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(2+1)-dimensional gravity over a two-holed torus, T²#T² Newbury, Peter R. 1993

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(2+1)-DIMENSIONAL GRAVITY OVER A TWO-HOLED TORUS, T²#T² By Peter R. Newbury B.Sc.Hons., University of Manitoba, 1990 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MATHEMATICS INSTITUTE OF APPLIED MATHEMATICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March, 1993 © Peter R. Newbury, 1993 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature)  Department of  Mathematics The University of British Columbia Vancouver, Canada - Date ^12.^OVJA. 1993 DE-6 (2/88) Abstract Research into the relationships between General Relativity, topology, and gauge theory has, for the most part, produced abstract mathematical results. This thesis is an attempt to bring these powerful theories down to the level of explicit geometric examples. Much progress has recently been made in relating Chern-Simons gauge field theory to (2+1)- dimensional gravity over topologically non-trivial surfaces. Starting from the dreibein formalism, we reduce the Einstein action, a functional of geometric quantities, down to a functional only of the holonomies over flat compact surfaces, subject to topological constraints. We consider the specific examples of a torus T 2 , and then the two-holed torus, T2#T 2 . Previous studies of the torus are based on the fact that the torus, and only the torus, can support a continuous, non-vanishing tangent vector field. The results we produce here, however, are applicable to all higher genus surfaces. We produce geometric models for both test surfaces and explicitly write down the holonomies, transformations in the Poincare group, ISO(2,1). The action over each surface is very nearly canonical, and we speculate on the phase space of dynamical variables. The classical result suggests the quantum mechanical version of the theory exists on curved spacetime. ii Table of Contents Abstract^ ii List of Figures^ v Acknowledgements^ vii Introduction^ 1 1 The Dreibein Formalism^ 5 1.1 The Dreibein Formalism of General Relativity  ^5 1.2 Chern-Simons Gauge Field Theory over the Poincare Group  ^9 2 The Einstein Action^ 13 2.1 Reduction of the Einstein Action  ^16 3 The Simple Torus^ 22 3.1 The Action over a Torus ^  22 3.2 Consistency of the Constraints  28 3.3 A Model of T 2 ^  31 3.4 An Alternative Approach to the Torus ^  35 4 A Two-Holed Torus^ 39 4.1 The Action over a Two-Holed Torus ^  39 4.2 Consistency of the Constraints  47 4.3 A Model of T 2#T2 ^  51 iii 4.3.1 Tilings  ^52 4.3.2 Construction of an Octagonal Tile ^  53 4.3.3 Holonomies ^  58 4.3.4 The Fields Uab and pa  ^61 5 Canonically Conjugate Variables and Quantization^ 64 6 Conclusions^ 72 6.1 Generalization of the Result  ^72 6.2 Future Research ^  73 Bibliography^ 77 Appendices 78 A Properties of Lorentz Transformations Uab^ 78 B Setting C'Ji ab 0 and e z a = 0 ^ 80 C Properties of {W} ^ 86 iv List of Figures 0.1 A simple torus, a 2-holed torus, and a g-holed torus, as they appear em- bedded in R3 . Also shown are the non-trivial loops, 2 around each 'hole', that cannot be continuously contracted to point.   2 2.1 The difference -y i — 72 between two homotopic curves -y i and 72 is con- tractible to a point. The holonomy of this trivial loop is the identity, so -yi and -y2 have the same holonomy.   15 3.1 (i) Labels for the sides and corners of the tile, and identifications of sides A&C and B&D. (ii) The neighborhood of a vertex where four tiles meet. 23 3.2 The transformations W across the identified edges (i)C&A, (ii)D&B. . . 24 3.3 Values for the 'phase' U at the corners of the tile.   34 3.4 Tiles of the tangent space over the torus are parallelograms in boost-trans- lation space^  36 4.1 (i) Label for the sides and corners of the tile, and identifications of sides. (ii) The neighborhood of a vertex where eight tiles meet. ^ 40 4.2 The transformations W across the identified edges (i)C&A, (ii)D&B, (iii)G&E, and (iv)H&F^  42 4.3 Circles of radius i- intersect the unit disk orthogonally. The arcs within the unit disk are parametrized in 0^  55 4.4 For various values of F, arcs (i) do not intersect, (ii) are tangent to one another, (iii) form an almost regular octagon about the origin, and (iv) intersect with angle 7/4^  57 v ^4.5 The octagonal tile lying on a hyperbolic hypersurface of constant time T ^ In Minkowski space, every point on the tile is the same distance T from the origin ^59 4.6 Values of the phase U on the corners of the r#T 2 's octagonal tile. The constraints guarantee U(1) is well defined.  ^62 5.1 (i) A principle ISO(2,1)-bundle over the base space r#T 2 . (ii) The 'phase space' is like an IS0(2,1)-bundle over the space of configurations of the surface.   67 5.2 J2 and J3 are generators of Lorentz boosts while Ji generates rotations Because the same eigenvalue 0 results from rotations by 0 + 2irn for all integers n, the coordinate 7/ 1  0 is periodic and the Hilbert space of wave functions has a "cylindrical" geometry.   70 BA An infinitesimal loop of generated by vectors T4 , Su with area ATAS. . . 