of the 1 3 0 form X*G = Q 2(z)g0. Since there is only the one equivalence class of spheres, there is no need to map to any comparison region, and all calculations can be done on the sphere. So, the action in the path integral reduces to S[X,go,G] = IJ d 2z ^ D etg o < g0,0 2(z)go > (4.20) S2 The inner product reduces even further since Z = |"d [q] j(G,Q) exp - J d2z ^Detgo Q 2(z) (4.21) S2 Now the functions Q 2(z) are simply positive functions on the sphere, and are determined by the square of an arbitrary function expanded over the complete orthonormal set of spherical harmonics on the sphere. The exponential in (4.21) is the inner product in the function space of spherical harmonics on the sphere, and so the exponent can be written (Q , £2) to denote the inner product. If one considers the target space to be of sufficiently high dimension and flat (d>dcrit), then the Jacobian in (4.21) is just a constant which can be pulled out of the integral and discarded, since the regulated integral is defined to be the integral over one cross-section. In this case (4.21) reduces to a straightforward gaussian integral with a non-degenerate inner product, and this gives a well-defined result Z = J d [q ] exp -(Q, £2) = N (4.22) S2 where N represents the chosen normalization for the integral. In the case where the dimension is less than the critical 1 3 1 dimension, or if the curvature in the target space is irregular, the Jacobian will be non-trivial. However, since the sphere has only one conformal equivalence class associated with g0, the actual form of the Jacobian can depend only on the function H2(z). The physical meaning of the Jacobian in this case is that for a given point in the base space of the embedding space quasi-fibration, it would provide a relative measure of the volume of the fiber over that point. Since each point in the base space can be characterized by the function Q 2(z), the Jacobian must be as well. Then the Jacobian can be written as a bilinear functional of a general expansion over the set of spherical harmonics in the form J(£2, £2), and (4.22) becomes well-defined and integrable using equation (4.17), giving Z = J d [ q ] J(Q , Q) exp -(Q , Q) = N Tr J (4.23) Even in the simplest case of the genus zero calculation, all of the key aspects of the dynamics of dimension are revealed. For target spaces above the critical dimension, the theory is trivial. As the dimension drops below dcrit non-trivial effects begin to appear, effects which should be calculable by non-perturbative methods as in (4.23). It also indicates how a massive or interactive theory can arise from a trivial theory since for d < dcrit the Jacobian could be written as an exponential, which would then appear as an interaction term in the original lagrangian. Thus, the reformulated approach to defining the path integral can provide answers to questions about critical dimensions, development of interacting theories from trivial theories, transitions to low-energy limits, and non-perturbative techniques, provided enough information can be learned about the structure of the quasi-fibration of the space of embeddings. To test further the applicability of the reformulated approach to defining the string path integral, the first non-trivial case of 1 3 2 the genus one tori will be considered. §6B. GENUS ONE - THE TORI The genus one example includes all of the aspects of the full theory, and it is here where the power of the reformulated approach becomes evident. In this example, all of the results on Riemann surface theory will be directly applicable, the decoupling of the conformal factors will be exhibited, and the development of a finite integral for flat target spaces above the critical dimension will be derived. All of the information necessary for the specification of the conformally inequivalent tori was included in appendix A, using the period parallelograms whose sides are generated by one vector in the defining fundamental region and the other the unit vector on the real axis. In the reformulated path integral (4.14) there is a double Teichmuller integral; each one must be represented as an integration over the period parallelograms generated from the fundamental region. The measure for these integrations must be chosen so that it respects the invariance of the theory under different choices of the fundamental region. The region selected for the calculations is defined by [see Fig.7] ~- o V o V + C cos 27tm(u — - v)sin 27tn — + D cos 27im(u - —- v)cos 27tn — mn y y mn v y V (4.26) o o o o Any (L2) function on the period parallelogram can be expanded in such a way, and since the path integral prescription requires that all possible paths be considered, the coefficients will be allowed to take on all values. The second restriction on the conformal factors is that they must be positive functions. Since the fourier expansion of any positive function may be written as the square of some other functional expansion [4-14], it suffices to take as the conformal factors the squares of all functions of the form (4.26), as the suggestive notation fl 2(z) for the conformal factors was meant to indicate. The measure D[q] is the gaussian measure associated with the non-degenerate inner product (Q , Q). Again, the problem has been reduced to a one-dimensional gaussian integral, thus 134 avoiding the problems discussed in Chapter 2. To perform the integrations in the reformulated path integral (4.14), the period parallelograms must be mapped onto a comparison region fi0, which will be taken to be the unit square for simplicity. The mapping will be the affine transformation taking the vector (u^Vj) in the fundamental region which defines the parallelogram fij, to the vector (0,1 ) of the unit square fi0, while keeping the other side ( 1 ,0) fixed, since it is the same for both fij and fi0. Define the map F a y -^ (4.27) The transformations are given by the equations below, where u,v are the coordinates on fij and x,y are the coordinates on fi 0 u. i , F(u,v) = (u - —v , — v) = (x,y) F (x,y) = (x + u.y , v y) = (u,v) (4.28) V i V. i l l Once the form of the affine transformation is known, it is a simple matter to pull-back the flat metric on to fi0, so that the map is an isometric map of onto fi0, as is required by the reformulated approach. The pull-back is found using the inverse function F '1 and the relation (F-1* 8)ab(x,y) = d^F^Fd^F-W^v) = Yab(x,y). The components have a particularly simple form V f(x,y) = sin 27tmx sin 2nny + sin 27tmx cos 2icny m,n + C cos 2jimx sin 27my + D cos 27cmx cos 27cny (4.31) By transforming the period parallelograms to the comparison region, the conformal factors have been completely decoupled from the metric terms, since the metric terms (4.29) are explicitly dependent on the defining vector (u^vp of the period parallelogram. Since the conformal factors represent the deformations of the tori which arise from the embedding into the target space, the fact that they decouple from the metric terms is a further indication that the interpretation of the theory as a theory of deformations of embedded surfaces, and the related concept that the intrinsic metric is just a comparison point for the energy functional is well founded. The physical significance of the deformations of the surfaces has been accentuated by this approach. Now that the form of the pull-back metric from the period parallelogram to the comparison region fi 0 has been found, it is possible to write out the action (4.15) in an integrable form. Recall for the moment the form of (4.15) 1 3 6 S[g°,Q,g°] = ijd 2z^Det(h;Vg; < hj^g;, a2(z)hj1*gj° > (4.15) B The pull-back metrics (hi"1)*gi° and (hj'1)*gj° are of the form (4.29), so to simplify the notation, let them be called yi and Yj respectively. Then the inner product in the action (4.15) can be written < Yj, £22(x,y)Y. > = < y . , Y > Q2(x.y) = n 2(x,y) (4.32) Now substituting in the