82 B.2 Two infinitesimally different paths to xri = y. . ^  83 vi Acknowledgements I would like to thank my supervisor, Bill Unruh, for guiding me through this project. His top-down insight met with my ground-up exploration to produce the results. I am also grateful to David Austin for reading this thesis from the point of view of a mathematician sympathetic with the pedagogical style of physics, and to Nathan Weiss for his critical reading of this thesis. Thanks to the gang in the Audx for putting up with all my questions. Finally, I want to thank Margaret for her unending support and confidence. Finding her was the most unexpected, but easily the greatest result of this whole endeavor. vii Introduction Using General Relativity (GR) to study of the dynamics of a spacetime generally follows a simple algorithm: Write down the action; break up the spacetime under a splitting — usually (3+1)-dimensional; re-write the action in terms of the dynamical variables on the spatial slices and constraints that govern the splitting; vary this action with respect to the dynamical variables; solve the resulting PDE's subject to the constraints for the compo- nents of the metric. In practice, even the first step of choosing an appropriate action can be daunting, let alone solving an often highly non-linear system of constrained PDE's. One approach to removing these problems is to study a different system! Rather than tackle the full 4-dimensional theory, which may describe physical space in some cases, consider the simpler case of merely 3-dimensional spacetime. This space is easily split into (2+1)-dimensions — 2 spatial and 1 temporal. While the results of these studies are clearly not physical, the techniques and problems that arise may shed light on the orig- inal (3+1)-dimensional case. fortunately, the mathematics of GR on (2+1)-dimensions is much simpler and the physics of the space is not (so) lost in mathematically difficult equations that much be examined. With 3-dimensional spacetime sliced into (2+1)-dimensions, a system is defined by the choice of the 2 spatial dimensions. The simplest cases are those where the space is a plane or some similar infinite region. More interesting problems arise when the 2 dimensions are wrapped up into compact surfaces, especially when the surfaces are not simply (topological) spheres. In this thesis we will consider first the torus T 2 and then a genus 2 surface, the 2-holed torus T*T 2 (See Figure 0.1), and the interplay between the differential geometry of GR and the non-trivial topology of this surface. 1 Figure 0.1: A simple torus, a 2-holed torus, and a g-holed torus, as they appear embed- ded in R3. Also shown are the non-trivial loops, 2 around each 'hole', that cannot be continuously contracted to point. In choosing to study the simpler (2+1)-dimensional case, we have given up the possi- bility of producing a theory which directly describes the dynamics of the space about us. The goal, instead, is to find a mathematical result that reveals some of the subtleties of GR on non-trivial spacetimes. Nothing helps to answer questions more than a working model — where would mechanics be without the simple harmonic oscillator, or quantum mechanics with the Stern-Gerlach experiment? These idealized models answer questions about physics without the clutter of experimental error or unsolvable mathematics. Real physical systems can then be studied as perturbations of the ideal model, and generaliza- tions to more complex models can be made from the simple ones. This is what we hope to achieve here. By constructing a simple, though unphysical, model on which we can see the roles of GR, topology, and ultimately quantum mechanics, more complex models may be built. Hopefully this will contribute to one day producing a theory of quantum gravity, one of the last stumbling blocks towards a Grand Unified Theory. This thesis is developed as follows. We recall first, in Chapter 1, the dreibein formalism for GR and see how this approach, together with gauge field theory, has recently renewed 2 interest in using (2+1)-dimensions to attempt to model quantum gravity. In Chapter 2, we reduce the Einstein action down to an (almost) canonical form. In this procedure, we see how the differential geometry drops out leaving only topological quantities: the holonomies associated with each non-trivial loop on the 2-dimensional surface. The result comes from carefully keeping the boundary terms in the action, rather than dismissing them as irrelevant, as far as the action is concerned. In Chapter 3, we construct a model for T 2 . Taking this model through the prescription of Chapter 2 endows it with a frame field and connection. The resulting action depends only on the holonomies of T 2 . An explicit, geometric model of the torus also shows that there are enough degrees of freedom that many different collections of holonomies are possible, some possibly admitting closed time-like curves. As a more complicated trial, Chapter 4 repeats the process over a genus 2 two-holed torus. The reduced action again depends only on the holonomies about non- trivial loops on this surface. We construct a model for r#T 2 by folding up an octagonal and calculate the holonomies of this particular construction. In Chapter 5, we speculate on the classical conjugate variables. The action we produce is not canonical and we make a qualitative interpretation of the phase space. While the results of Chapter 2 are classical, the form of the "conjugate variables" strongly suggests the quantum mechanical version of the model lives on curved spacetime. Finally, in Chapter 6, we see how the results may be generalized to higher genus surfaces, and where further research can be done on the link between gauge field theory and General Relativity. This is a subject rich in both Physics, through GR and the equations of motion, and Mathematics, through topology and gauge theory. For the purpose of this thesis, we are considering (2+1)-dimensional gravity over topologically non-trivial surfaces as a Physics problem. It can equally be approached as an example of differential geometry over surfaces with non-trivial fundamental groups. The relations we encounter and choose to interpret as topological constraints imposed on the space are none other than the 3 representation of the fundamental group of the surface in the gauge group. The particular collection of holonomies of the surface, a subset of the gauge group, obey relations which demonstrate the structure of the fundamental group. So while this thesis emphasizes the Physics interpretation of the results, the Mathematically-minded reader can translate these same results into Theorems, Proofs, and Corollaries. Finally, a word on whom to attribute the results of this thesis. The ideas of Chapter 2 are developed as a special case of the theory of GR, and not as an interesting observation of the manipulation of the dreibein and connection. The difference between these two approaches is that the former requires an understanding of and experience in the me- chanics of GR, gauge theory, and even index manipulation. For these reasons, Chapter 2 is a reproduction of work done by my supervisor W.G. Unruh, and to him the results should be attributed. The calculations in the rest of the thesis, the "easy part", were carried out by both of us independently, in the sense that we ended up with two stacks of paper, although I was prodded in the right direction at several stages. Therefore, this thesis should be viewed as a report of Unruh's exploration of (2+1)-dimensional gravity in the dreibein formalism, annotated and "demystified" to allow graduate students like myself to understand and appreciate the results. 4 Chapter 1 The Dreibein Formalism The basis for studying spacetime dynamics in classical GR is the Einstein action. 1 This action is a complicated functional of the metric components g . An alternative approach is to use the tetrad, or vierbein, formalism. In this chapter, we see how in only (2+1)- dimensions, the triad, or dreibein, variables greatly simplify the Einstein action. We also review recent work that recasts the dreibein approach to GR in a Chern-Simons gauge field theory. 1.1 The Dreibein Formalism of General Relativity The most common approach for finding the metric on a spacetime is the Lagrangian formulation with the Einstein action I = where g is the metric determinant and R, the Ricci scalar, is the twice contracted Reiman- nian curvature tensor. Various constants that often appear before this action, for example [1] or ifir [2], can be ignored when using variational principles. Equations of motion are found by variations of the action with respect to the metric components, g These equations of motion are generally highly non-linear PDE's, owing in part to the square root VIM. The advantage of the dreibein formalism is that is supplies a kind of square root for gµ„ by using the components of the frame field as variables. The dreibein is 'This action is variously referred to as the Einstein [4], Hilbert [3], and Einstein-Hilbert [1]. 5 Chapter 1. The Dreibein Formalism^ 6 the collection of three vectors^where indices ,a, v,^= 0,1, 2 are the tangent-space components. We will use i, j, = 1, 2 to indicate the spatial components. The indices a, b, .= 1, 2, 3, sometimes known as the Lorentz indices, merely label the vectors in the frame. The metric on the spacetime is defined by e gab, = epa eva hence giving a kind of 'square root' to gov . The vectors must remain orthogonal in the Lorentz space: a pcbe e = ab where qa b =diagf— + +} is the usual Minkowski metric in (2+1)-dimensions. All raising and lowering of a, b indices is done with qab. A new connection co: b is required to account for curvature in this mixed space. Covariant differentiation is defined by Dtheva = a^„ a^, a b ^ ip,v'A^"1 b'y (1.2) where ['A are the usual Christoffel symbols. We demand that lab be constant in it'sAv Minkowski space, which forces Witab to be antisymmetric: 0 = Doab ab = +WA a cil cb + Wit b cri ac = ab^ba The spin connection co: =- 2615ibc Wizbc is sometimes used in place of the connection, al- though throughout this thesis we will remain with co: b • We can now re-write the Lagrangian in terms of the variables e and co. The Ricci scaler R is found by contracting the "internal" curvature tensor R it, ab with the dreibein: R^R^= aellb R /Iva!) (1.1) 2 Here and in all that follows, we adopt the Einstein summation convention of summing over repeated indices, both latin and greek. Chapter 1. The Dreibein Formalism^ 