appropriate components in term of u^v^UjjVj for the respective metrics gives ( y ) (Yj)ab= - r ( uf + v f • 2uiuj + uj2 + vj) and y /DetY = vi (4-33) The action can then be written in its final form S[u.,v.,u.,v.,£2] = 2“ (u^ + vf - 2u.u. + u? + v2)Jdx dy £22(z) (4.34) The terms in front of the integral pull out because they are independent of x,y, and it is evident that the integral is just the inner product (£2 , £1). The calculation of the full partition function for the genus 1 case can now proceed. In the instance where the target space is flat and the dimension is sufficiently high, the Jacobian in (4.14) is just an infinite constant which is disregarded, since the regulated integral is defined to be the integral over one cross section, giving the finite integral ffd u . dv. frau.'du. dv.av. r r j j (4.35) z=2 2 2 JdM exp • s[Ui>vi-Vj’n] u.1 + v. 1 U. j + V. j where S is given by (4.34), and the range of the u variables is -V 2 ^ u ^ /2» and the range of the v variables is V(1 - u2) < v < The integration in £2 is just a one (functional) dimension gaussian integral based on the non-degenerate inner product (£2 , £2) and the integration in the u^v^u^vj 1 3 7 is convergent. The integral cannot be written in terms of simple functions, but can be calculated numerically. There is no conformal anomaly present,and the only way dimensional considerations appear is through the Jacobian factor, which only becomes non-trivial for d < dcrit, at least for flat target spaces. Further, because of the careful choice of the measure for the integration over the metrics, the integral is invariant under a transformation to any other fundamental region for the defining period parallelograms. Since this is the equivalent of a gauge transformation, the reformulated integral retains its gauge invariance, even though the initial formulation would have been termed gauge-fixed. Of course, for d < dcrit the Jacobian factor will be non-trivial, and the full form of the path integral must be considered, giving As discussed in the previous section, there is some reason to believe that the Jacobian is dependent only on Q2. If further mathematical investigation proves this to be the case, then the theory is trivially integrable even for d < dcrit since the Q. integral can be done by converting the Jacobian into a bilinear functional and using equation (4.17) (4.17) If the Jacobian cannot be restricted to this simple form, but can be separated into a term dependent on Q, and a term dependent on gj°, i.e. j(G,Q,g°) = A(n, a) F(g°) = a (q , n) f(u .,v . ) (4.37) then the Q. integral should be calculable, and since the integrand is a decaying exponential in the u and v variables, there is a large class of functions F for which the integral would exist and be finite. This formulation also offers the possibility of calculating various 1 3 8 correlation functions based on deformations of a locally embedded surface. This simply means that for surfaces embedded in such a way that the pull-back metric has conformal factor A1 to go through a transition to a surface with pull-back A^, one would calculate _ ffdu. dv. ffdu. dv. r r , Z = J I —H l-y1—y D[n]AA2 exp-S[ui,vi>uj,vj>£l] (4.38) u. + v. J u. + V. 1 1 J J this would then be calculable using (4.17) and defining a bilinear functional A(f2,Q) in such a way that A(Aj,Aj) = 1 and zero otherwise. In this way string theory becomes a physical theory of surfaces moving and deforming in space. 7. GENERAL RELATIVISTIC ASPECTS The reformulation of Polyakov string theory was initially developed to stress the geometric aspects of the theory, and allowed all of the objects on the theory to be expressed in terms of metrics on the Riemann surfaces. This raises the hope that the reformulation will allow one to consider the connection between string theory and general relativity in its full geometric character. Primarily, one would like to consider the relation between the diffeomorphism group of the target space and the structure of the embedding space quasi-fibration. This is not a trivial problem, however, the impact of general relativity in its full geometric form is only seen when the full diffeomorphism group of the target space is considered. Since the reformulated theory is based on the quasi-fibration of the embedding space, it is sufficient to consider the effect of the target space diffeomorphisms on the quasi-fibration. If the group DiffiM) leaves the 1 3 9 quasi-fibration invariant, then the full reformulated theory will be invariant under the invariance group of general relativity. Let the space Emb(W,M) represent the space of embeddings of W into M, where G is the metric on M. To prove that the elements of Diff(M) map Emb(W,M) onto itself, consider a function YeEmb(W,M) and a diffeomorphism feDifftM). Since f is a diffeomorphism, it is a one-to-one map, and denoting the image of W under the map Y by ImY(W), f restricted to the images is a one-to-one map, i.e. f:ImY(W)—»Imf#Y(W) is one-to-one. Since Y must be an imbedding or immersion to be an element of Emb(W,M), and f is one-to-one, the composite function f ° Y must also be an imbedding or immersion, and hence, f»Ye Emb(W,M). Further if G is the C°° metric on M, then f*G is also C°° in the region of ImY(W), so the pull-back metric (f» Y)*G = Y*(f*G) is a non-singular metric on W. Therefore, for any element f of DifiKM), and any element Y of Emb(W,M), the composite function f » Y is an element of Emb(W,M), so the diffeomorphism group of the target space leaves the embedding space invariant. Since the quasi-fibration is over RiemMet(W)/Diff(W), and this is independent of the target space, the full integral is invariant under DififiM), i.e. in the path integral (4.14) d|i(...) and J(...) are invariant since the quasi-fibration is invariant. The net effect of Diff(M) is to mix up elements of the fibers, but the mixing is such that the quasi-fibration is invariant. In another sense, the path integral can be viewed as a means of defining a vacuum to vacuum amplitude in a small region of a curved space, according to the deformations about a given embedding which can arise in the region. Define a local iacobian as a functional delta function which takes the value 1 for all embeddings that can arise from deformations of the given embedding, and zero for all embeddings which are excluded in the region. If in the neighborhood of a given embedding 140 the local jacobian is known, the regulated path integral would be the integral over just one cross section, where the Jacobian removes any excluded functions. Then if one makes a diffeomorphism of the target space the regulated integral would not be invariant in general, because under a map f e DifftM), the image of W can be taken to a region of the target space where the curvature is very different, introducing obstructions to the embeddability of surfaces, and thus changing the excluded deformations of the given surface. This would change the local jacobian and the integral would not be invariant. The second intriguing possibility which arises as a result of the reformulation of Polyakov string theory is that it may eventually lead to a satisfactory answer to the question of how string theory can generate the structure of spacetime. To do this the theory must be formulated in a way which is intrinsic to the string, and since the reformulation is based on the intrinsic metric and the pull-back metrics, it satisfies this criterion. In the case where the target space is flat of dimension d > dcrit, the regulated path integral in the reformulated theory is in fact independent of the target space. One might then reverse the argument and postulate that the strings behave according to the regulated path integral with J = const., and the spacetime metric which arises in the neighborhood of a string or ensemble of strings is in fact induced by the strings. That is, to a given string would be associated a particular energy which would then determine the conformal factor 0 2(z). Then since the string would have to be embedded in such a way that the induced metric had a conformal factor Q 2(z ), the structure of the spacetime in the neighborhood of the string would have to be such that it could "support" the embedding of the string, i.e. it would have to be of the right dimension, metric structure, etc. In this way then, the strings would produce the structure of spacetime, and there would an interaction between the spacetime structure and the string structure. 