7 While Roz, aP is the curvature of spacetime, R iwab is the curvature that relates the space- time (av) to the internal space (ab). The curvature is exchanged between the two spaces via the dreibein: = eacpbRtwab. A flat internal space, R iLva b = 0 implies a flat spacetime, Ritv P = 0. In this mixed spacetime-internal space, the Lagrangian becomes Og iR = ( Epvp a^c, \( ^df C abC)k e deaf  -"Aa ) • By the associativity of addition, we can re-arrange the summed Lorentz indices: = f twP Eadf epaevb epc ObeucRAu df E""Eadf e tLa j pA d;RA, df by the orthogonality (1.1), leaving simply = cpvp cab, e a R be .A The curvature tensor can be calculated by looking at the failure of covariant deriva- tives to commute. Given a vector field Va, the curvature tensor is defined by [Do , DI,JVa — Rk,,,bVb With the definition of covariant derivative (1.2), it is simple to calculate Alp ab^„ ab^, a , db -"pv^'11 ,v^"Iv d'p • Finally, with volume form fluiPd3x =^A dxv A dxP, we arrive at the expression a( I = fabcep kWv bc  ,p b dWp dc )ale A dxv A dxP (1.3) The great advantage to this formulation of the problem is that we will nowhere encounter below a raised tangent-space index ,a. This allows us to work in the more general case where gi`u may not exist; that is, where singularities are allowed. Furthermore, on so- lutions to the equations of motion, the variation of the action with respect to the e: Chapter 1. The Dreibein Formalism^ 8 variables must vanish. From (1.3) we clearly see that these solutions are those connec- tions for which dw w A w, the Ricci curvature, vanishes. Now in three dimensions, the full Reimann tensor can be written in terms of the Ricci tensor and the Ricci scalar. Hence a vanishing Ricci tensor, which immediately produces a vanishing Ricci scalar, in this particular case also implies the Riemann tensor vanishes: the solution spacetimes are truly FLAT. This is one of the main reasons why (2+1)-dimensions are studied — the geometry does not contribute to the physics of the spacetime, thereby leaving topological considerations more apparent. In analogy with the ADM formulation [7] of (3+1)-dimensions, the action (1.3) is re-written in terms of the canonical variables and their conjugate momenta. Because of the antisymmetry of the volume form, neither e 0a nor c;)0 ab appears in the action, where ( ) indicates differentiation with respect the time coordinate of the spacetime. Without conjugate momenta, these variables are constant in time. In the language of variational calculus, e oa and c00% are Lagrange multipliers of constraints which govern the way the 2-dimensional slices of 'space' evolve in 'time'. Explicitly, we find I = —2 fdt P2 x Eije j aci7j, •eoa cij(wi bc j^w j b dwi dc wi bdw j dc)idt fd2 X eabcf ^wo bc fij ( ci a^j^e^eidw j ad^eidwi ad il The constraints, proportional to e oa and wo bc , are recognized as Rij bc^0 (1.4) = 0 ,^ (1.5) respectively. These two constraints, which tell us the spatial slices are flat and torsion free, greatly simplify the system, as we will see in chapter 2. Chapter 1. The Dreibein Formalism^ 9 We see that of all the ways to attempt to describe the geometry of 3-dimensional spacetime, the result produced by the theory of GR is fairly simple. In fact, most of the gymnastics of differential geometry have disappeared! It is no wonder, then, that this "simple" result can be reproduced from a quite different abstract approach, Chern-Simons gauge field theory. 1.2 Chern-Simons Gauge Field Theory over the Poincare Group In recent work that rekindled interest in (2+1)-dimensional GR, Witten [1] recognized the Einstein action written as (1.3) as a Chern-Simons action of the the gauge field theory for the Poincare group. This gauge group is the collection of all Poincare transformations, consisting of a Lorentz transformation followed by a spacetime translation: Va tlab V b Ta . (1.6) The Lorentz subgroup of the Poincare group, more often associated with Special Rela- tivity, has disconnected components corresponding to proper, orthochronous transforma- tions of the connected component of the identity, and components connected to parity-, time-, and total-inversion. The Lorentz transformations we will deal with here are re- stricted to the first of these components, the only component which represents physical transformations. An ISO(2,1) representation (Inhomogeneous Special Orthogonal) of the Poincare group is the collection of 4x4 matrices \^0 [PIT] = Tatiab where tiab is an SO(2,1) Lorentz transformation. Properties of these matrices liab that will be used below are found in the Appendix A. Transformations of 3-dimensional vectors Chapter 1. The Dreibein Formalism^ 10 Va are carried out by appending a fourth component, of fixed length 1, to the 3-vectors forming VA = (Va, 1): /^\ /^\^/^\ tiabV b + T a=Uab Ta V' Ik 10 The 3-dimensional component of these resulting 4-vectors reproduces the 3-dimensional Poincare transformation on Va. The Poincare group has generators / 0 0 0 1 /^0 0 —1 /0 1 0 0 0 —1 0 0 0 0 0 1 0 0 0 J1 = J2 = J3 = 0 1 0 —1 0 0 0 0 0 0 0/ 0 0 / \ 0 0/ Under exponentiation, these generate, in Minkowski space where xa = {t, x, y}, rotations in the xy-plane, boosts in the (—y)-direction, and boosts in the x-direction, respectively. While these 4 x 4 matrices generate the Lorentz subgroup of ISO(2,1), the upper left 3 x 3 submatrices by themselves generate the pure Lorentz group SO(2,1). The generators of translation in the t-, x-, and y-directions are, respectively, / 11 / 0 " / 0 1 0 0 0 1 0 0 P1 = P2 = P3 = 0 0 1 \ 0 0 / \ 0 0 / 0 0 / The generators of this group obey the Lie algebra [fa , 'Id .