142 CH. 5 - SUMMARY 1. CONCLUSIONS The line of research presented in this thesis began with a critical look at the measure for the embedding integration in the standard approach to Polyakov string theory, and proceeded to consider string theory in an increasingly general form, first removing the linear span restrictions on the gaussian measure in chapter 2, then generalizing the theory to curved target spaces in chapter 3, and finally reformulating the theory in a manner applicable to curved target spaces. At all stages, the theory was considered without the addition of cosmological terms to the action, and the final reformulation was a non-perturbative theory. In chapter 2, the gaussian measure imposed on the embedding integration was shown to be highly dimension specific, since it restricted contributions to the path integral to be from surfaces with a linear span equal to the number of target space dimensions. This is a natural outcome of the fact that the laplacian operator is degenerate on the constant functions, which is connected to the existence of sets of measure zero with respect to the gaussian measure. As a consequence, any embedded surface which could be contained in a box of fewer dimensions than the number of target space dimensions would be in a set of measure zero, and would not contribute to the integral. To derive a measure which included no assumptions about the linear span of the embedded surfaces, it was necessary to develop a variable dimension field theory. This work may have applicability outside the area of string theory, and may provide a framework for the investigation of a variable dimension formulation of other theories based on degenerate operators. The infinite 143 sum over the dimension resulted in an effective compactification of the theory, with the result that the naive scaling limit gave the usual [DetA]'1^2 answer, and the corrections to that result in the full theory arose non-perturbatively. The discussion in the third chapter considers the extension of string theory to curved target spaces, and adopts the energy functional interpretation of the action. This provides a physical interpretation of classical solutions of the string equations of motion in terms of harmonic maps and minimal surfaces, and introduces the mathematical concept of the tension field of an embedded surface. The interpretation of the action as the energy functional gives a physical explanation for the existence of conformal invariance, since then the intrinsic metric serves simply as the comparison point for energy calculations. It is shown that only trivial solutions of the classical equations of motion exist in a euclidean target space, which provides motivation for seeking a formulation of string theory applicable to curved target spaces. At a simplistic level, it is shown that classical solutions for embeddings into S3 can be used to develop a toy model for strings in S3 using background field techniques. A more involved calculation arises when the toy model is used to create a theory of cosmological membranes in a hypothetical S3 x R spacetime, which seems to indicate that in curved space using the time-sliced gauge, it is possible to find a non-trivial membrane theory without the addition of a cosmological term to the action. However, the work in chapter 3 only outlined such an approach, so a great deal of further work would be required before drawing any hard conclusions. Finally, the background field calculations exhibit different growth in the exponent of the [Det A]'1/2 factor from the standard gaussian based approach to defining the embedding integral in euclidean space Polyakov string theory, providing 144 further motivation for the development of a formulation of string theory with a dimension non-specific measure. The culmination of the line of research developed in this thesis comes in the fourth chapter, where a new formulation of string theory is presented. The formulation stresses the geometrical objects in the theory, with particular emphasis on the pull-back metrics arising from the embeddings. In keeping with the interpretation of the action as the energy functional, the path integral is developed in a "gauge fixed" manner which treats the intrinsic metric as the comparison point of the energy calculations. However, because of a careful selection of the measure for the Teichmuller integrals, the path integral retains gauge freedom under transformations to different fundamental regions. The embedding integration is converted to an integration over pull-back metrics through the development of a quasi-fibration of the space of embeddings, and the observables of the theory become the functions which represent the deformation of the image surface caused by the embedding. The measure for the embedding integration satisfies all of the dimension non-specific requirements suggested by the work in chapter 2, since any embedding which produces an image with a well-defined pull-back metric contributes to the integral. The quasi-fibration allows for the development of an entirely new method of regulating the infinities in path integrals by defining the regulated integral to be the integral over one cross-section of the canonical family of cross-sections in the quasi-fibration. The family of cross-sections arises from deformations of a fiducial surface on each cross-section, so the concept of locality in the new formulation refers to a comparison of a given embedded surface with those into which it can be deformed. The theory leads naturally to a new concept which is termed the "dynamics of 145 dimension", since it provides a mechanism for dynamic changes in the theory as a function of the target space dimension or curvature, and can give rise to effective massive terms or interactive terms in the action. The notion of critical dimension is given an alternative interpretation as the dimension above which the quasi-fibration decomposes into the canonical family of cross-sections. The theory provides for the non-perturbative calculation of correlation functions between the new observables. The new formulation of the theory can be interpreted as the complete generalization of the background field technique, albeit in a non-perturbative manner. Here the embeddings which produce the constant curvature surfaces play the role of the classical solutions in the background field expansion, and the fluctuations in the background field technique are analogous to the conformal part of the pull-back metric, £22, since Cl2 represents the deformation of the image surface away from the constant curvature surface. Of course, this formulation considers the constant curvature embeddings for aU conformal classes of inequivalent surfaces (thus improving the genus one toy model-type approach), and the deformations arise naturally, rather than being put in by hand. In fact, this approach may provide some clarification about the applicability of the background field techniques. Needless to say, there are many difficulties associated with the new formulation. It was evident from the work in chapter 4 that the determination of the structure of the quasi-fibration of the embedding space is extremely difficult, making it imperative that further mathematical research in the area of isometric imbeddings and immersions be pursued. The calculations presented for the genus zero and genus one surfaces exhibit all of the desired properties of the new formulation. The inner product between the pull-back metric and the intrinsic metric can 146 be calculated so that the action can be expressed in terms of the parameters of the theory. The measure gives explicit support to all surfaces with a non-degenerate pull-back metric. The embedding space functional integral which appears as a d-fold product of gaussian integrals in the standard euclidean space formulation of string theory is reduced to a single gaussian functional integral over the conformal factors arising from the pull-back metrics. This eliminates the problems with determining the functional Haar measure in chapter 2. The resulting integrals are well-defined and finite when normalized. The double Teichmuller integration in the genus one case should be calculable, and although the final results for the (d > dcrit) genus one path integral are not expressible in terms of simple functions, it should be amenable to numerical techniques. The comparison with the genus one results of Polchinski shows that the new formulation is inherently different from the standard approach, which should not be surprising since the new formulation does not add a cosmological term, develops a different measure and uses a new geometric regularization scheme. It should be noted that in the Polchinski results it does not seem feasible to perform the Teichmuller integral, and the new formulation has the added advantage of allowing one to consider the form of the theory for any target space dimension, after invoking the alteration of the theory required by dynamics of dimension. Finally, because of its geometric character based on metric structures, the new formulation allows one to consider the relation of general relativity with string theory. It becomes possible to consider the effect of target space diffeomorphisms on string theory. When considered in a small patch of the target space, these questions give rise to the notion of a local jacobian which is not invariant under diffeomorphisms of the 147 target space. This simply expresses the fact that one would not expect the local results for observables in a small patch to remain the same if the conditions of curvature, dimension, etc. change in the patch. By pursuing the idea of local jacobians, it may be possible to consider interactions between the string geometry and the spacetime geometry. 2. SPECULATION It is hoped that extensions of the ideas presented in this thesis will prove fruitful in the years to come. The formulation of the variable dimension field theory may allow one to extend the concept of dimension to include fractional dimensions. Since the results in chapter 2 involve gamma functions, and since these can be extended beyond the positive integers to include real arguments, the extension to fractional dimension might be straightforward. However, before conjecturing such an extension, it will be necessary to look carefully at the support of the measures. The work in this thesis raises fundamental questions about the observables in a theory. The point of view put forward here is that the observables should be geometric objects. Further, the discussion of sets of measure zero and the idea of dimension specific gaussian-based functional integrals suggests that there may be a question about what is being observed, e.g. is a constant embedding of a string considered a point or a string? More significantly, if the concept of dimension can be extended to fractional dimensions, the original quantum mechanical question of observability arises, since any detector which humans envision building is based on integer dimensions. The idea of using pull-back metrics to calculate the theory should 148 be extendable to higher genus surfaces. Since the action is considered as a global inner product and is hence additive, it may be possible to create higher genus surfaces by pasting together simpler surfaces. This could possibly allow for the development of a field theory of strings. Finally, it may prove to be reasonable to restrict the embedding integration to be only over the space of imbeddings, i.e. without self-intersections, since if the image surface is allowed to self-intersect, it is questionable as to whether or not the surface belongs to the higher genus surfaces. In fact, this is not a new question at all, as this quote from the 1949 paper by Chen suggests [5-1] "...the branch points have the physical significance that they can not be immediately demonstrated by soap films. As soon as it looks unavoidable for the surface to have branch points, or at least self-intersections, the film either form a surface of higher topological structure without self-intersections, or there appear lines of discontinuity of the first derivative along which several pieces of the surface cut at certain angles to form a state of equilibrium" This physical statement would seem to suggest that embeddings which produce self-intersections of a given genus Riemann surface should be considered as higher genus surfaces. This restriction was naturally encoded in the new formulation of string theory developed in this thesis since the quasi-fibration was considered for imbeddings, so it is hoped that future discussion will clarify the question of self-intersecting surfaces. 1 4 9 APPENDIX A ■ RIEMANN SURFACES Riemann surfaces are generally defined as one (complex) dimensional connected complex analytic manifolds. However, since the metrical properties are primarily of interest here, an equivalent definition of Riemann surfaces will be used which develops this connection more clearly. The discussion will follow closely the presentation in the lectures by Earle published in [A-l]. Since Riemann surfaces are manifolds, they can be given a global atlas of coordinate charts, so it is no restriction to look at planar regions. On a plane region D, any Riemannian metric can be written in the form ds2 = E dx2 + 2F dx dy + G dy2 (A.l) where E, F, and G are real-valued functions in D such that E and EG-F2 are positive everywhere. At its most basic level, a connected two (real) dimensional surface and a Riemannian metric are all that are needed to define a Riemann surface. The need to clasify equivalent surfaces, however, requires that one must look further. Consider another planar region D*, and a sense preserving diffeomorphism, w:D—>D*. Let the metric on D be called y, and the metric on D* be called yv Then w is a conformal map if Y(x,y) is proportional to y1(w(x,y)) at every point (x,y) in D. Since the proportionality constant can vary from point to point, it is evident that they can differ globally by a positive function of (x,y). Here it is probably easiest to introduce one complex coordinate z = x + iy to replace the two real coordinates, as is usually done. Then one can summarize these statements by saying that if y and yx are metrics on a region D, then they are conformally equivalent if y = n2(z)yj for some positive function G2(z) on D. It is important to 150 realize that this is an equivalence statement for metrics on the Riemann surface, and although it is true that these equivalent metrics produce equivalent surfaces, it is not the only way to produce equivalent Riemann surfaces. Here the conformal equivalence arises from an identity map of D onto itself which only changes the metric by the conformal factor Q (z). Thus an equivalence relation has been defined,and these conformal equivalence classes of metrics are termed conformal structures. The discussion has been for a plane region; however, the global atlas on the Riemann surface can be used to extend these definitions to all the coordinate charts, and it is apparent that the coordinate maps preserve the conformal structure. The equivalence relation defined for metrics on Riemann surfaces leads to the the first generalization of the definition