= Cabcf c Chapter 1. The Dreibein Formalism^ 11 Val Pb] = EabcP c [Pal Pb] = 0 In [1], Witten observes that in taking the 1-form A = Apdx 1` with components A, = coaP, c.o4aJa as a gauge field, the Chern-Simons action /cs = 2 f Tr(A A dA IA A A A A) M exactly coincides with the Einstein action (1.3). We see that ISO(2,1) is the gauge group by varying the gauge field A. The infinitesimal transformation of A, generated by a parameter u = pa Pa + TaJa with infinitesimal pa, Ta is defined as 8A„ = u,„ [A,, 'a] . This variation is the covariant derivative of the field with respect to the connection A,. Varying the field in this way produces (SA„ = 8e: Pa + ( where Se a =^_ fabcembrc — cabc„4 bpc ,^a —Ta — abc bTp,^c • Witten notes that by setting pa = Vaq, for an infinitesimal spacetime vector -PL, the difference between these transformations of E lla  and co t:, and those generated by infinites- imal Lorentz transformations along with infinitesimal diffeomorphisms (translations!) in the —V 4L direction, is simply a Lorentz transformation. Since Lorentz transformations Chapter 1. The Dreibein Formalism^ 12 are part of the gauge group, any gauge invariant quantity, like the action, will be unaf- fected by this difference. That is, the Chern-Simons action in the gauge field AN, over the Poincare group is the same as the Einstein action of GR. Gravity can be re-expressed as a gauge field theory, which greatly increases the chances of finding a quantum mechanical version of the theory. Though this thesis is based on classical GR, and not quantum or gauge field theory, we will see the importance of the Poincare group in the results that follow. Chapter 2 The Einstein Action In this chapter we derive the main result of this thesis: the Einstein action over a compact surface can be written explicitly in terms of the holonomies on the surface. Before we produce this result, let us briefly review holonomies and boundary terms. Intuitively, a holonomy is the failure of a vector to return to its original orientation after being parallel transported around a closed path, or loop. We will define holonomy as the transformation which carries the initial vector onto the final. While in general a holonomy is an automorphism of the tensor fields over a manifold, the holonomies encountered here are simple transformations on the tangent space. Curvature, also a measure of the changes in parallel transported vectors, is closely related to the holonomy of trivial, or contractible, loops. When a surface is flat, the holonomy of all trivial loops is simply the identity — no change occurs in a vector when it is parallel transported about a flat surface. When curvature is present, the holonomy depends both on the base point of the loops and the shape of the loops itself: parallel transport around "long and twisting" loops will alter a vector more than parallel transport around "small and simple" loops. As we saw in Chapter 1, the surfaces we deal with here are flat, and the presence of interesting holonomies seems unlikely. However, the surfaces we deal with are also topologically non-trivial, and are covered in non-contractible loops, those paths which get "caught" on one or more holes formed by the surface (See Figure 0.1). The holon- omy over these non-trivial loops may not be the identity, even though the surface is 13 Chapter 2. The Einstein Action^ 14 everywhere flat. The topological degrees of freedom can generate non-trivial holonomies. Assuming hereon that "surface" means a flat compact surface with interesting topology, we recall that all homotopic curves have the same holonomy. Two curves which can be smoothly deformed into each other differ by a contractible loop (See Figure 2.1). This path does not contribute to the holonomy as the surface is flat, so the holonomy of the two homotopic paths is the same. Therefore all holonomies on a surface will be known once those about a few representative non-trivial loops are known. Just as the genus 1 torus has 2 distinct classes of incontractible loops, a genus g surface, like a g-holed torus, has 2g generating loops. By finding, or specifying, the holonomies about each of these 2g loops, the geometry of the surface is completely determined, at least up to global gauge transformations. These are the transformations which transform the entire surface while leaving the action (or any other gauge-invariant function) invariant. It is these 2g holonomies which will play the role of the dynamical variables in the expression for the action we derive from the Einstein action (1.3) in this chapter. Connected-sum surfaces, like those in Figure 0.1 have no boundary. How then, do we handle boundary terms that arise in the action? The answer comes by looking at the way these surfaces are constructed. Generally a genus g surface is constructed from a 4g- sided