of a Riemann surface. That is, a Riemann surface is a connected two (real) dimensional surface on which a conformal structure has been defined. This is not really a restriction of the definition of the surface, it is just a classification, since conformally equivalent metrics produce equivalent surfaces. Introducing the notation [y] to represent an equivalence class of metrics, a Riemann surface can be denoted (M,[y]) where M represents the topological structure of the manifold. It was alluded in the previous paragraph that even a conformal structure is not restrictive enough to classify equivalent surfaces. This essentially means that two Riemann surfaces may be equivalent even if they have different conformal structures. Let (M,^]) and (M,[y2]) represent two Riemann surfaces with the same topological structure M (i.e. same genus,etc.). Let yx e [yj and y2 e [y2] be representative elements of their respective conformal structures. Then the Riemann surfaces (M,^]) and (M,[y2]) are conformally equivalent surfaces if there exists a sense preserving 1 5 1 diffeomorphism f:(M,[y1])^(M,[y2]) such that the pull-back metric is conformal, i.e. if f*y2 = ^(zfy. It is an obvious extension that this property holds for any representative elements of the respective equivalence classes. Thus when the phrase "conformally equivalent surfaces" is used, it is used in the sense above, where the important distinction lies in the word surfaces. This distinction between equivalent surfaces and equivalent metrics will be important in the work that follows. It is probably evident that in defining the above equivalence relations, the net effect for a topological surface M is to factor the space of Riemannian metrics, RiemMet(M), by the group of conformal transformations, ConfiM), and the group of (sense preserving) diffeomorphisms, Di£f1'(M). In fact, the factorization is by the semi-direct product of these groups, ConfiM) (S) Difff(M). The quotient space RiemMet(M)/j-£on^ ^ ^ DifPtM)] ca^eci the space of Moduli, or the Riemann space, and is shown in a proposition by Earle to be in canonical one-to-one correspondence with the set of equivalence classes of closed Riemann surfaces of genus p.[A-l, p.146] The genus simply designates the overall topological structure of the Riemann surface, and can be seen to be equal to the number of handles on a closed Riemann surface. For completeness it should be mentioned that if a Riemann surface has a non-empty boundary (i.e. an edge), then it can be naturally mapped to a closed Riemann surface (by the inclusion map) which will have the number of loops corresponding to the genus of the surface. The use of the group Difi4‘(M) causes some difficulties, so following the suggestion of Teichmuller, one can divide instead by the group of diffeomorphisms connected to the identity, DifP(M), which is a normal subgroup of DifPtM). Factoring RiemMet(M) by Conf(M) (s) DifP(M) produces the 152 Teichmuller space. The Teichmuller space is a finite dimensional space of (complex) dimension 0,l,3g-3 for surfaces of genus 0,1,and g>2 respectively. The parameters specifying these degrees of freedom are called the Teichmuller parameters. The discussion from this point will be restricted to defining closed Riemann surfaces and their covering surfaces. It is an important result in Riemann surface theory that there are only 3 distinct simply connected, constant curvature covering spaces for closed Riemann surfaces of any genus. Conversely, any Rieman surface can be produced by an appropriate factorization of its covering manifold by an discrete automorphism. The three constant curvature covering spaces are the complex number sphere, the complex plane, and the upper half of the complex plane or, equivalently, the Poincare disk. The complex number sphere is just the complex plane plus the point at infinity, C + °o. It has constant curvature, K = +1, and the metric is given by 2 4(dx2 + dy2) ds = ------where z = x + ly 2 2 J (1 + Izl) The complex plane has constant curvature, K = 0, and it has the usual flat metric ds2 = dx2 + dy2. The upper half plane has constant curvature, K = -1, with the metric given by dz/Im(z), where Im(z) is the imaginary part of z. Equivalently, the constant curvature K = -1 space can be represented by the Poincare disk, with the metric written 2 2dz ds = - By considering the automorphisms of the covering manifold, it will be shown how the compact Riemann surfaces are formed. The constant curvature covering manifold must be factored by a discrete (and hence discontinuous) automorphism. A discrete 1 5 3 automorphism is a map of the constant curvature covering manifold onto itself in such a way that it leaves no points fixed (fixed point free). If one considers a sphere, it is apparent that any non-trivial (oriented) map of Sn->Sn must leave at least one point fixed, so there are no fixed-point free automorphisms of the sphere, and the complex number sphere only covers the Riemann sphere.[A-2, p.2] Thus, this is the only constant curvature genus 0 Riemann surface. The complex plane is the covering surface for genus 1 Riemann surfaces. The fixed point free automorphisms of the complex plane are translations with one or two generators. If there is only one generator, z—> z + mcOj for integer m, and C0j a complex number. Factoring the complex plane by this equivalence relation has the effect of identifying the edges of strips of width I C0j I, perpendicular to cOj. This obviously rolls the plane up into a cylinder. For the translations with two generators, z-» z + mcOj + nco2 for m, n integers, and coj and co2 independent complex numbers, i.e. ImC^cOj) > 0. This equivalence relation produces a tesselation of the plane by parallelograms, where opposite edges are identified. Each parallelogram is called a period parallelogram, and C0j and co2 are the periods. This is depicted pictorially below, and the points equivalent to p are marked with an x. 154 Under the identification of pairs of opposite sides, the plane is rolled up into a torus. For higher genus surfaces, g > 2, the constant curvature covering space is either the upper half plane or the Poincare disk. The two volumes by Siegel are very good references for the material in this section.[A-3, esp. Vol.II p.89] The group of discrete automorphisms for the upper half plane is the group of Mobius transformations 3.Z + b t j z —»------r where a,b,c,d e JK an d ad - be = 1 cz + d The transformation group is the group SL(2,R)/Z2, where an element T g SL(2,R)/Z2 acts by the relation a b az + b then r : z —» if r = cz + d Equivalently, the Poincare disk can be used (there is a simple conformal transformation between the upper half plane and the Poincare disk), in which case the generators of the discrete automorphisms are given by az + b _ z - » -gz + for a,b complex and aa - bb = 1 1 5 5 The effect of factoring the covering space by elements of this group is to tesselate the upper half plane or Poincare disk by non-euclidean polygons. The polygon will have 4g sides corresponding to a surface of genus g; the sides are identified pairwise by the factorization of the covering space by the group, forming the genus g surface. This identification comes about because the group elements are non-euclidean motions of the upper half plane or Poincare disk which carry the associated sides into each other. The group of these motions produces a tesselation of the covering surface which maps down to the surface under the factorization by the group. Such tesselations are difficult to draw, but below is a diagram depicting the non-euclidean polygon corresponding to a genus 2 surface, where the complicated tesselations along the rim of the Poincare disk have been omitted. The identified sides are labelled by the same letter, and the inverse indicated by the -1 exponent indicates that the corresponding side is identified with the reverse orientation. Figure 5 - Genus 2 So, the factorization identifies a with a'1, b with b'1, and c with c'1, producing the desired genus two surface. It has now been shown how factoring the constant curvature covering space by a discrete automorphism produces a Riemann surface, i.e. by identifying corresponding sides in the tesselations. However, the 156 earlier discussion indicated that simply defining a surface does not guarantee that another surface is not equivalent to it. The analogous statement from this tesselation-based point of view is that choosing all possible tesselations does not correspond to choosing only inequivalent Riemann surfaces of a given genus, since some of these surfaces are equivalent. This can be illustrated most clearly for the case of genus one tori. Recall that for tori, the group of discrete automorphisms of the complex plane has two generators, so z —»z + mojj + nco2 for C0j and co2 linearly independent, which is equivalent to saying that the imaginary part of their ratio is non-zero. One can then impose the normalization condition that cot = (1,0), and then the period parallelogram is characterized by x = co^cOj with Im x > 0. This produces a tesselation of the complex plane as before. However, note that one can choose another basis for an equivalent tesselation by taking linear combinations of (Oj and co2.[see e.g. A-2, p.197 or A-3, Vol.II p.90] This gives two new (equivalent) generators co =aco + pco and co = yco +8co 1 1*2 2 ' 1 2 where a,p,y,S integers and aS - py = 1. Then imposing the normalization condition, one finds co yco +Sco o x = 2 * i 2 _ v+8x with aS - Py = 1 (A.2) coi acoi + pco2 a + px From (A.2) it is evident that any choices of x,x' which are related by the transformation above produce equivalent tesselations of the plane and, hence, equivalent tori. If a,p,y,S are taken to be elements of a matrix, then these matrices are elements of the elliptic uni-modular group SL(2,Z),and the action is defined for T e SL(2,Z) to be 157 a p r = then r : x ^ - ^ £ l .Y 5. y+5x These considerations imply that if one wants to generate all conformally inequivalent tori (in the sense of surface equivalence), one need only specify that the first generator be the unit vector on the real axis, and choose the second vector in a restricted portion of the upper half plane. The restriction will be defined so that no two vectors in the chosen region can be related by an elliptic uni-modular transformation, whereas for any fixed vector, x, in the chosen region, all x' equivalent to x can be reached by a uni-modular elliptic transformation. The chosen region will be called the fundamental region. Thus one must consider the action of the group SL(2,Z) on the upper half of the complex plane. The group can be generated by the transformations z -» z + 1 and z —> "Vz and can be seen to act discontinuously, dividing the upper half of the complex plane into fundamental regions. It should be noted that any fundamental region is sufficient for the definition of all inequivalent tori, but the one commonly chosen as the defining region is the region of the upper half plane above the unit circle, and between Re(z) = V2 and Re(z) = -V2. This region is cross-hatched in the diagram below, where several distinct fundamental regions are outlined. 158 Figure 6 - Fundamental Regions (Genus 1) Consequently, it is fairly easy to specify the period parallelograms for all conformally inequivalent tori. It should be noted that the metric on each period parallelogram is just the flat metric inherited from the complex plane. Notice that considering the original space of all Riemannian metrics defined on a topological genus one surface would have required working with the full infinite dimensional space, but by restricting to the space of conformally inequivalent tori the problem is reduced to a space of one complex dimension or, equivalently, 2 real dimensions. The determination of the fundamental region for higher genus surfaces, g > 2, is not as clearly defined as for the g = 1 case, although it is an area of current research by mathematicians. It is true, however, that given a particular higher genus Riemann surface, one can determine the fundamental polygon in the constant curvature covering space which projects down to form this surface. This process is achieved by making a canonical decomposition of the Riemann surface along the fundamental periods of the surface, i.e. for a surface of genus g, there are g handles, and on each handle there are 2 homotopically distinct closed loops - one around the inner radius, and one around the cross-section of the handle. These closed loops are the periods, so there are 2g periods for a genus g 1 5 9 surface, as depicted in the diagram below; Notice that if the surface is cut along these fundamental periods, the surface remains connected, and has 4g edges. Such a decomposition process is carefully reviewed in Siegel [see A-3, Vol.I p.49, p.106], and is termed a canonical dissection. When the canonically dissected surface is lifted to the constant curvature covering space, the simply connected dissected surface is mapped onto the fundamental polygon. The 4g edges of the canonically dissected surface map onto the 4g sides of the polygon in such a way that the edges that were on opposite sides of the cut in the dissection process map onto corresponding sides, e.g. a, a-1, of the fundamental polygon. In this way, the higher genus Riemann surfaces may be generated by choosing fundamental polygons in the constant curvature covering space (either the Poincare disk or the upper half plane), and these in turn may be generated by selecting the non-euclidean segments which form the sides of the polygon, much in the same way as this is done for the genus one tori. However, it is still an open question as to how one should select these segments so that only inequivalent surfaces are specified. In any case, it is known that the dimension of the space of inequivalent Riemann surfaces is finite of (complex) dimension 3g-3 for the higher genus surfaces. These are the Teichmuller parameters discussed earlier. 1 6 0 The discussion in this appendix can be used to express all of the conformally inequivalent genus one surfaces as surfaces defined over the unit square. This is achieved by isometrically mapping the defining period parallelograms to the unit square, and using the deformed metric on the unit square to define the inequivalent tori. The mapping will be the affine transformation taking the vector (u^v^ in the fundamental region which defines the parallelogram fij, to the vector (0,1) of the unit square fi0, while keeping the other side (1,0) fixed, since it is the same for both fij and fi0. Define the map F a y -^ The transformations are given by the equations below, where u,v are the coordinates on fii and x,y are the coordinates on fi0 u. i , F(u,v) = (u - -^-v , — v) = (x,y) F (x,y) = (x + u.y , v.y) = (u,v) (A.3) i i Once the form of the affine transformation is known, it is a simple matter to pull-back the flat metric on to fi0, so that the map is an isometric map of J3i onto fi0. The pull-back is found using the inverse function F'1 and the relation (F'1*8)ab(x,y) = d^F’W^F-^Ku.v) = Yab(x>y)- The components have a particularly simple form Y = 1 Y = u. Y = u2 + V2 (A.4) Thus for any period parallelogram, the metric is of the form (A.4) when pulled back onto the unit square fiQ. APPENDIX B - HARMONIC MAPS AND MINIMAL SURFACES In this appendix, the technical statements of the theorems summarized in chapter 3 are collected. In one of the first existence theorems, Eells and Sampson [B-l, p.125, see also p.158] prove Theorem: Let W have non-negative Ricci curvature, and let M have non-positive Riemannian curvature. Suppose M is compact or satisfies certain conditions at infinity. Then any continuous map X:W—»M is homotopic to a totally geodesic map. Furthermore, (1) If there is at least one point of W at which its Ricci curvature is positive, then every continuous map from W to M is null homotopic, and hence every harmonic map is constant. (2) If the Riemannian curvature of M is everywhere negative, then every continuous map from W to M is either null homotopic, or maps W onto a closed geodesic, and hence the same is true for any harmonic map. For example, this implies that if the Riemannian curvature of M is non-positive and if the scalar curvature of W is negative at some point, then there exists no harmonic immersion of W in M. Hartman went even further and discussed the question of homotopies of harmonic maps.