polygon with pairs of sides identified, folding up the polygon, creating the surface. The matched points on the boundary of the polygon come together to form a seamless, boundary-less, surface. Another way to view this construction is to tile some infinite plane with these polygons, so that moving off one tile onto an adjacent one is the same as travelling off a single polygon and reappearing at the identified point on the boundary. The boundary can be re-inserted into the closed, compact surface by slicing the surface open. This does not affect integration over the surface: A continuous function has the same value at two identified points on the boundary. Since the outward-pointing normals from the two identified boundary points have opposite orientations, any contribution to Chapter 2. The Einstein Action^ 15 Figure 2.1: The difference 'Y1-72 between two homotopic curves 7 1 and 72 is contractible to a point. The holonomy of this trivial loop is the identity, so 7 1 and 72 have the same holonomy. Chapter 2. The Einstein Action^ 16 a boundary integral along one boundary component is exactly cancelled by the matching boundary's contribution. The converse to this situation need not by true, though, as we will see below. By starting with the tile and matching pairs of sides, we are no longer guaranteed that functions are continuous across the boundary. The discontinuities we allow are exactly the holonomies that appear in the action. 2.1 Reduction of the Einstein Action The dynamical variables that appear in the Einstein action (1.3) are the dreibein e ma and the connection co: b . Let us assume there exists a collection of e fia(xv) and woa b (xv) which are continuous over the surface, or equivalently, continuous on the tile, even across the identified boundaries. Because only the torus T 2 can support a non vanishing tangent vector field [6], we must allow the possibility of the dreibein becoming singular on the tile. First consider the symmetries of the action. The action, as the Correspondence Principle implies, is Lorentz invariant. Transform the variables under , a Fi a WA b Uab ettb co1:d audb uc a (ucb,m )^ (2.1) Some index manipulation along with the properties of U (Appendix A) show the action is the same functional of e and L. Recall from the (2+1)-dimensional splitting of spacetime that the space is subject to the constraint Ri j ab = 0. This implies that on any coordinate patch, we can find a particular Lorentz transformation Uab (e) such that c" -;.5i ab = 0 (See Appendix B). Note that only the spatial components of (.7., can be made to vanish because only the spatial Rid ab = 0. Chapter 2. The Einstein Action^ 17 In this new coordinate system, consider a transformation in the internal space gener- ated by a function pa (x"): et: eµ^DµAtpa^ (2.2) where h is the covariant derivative with respect to the connection a While is is assumed that e if is continuous across the boundary, it is not necessarily true that e lia is continuous. With this transformation, the action becomes I =^fabc (6Aa noa)fiupbc a A de A dx f) M where M is the (2+1)-dimensional manifold and R is the curvature, also written in terms of a An integration-by-parts on the second term gives I = e t: fcpbc axt.,, A de A dxPi fabc M + I Ca bc pa ft,bc dx"A deI— I Eabc pa b[1Wc de A dxv A dxP .it4- The last term vanishes by the Bianchi Identity. A second constraint imposed on the system by the (2+1)-dimensional splitting is D[ i cj]' = 0 in the original coordinate system. This constraint implies we can find a pa(xv) such that the spatial e i a = 0 (See Appendix B). The first term of the above action vanishes leaving only the boundary integral. The boundary here includes the spatial boundary where the identification takes place and also the initial and final hypersurfaces in time. The integrals over these temporal boundaries, however, come from the term bo(pa fiv \ ,x o Eabc^ pbc)a A de A dxP .im This total time derivative can be removed form the Lagrangian as it adds only a constant to the action and has no effect on the equations of motion found by variational principles. = E edges^g Cabc (Pla ^be^a^bc4.7610 — p2 632 o ) dx i ed e . Chapter 2. The Einstein Action^ 18 This leaves only I = Eabc pa w bc• — be^ib dcjo dc^ob i dc)dx i where 0 represents the spatial boundary. As the spatial components (Di ba vanish, the action is further reduced to I =Ea bc^bcdxp . j One more integration-by-parts gives I =PaCab Pa coobcdxi — Ja The boundary term of this integration vanishes, being the boundary of a boundary. Now, each side of the 4g-sided tile is attached to another tile by the identifications used to get the topology. Thus each distinct edge of the tile contributes twice to this integral. Call the two matching sides '1' and '2'. Summing over the 2g different edges gives a Recall from the transformation (2.1) o cd = w oab uc a udb licb udb^ (2.3) Just consider the first term, in coo . As wo is assumed to be continuous across the boundary, wiobc = w2obc; we can write just w o bc. This first term of the action is bd tiice — p2 a i^bd^de dx iCa c (P1 edges edge As U1 and U2 are both Lorentz transformations, there is a Lorentz transformation between them. Define ui ab = wac u2 c b^ (2.4) Chapter 2. The