[B-2, p.675] He proves Theorem: If X^q and X^ are homotopic harmonic maps, then there exists a C°° homotopy X^fz.u^W x [0,1] —> M from X^f-,0) = X^q to X^C-,1) Xooi with the properties: (i) for a fixed u, X J intu )iW —»M is a harmonic map and (ii) for fixed z, the arc X^feu) for 0 < u < 1 is a geodesic arc with length independent of z, and u proportional to the arc length. 1 6 2 Hartman then goes on to prove intermediate results about neighborhoods of points of homotopic harmonic maps lying on geodesics in M, but the final result is a good summary: Theorem: Let X^W —> M be a harmonic map and suppose that all the sectional curvatures of M are negative at every point of the image of W, X^CM). Then a necessary and sufficient condition that there exist a harmonic map * X^ but homotopic to X^ is that either (i) X^M) is a point or (ii) X^CM) is a closed geodesic y. In the latter case, all harmonic maps homotopic to X^ are obtained by a "rotation of X^.. (i.e. by moving each point X^Cz) a fixed oriented distance u along y) and, conversely, every "rotation of X^" is a harmonic map homotopic to X^. The importance of this work is that it essentially means that if M has negative sectional curvature, harmonic maps from W into M are essentially unique in their homotopy classes [B-3, p.646]. This is obvious since there is no significant difference between a solution rotated around the geodesic in case (ii) or a mapping into any particular point in case (i). When considering target spaces of non-negative curvature, however, different approaches must be employed. The early work by Eells and Sampson, and Hartman pointed out the importance of closed geodesics in the target space for the existence of harmonic maps, and this led naturally to the consideration of the fundamental groups as a means of generalizing the early results to extend to target spaces of non-negative curvature. The importance of homotopically non-trivial maps is evident, and Schoen and Yau [B-4, p.127] prove that a non-trivial extremal map always exists if the original map X (which will be deformed to an extremal map) is not homotopically degenerate (i.e. contracts to a point). Since the concept of homotopic degeneracy is inherently linked to the fundamental group of a manifold, 1 6 3 it is obvious why mathematicians would look for generalizations based on the fundamental group. Schoen and Yau prove [see B-4, p.135] Theorem: Suppose M is a compact Riemannian manifold, WQ is a compact surface of genus g > 1, and This result is later extended to include the g = 1 case.[see B-4, p.138] Following along these lines, Sacks and Uhlenbeck [B-5, pp.18-19] prove Theorem: If M is compact and tc2(M) = 0, then there exists a minimizing harmonic map in every homotopy class of maps in C°(W,M). Theorem: Every conjugacy class of homomorphisms from TCjCW) into is induced by a minimizing harmonic map from W into M. In a later paper, Sacks and Uhlenbeck extend their previous results to prove [see B-3, p.646] Theorem: Let W be a closed topological surface of genus larger than one and let Y:W—>M be a continuous map such that Y#:711(W)-^tc1(M) contains no non-trivial element represented by a simple closed curve in its kernel. Then there exists a conformal branched minimal immersion X:W—>M with the same action on TCjfW) as Y such that A(X) < A(X'), for all branched iimmersions X':W—»M with the same action on ^(W) as Y. If 7T2(M) = 0, then X can be chosen homotopic to Y. It is interesting to note that one important class of manifolds which satisfy rc2(M) = 0 are the finite dimensional Lie groups.[B-6] The above results show that whenever there is an injection (one-to-one map) 164 between the fundamental groups of the Riemann surface and the target space, the minimal immersions exist. 165 APPENDIX C ■ THEORY OF ISOMETRIC IMBEDDINGS AND IMMERSIONS In this section certain pertinent results on the theory of isometric imbeddings and immersions of Riemannian manifolds into higher dimensional Riemannian manifolds will be cited, and since the work in section 5 of chapter 4 will require manipulations based on these results, it will be necessary to outline some of the methods of proof. Since many of the proofs are very elaborate, whenever possible the summarizing statements of Gromov and Rokhlin[C-l] will be used. The work will be discussed for the case of isometrically imbedding and immersing into a higher dimensional euclidean space, since the analysis is easier to formulate in that framework. To state the problem clearly, the isometric immersion of an n-dimensional Riemannian manifold (W,g) into Rq will be considered, where g is the metric on W, and since the desired immersion is to be isometric, the pull-back metric induced by the immersion must be equal to g. This is equivalent to finding q functions <{>,,...A ,6 4 mapping V—»R such that g = (dc^)2 + (d 3({). where d(j). = 2 a—~ dxp (C.l*) 1 p=i dxp Notice that this is just the standard form of a pull-back metric, but this notation will prove to be useful. If such functions can be found, then the desired isometric immersion is given by X:(W,g)->((j)1,..., <{)q). The functions Theorem: A compact n-manifold with a Cr positive metric has a Cr isometric imbedding in any small volume of Rq with q = 3sn + 4n, where sn = l/^nfn + 1), provided that 3 < r < ©o. This value for q is not the lowest value of q for this range of r. However, Gromov and Rokhlin show that the lowest value of q (at least for the case r = °°) does not exceed sn + n.[see C-l, p.7] The proof of Nash's theorem is very difficult, but it hinges on two related propositions. The first is commonly called "Nash's implicit function theorem", and in the language of Gromov, the second is called "the approximate imbedding theorem". A good summary of the method of proof is given by Gromov [C-l, p.11-12], restricted to the case of r = for simplicity, and the discussion will follow this closely. Consider an n-dimensional C°° smooth manifold, W. Let G be the space of all C°°-Riemannian metrics on W (with the C3 topology). Nash's implicit function theorem states that for every metric induced by a free C°° imbedding W—>Rq, there is a neighborhood in G such that every metric in that neighborhood is induced by a C°° imbedding W—>Rq. Note that this is only a local statement. The approximate imbedding theorem states that the set of metrics induced by C°° imbeddings W-*Rq with q = 2sn + 2n is everywhere dense in G. The proof of the theorem relies critically on these 167 two properties. Several definitions will be needed in order for the outlines of the proofs to be clear, so they will be given here for easy reference: short (strictly short) - A differentiable immersion f:W,g—»M,G is said to be short (strictly short) if the induced Riemannian metric f*G on W is such that the form g - f*G is non-negative (positive). free map - If W is an n-dimensional manifold and f:W-»Rq (where f, W are at least C2), then f is said to be free if the vectors formed from the first and second derivatives of f are linearly independent at every point of W or, equivalently, if u^ are local coordinates around a point v e W, then the span of the vectors df(v) (i = 1,2,...,n) d2f(v) (i,j = l,2,...,n) 3u. 3u. 3u. 1 i j is equal to n + sn at all points v in W. The first definition will play