Einstein Action^ 19 This maps U on one side of the boundary onto U on the other side, in some sense carrying U across the boundary. Observe that = (V V c ati2da )(W celk e = (Wc dWce) 142 datt2 e b,i be + Wed /122 w'e7i u2 e b With C3i ab vanishing in (2.1) and co lobc = co2obc, Hence Or uida^= u2da u2 dbi wc d u2da wce,i u2 eb = 0 , ei = 0 : (2.5) the W transition matrices are constant on each time slice. Substitute into the action for U2 in terms of W and U1 . Property (A.4) then reduces this term of the action to simply E f (pi ai wab p2^bd Ce wode dx edges^ i edge (2.6) Before we calculate the second part of the action from the UU term in (2.3), not the following. Recall Coo de does not appear in the action by the antisymmetry of the volume form. Thus wo de is a Lagrange multiplier. The UU term contains no CA) o de so the constraint associated with this Lagrange multiplier comes from the first term of the action alone, namely Pic; — wa bb p2,, = 0 • We found in (2.5) that Wab is constant on each time slice, so this constraint is equivalent to (2.7) w bp2 19 + Ha = pi a^ (2.8) Chapter 2. The Einstein Action^ 20 where Ha is constant. Under the identification of points that generates the topology of the surface, p 1 and p2 are two vectors sitting at the same point (in the same tangent space). The relation (2.8) shows these two vectors are related by a Poincare transformation (see 1.6)). Another way to compare vector P 1 with vector p 2 is to parallel transport p 1 across the tile between the two identified points on the boundary. The resulting vector is defined to be p 2 . Now (2.8) shows the holonomy of this loop in a Poincare transformation. We have seen that the W are constant on each time-slice and that the holonomies are path independent. Thus p on the whole of side 1 maps onto p on the whole of side 2 under this Poincare holonomy. Now consider the remaining term in the action, a 7j b^cd^a 71. b 71 cdf^Eabc „ L41 d 71^— p2,, (42 d 442 edges edge From (2.7) and (2.4) we substitute dx . P2 a,i = Wb a Pl b /42 cd = wy c ul yd Because Wab does not necessarily vanish (only the spatial derivatives do), we cannot yet • substitute for /42 b  d. These substitutions give cdE^cab, (pi a 141 b, i^d"1 edges edge • Wea P1 e ,i U2 b  dWv c tOd ) dx i . Replace E abc under (A.4) in the second term. Then (A.1) and (A.3) reduce the action to Now edges ledge b^wb 71.^\ 71^iEabc^At'fl d — YV (-4 2 041 dx . wbx 1:12d = wbs u2 x d Wbs u2 xd = ui b d — wbx wy sui yd Chapter 2. The Einstein Action^ 21 giving E Eabcp1a,i wbx wcx ch i edges edge Recall the Wab are constant and hence can be pulled out of this integral along the edges of the tile, leaving only • • f p i a 'i dx iedge This integral, simply the difference in p i between the ends of edge 1 of the tile, we denote by Ap i a. Finally, we produce the action on the tile. By imposing the constraint (2.7), which introduces the constant Ha, the first term of the action (2.6) vanishes. All that remains for the action is the sum over representative edges = E Api a wbx wcx cabc^ (2.9) edges By constructing a tile and assigning on the tile fields of W and II, we can explicitly calculate this action for a torus T 2 and later, the more complicated but interesting two- holed torus T2#T2. Chapter 3 The Simple Torus An elegant, but not particularly profound result of differential geometry is the observa- tion, here attributed to Carlip', that "a flat connection is determined, uniquely up to gauge transformations, by its holonomies around the nontrivial loops..." Mathematically, this is a concrete and definitive corollary, distilled from a much larger theory. It is not an explicit statement of the physics of the system, though, for it still deals with abstract ISO(2,1) transformations across some surface or region with identified points. In this Chapter, we see how the result (2.9) of Chapter 2, which asserts that the action in a function(al) of the holonomies (supporting Carlip's statement), is manifest on a simple 1-holed torus T 2 . This is relatively easy to do, as the flat genus g = 1 torus has only two independent holonomies, and can be constructed by identifying pairs of sides of a square in the Euclidean plane R 2 . 3.1 The Action over a Torus Each surface of constant time, the 2-dimensional spatial slices of (2+1)-dimensional space- time, is tiled with squares, or more generally parallelograms. To calculate the action on this tile, let us first label its components as in Figure 3.1(i). We call the sides A, B, C, D and the corners 1,2,3,4. The identification of sides A&C and B&D is indicated by the 1 [4], p. 2649. 