a major role in many of the proofs, whereas the second mainly serves as a technical device. The proof makes use of the fact that any n-dimensional C°°-smooth manifold can be freely C°° (differentiably) imbedded in Rq with q = sn + 2n (this holds, in fact, for all Cr functions with r >2). Now let W be the manifold and endow it with a metric g. By virtue of the statement above, W can be freely imbedded into the sn + 2n dimensional euclidean space (not necessarily isometrically). Let f be the imbedding function and let 8 be the flat metric in Rq. The imbedding can be made strictly short by multiplying the coordinate functions by a suitable number, so assuming this has been done, one can specify that g - f^S = g2 is a Riemannian metric. Let f2 be any imbedding W—>Rp with p = 2sn + 2n, and let gx' = g - f2*8. If the difference g { - P5 = g2 - f2*5 is sufficiently C° small, then gx' is 168 a Riemannian metric, and if the difference is sufficiently C3 small, then Nash's implicit function theorem applied to f*5 implies that gj is induced by some C°°-imbedding f^W—»Rqwith q = sn + 2n. But the approximate imbedding theorem says that the difference g2 - f2*5 can be made as C3 small as is required by suitably choosing f2. So the desired C°° imbedding which is isometric is given by ^©f^V-rft s + 2n ©R 2s + 2n =R 3s + 4n (4.19) In this way any compact Riemannian manifold may be isometrically imbedded into an euclidean space of dimension 3sn + 4n. Nash also extends his work to the case of non-compact manifolds in a fairly crude way which allowed the required dimension to increase greatly. Even so, he proved that any Cr (3 < r < °°) non-compact Riemannian manifold can be isometrically imbedded into Rq with q = (3sn + 4n)(n + 1). These early results have been improved by Gromov and Rokhlin. In [C-l] as an exercise, they show that any compact n-dimensional C°° Riemannian manifold can be isometrically C°° imbedded in Rq with q = sn + 4n + 5 = ^ (n 2 + 9n + 10), and the imbedding is free. The dimension is reduced still further in [C-3] and extended to include non-compact manifolds in an equivalent way with the result that the dimension can be reduced to sn + 3n + 5, including the non-compact case. Since the construction in [C-l, p.12] is somewhat representative of the methods that follow, it will be described here. Let W be a compact C°° smooth manifold and (J) a C°° function on W. It can be shown that if a C°° imbedding W-»Rq (for q > sn + 4n +5) induces a Riemannian metric g0 on W, then the metric g0 + (d(}))2 is induced by a free C°° imbedding W—»Rq with the same q. From this it follows that any metric of the form g0 + (d^)2 + • • • + 169 (d Riemannian manifolds for which all the sectional curvatures at all points are not positive, stating that such manifolds cannot be isometrically C4 immersed in R2n_1. Thus for n = 2, putting together the statements in the last two paragraphs, the minimum dimension in a flat target space to guarantee the existence of isometric imbeddings of Riemann surfaces with an arbitrary Riemannian metric must be greater than or equal to 4, and less than 10. Having thus gained some perspective on the restrictions which apply to isometric embeddings of Riemann surfaces, it will be necessary to consider several technical theorems in Gromov and Rokhlin [C-l,ch.2], which are applied to the determination of the fiber volumes in the quasi-fibration of the space of embeddings and the form of the Jacobian factor in chapter 4, section 5. A new notation will be introduced to represent the pull back metric induced by a map f, i.e. the previous notation for the pull-back f*5 will now be written g(f), literally the induced metric as a function of the map f. For simplicity, the theorems will be listed according to the theorem numbers in [C-l],with the page numbers cited: Theorem 2.6.2 [p.28] - For every free C°° imbedding (free C°° immersion) f of a compact C°° manifold W in there exists a C°° neighborhood of the metric g(f) such that any metric in this neighborhood is induced by a free C°° imbedding (free C°° immersion) W—>Rq. Theorem 2.6.3 [p.29] - For every free Ca imbedding (free Ca immersion) f of a compact Ca manifold W in R^ and any quadratic differential Ca form h:T(W)—>R there exists an e > 0 such that for any real number c with Id< 8, the form g(f) + ch is induced by a free Ca imbedding (free Ca immersion) W—»R9. 172 Theorem 2.7.10 [p.33] - If a compact (n-1 )-dimensional C°° (Ca) Riemannian manifold WQ has a free isometric C°° (Ca) imbedding in R^ with q > sn + 2n -1, then for sufficiently small a the product W0 x (-a,a) also has a free isometric C°° (Ca) imbedding in R9. This is also true for q > sn + 2n - 2 if W0 has no closed components. Theorem 2.7.11 [p.34] - If a compact n-dimensional C°° (Ca) Riemannian manifold W has a free isometric C°° (Ca) imbedding into Rq for q > sn + 4n + 5, then for sufficiently small a the product W x Int D2(a) also has a free isometric C°° (Ca) imbedding in R^. [Note: Int D2(a) means the interior of the unit disk of radius a] Theorem 2.9.2 [p.36] - If W is a compact C°° (Ca) Riemannian manifold with metric g, then there exist real-valued C°° (Ca) functions (jjj,* • •, (j)m and positive C°° (Ca) functions a1}* • *, 0^ on W such that g = a/dcjij)2 + • • • + am(d define a C°° (Ca) map f^:W—>R2m with g(f^) = g + X 2h, where h = (do^)2 + • • • + (dam)2. Theorem 2.9.4 [p.36] - A compact C°° (Ca) Reimannian manifold can be isometrically C°° (Ca) imbedded in an euclidean space of sufficiently high dimension. Theorem 2.9.5 [p.37] - For every free C°° (Ca) imbedding fQ of an n-dimensional compact C°° (Ca) manifold W in Rq with 173 q > sn + 4n + 5 and any C°° (Ca) function <{):W—»R there exists a free C°° (Ca) imbedding f:W-»Rq such that g(f) = g(f0) + (d({))2. Theorem 2.9.6 [p.37] - A compact n-dimensional C°° (Ca) Riemannian manifold has a free isometric C°° (Ca) imbedding in Rq with q > sn + 4n + 5. Since an understanding of these theorems is crucial to what follows, an outline of the proofs to theorems 2.9.4, 2.9.5, and 2.9.6 will be given: Proof of Th.2.9.4 - Let W be a compact C°° (Ca) Riemannian manifold with metric g. For sufficiently large k it is known that there exists a free C°° (Ca) differentiable imbedding fQ:W->Rk, and fQ may be taken to be strictly short (see comment in proof of Nash theorems). Then g - g(f0) is a Riemannian metric, and by Theorems 2.9.2 and 2.9.3 above, for some m there exists a family of C°° (Ca) maps f^:W—>R2m, b*0 with g(P) = g - g(f0) + Ar2h where h is some quadratic differential form on W. If X is sufficiently large, then by Ths. 2.6.2 and 2.6.3 there exists a C°° (Ca) imbedding f^W —»Rk with g(f^) = g(f0) - V"2h and then the C°° (Ca) imbedding f* © f X:W-> Rk © R2n> = Rk+2m induces the metric [g(f0) - Ar2h] + [g - g(f0) + Ar2h] = g. Proof of Th. 2.9.5 - Let W be a Riemannian manifold with metric g = g(f0). By Th. 2.7.11 one can construct a free isometric C°° (Ca) imbedding fj:W x Int D2(a) —> Rq for some a > 0. [Note: in the proof of Th.2.7.11 Gromov and Rokhlin regard WxO = W and hence fj Wx0= f0]. Define a map 0:V-»Int D2(a) by 0:W -> ( 4 cos^<|>(v), \ sin ^ Proof of Th. 2.9.6 - Let W be the manifold given in the theorem, with metric g. It is known that there exists a free C°° (Ca) differential imbedding f0:W->Rq for q = sn + 4n +5 which can be taken to be strictly short. Then g - g(fQ) is a C°° (Ca) Riemannian metric, and by Th. 2.9.4 there exist C°° (Ca) functions (j^,* • *,(J)m:W—>R such that (d^)2 + • • • + (d<|>m)2 = g - g(f0). Apply Th. 2.9.5 successively m times to get g(fQ) + (d^)2, • • • , g(f0) + (dt^)2 + • • • + (d<}>m)2 which are induced by free C°° (Ca) imbeddings W -»R^ with q = sn + 4n + 5, and obviously the last metric is the same as g. These theorems will allow for the determination of many properties of the fibers in the fibration of the space of Embeddings. 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