22 Chapter 3. The Simple Torus^ 23 Figure 3.1: (i) Labels for the sides and corners of the tile, and identifications of sides A&C and B&D. (ii) The neighborhood of a vertex where four tiles meet. arrows on the boundary. Folding up the tile by gluing together the identified sides with the indicated orientation creates a torus, T2 . The same gluing information can also be exhibited by looking at the neighborhood of the vertex where the 4 corners come to- gether, or where 4 tiles meet on the plane (See Figure 3.100). We produce this Figure by "bootstrapping" around the vertex, a method we will employ frequently in what fol- lows. Starting form the region about the vertex labelled 1, sides A and C are identified as part of the topology generating gluing. Adjacent to side C is side D, and these two sides meet at corner 4. Next, side B is glued onto D, side C is adjacent to B, and corner 3 lies at the intersection of these two boundary components. We continue in this way, labelling each side and each region of the vertex, showing all the gluing of Figure 3.1(i). Each line, like the one between A and C, is one of the boundaries we say has p i , Cj i on one side, and p2 , w2 on the other. The corresponding edge along which we calculate Ap i a is, in this case, the line that runs out from the 1-4 region of the vertex and returns to 1 2 3 4 D WDB 1 2 (i)^A 3 C WC A 4 Chapter 3. The Simple Torus^ 24 Figure 3.2: The transformations W across the identified edges (i)C&A, (ii)D&B. the vertex though the 2-3 region. Across the boundary between sides A and C we find from (2.8) WC A a b PC b HCA a = PA a • Here Wc A is the Lorentz transformation which relates WcAtic = 11A and HcA is the constant defined by (2.7). This relation holds all along the edge between sides A and C so we push the result down to the 1-4 region of the vertex. Evaluated at the vertex we write WCAab PC b (4) HCA a pAa(1) • The term pc a (1) is really the limiting value of pca as the vertex is approached along edge C. There is a similar relation at the other end of this CA boundary, the end in the 3-2 region of the vertex. The relations across each of the four edges of the tile can be written down by looking at Figure 3.2: Chapter 3. The Simple Torus^ 25 WCA ab PC b (4) HCA a • PAa(1) Figure 3.2 (i) WCAab pc b(3) HcAa PAa(2) Figure 3.2 (ii)^ WDBab PD b (4) + IIDB a pBa(3) To produce the action we must evaluate the Apa by integrating along the identified edges. Each edge contributes twice, recall, but the return integral along the 2-side is taken care of with the WW terms in the action. We need only consider the 1-sides, from corners 1 -4 2 and 2 3. The action is simply {(PA a ( 2) PA a ( 1 )) WCA bxWCA cs (PB a (3) PB a ( 2)) WDB bs WDB c  Eabc Now it is not true that, say, pc (4) = pA (1), even though these two functions are evaluated at the same point, under the identification of sides A and C. There is the discontinuity is pa defining 14/cA and licit . It is true, however, that at vertex 4, pD(4) = pc (4), as these are evaluated at the same point on the tile, without any identification of points required. Call the corner value just p 4 . Similar relations hold in each region of the vertex: P A(1) PD( 1 ) ^ PB(2 ) = PA(2 ) = P2^ (3.2) PC(3 ) = PB(3 ) = p3^PD(4) = pc(4) = p4 With these relations, along with the W and II above, we can express p in each region of the vertex in terms of, say, p l . By bootstrapping around the vertex from region 1, jumping across boundaries with (3.1) and around corners with (3.2), we evaluate p3(pl), for instance, in vector- rather than component-form, as P3 = WC A l (P2 — HC A) = IIVCA-1 (WDBP1 IIDB HCA) • (3.1) WDB ab PD b (1) + HDB a pBa(2) Chapter 3. The Simple Torus^ 26 Note that just as we define Wc A to jump from side C to side A, WAC jumps from side A to side C. Each of the W is invertible, though, so WAC WC A l • In this way we find P2^WDB(P1 11DB) P3^WCA-1 (WDBP1 ilDB — TIC A) P4^WC A-1 (pi - HCA)  (3.3) We arbitrarily chose to write p 3 = P3(P2 (pi )), but we equally could have chosen p3 (p4 (pi)), or even p3(p4(pi (p2 (p3(p4 (pi (• • •))• In order that the results be consistent, it must be true that the transformation giving a complete circuit of the vertex is the Identity: Pi (P4 (P3(P2(Pi )))) = P1 • As the transformation is in general a Poincare transformation, this means the Lorentz part is the Identity, while the translation vanishes. Jumping all the way around the vertex produces TAT -1 Tv- -1 1,, pi^WCA vvDB vvCA vvDBP1 + WC AWD B-1 WC A-1 H D B LT -1^-11-r WCA WDB vvCA 11CA — WCAWDB HDB + FICA • Hence the Lorentz transformations must obey the closure relation TAT -14,T7 -14,17 ^ WCA vvDB vvCA vvDB 1 •^ (3.4) This "constraint" is the representation of the fundamental group 7 1 (r) in the gauge group ISO(2,1). It is equivalent to the condition WCAWDB WDBWCA Chapter 3. The Simple Torus^ 27 so that the two Lorentz transformations commute. Applying the relation (3.4), we reduce the translation constraint to simply ^(1 WCA) HDB = (1 — WDB) HCA •^ (3.5) We will further analyse this constraint below, and how much, or how little, it further constrains the system. First though, we calculate the action. With the relations (3.2) and (3.3), we can express the action in terms of p 1 (or simply p), W, and II, plus the constraints (3.4) and (3.5). We first introduce notation to remove the cumbersome term viib wcxcx^abe Define {W}c, = Flibxwcscab, • We now adopt a 0-index notation so that H a vox wcscab, Ha { w} a^ll{w} The algebra of reducing the action is greatly simplified by the following properties of {W}, the details of which are found in Appendix C: (un){tair} = n{W} 11{14i .w2}=  (C.1) (C.2) (C.3)H ({ 4 } 0721) _ (H72 -1 11) { 1472 -1 144} With this new notation, the action, without the constraints yet imposed, reads I = (I/K , p){WC A-1 WDB-1 WC AWD B} — (WC A l llc A) {WD B} + ( wDB-1 11DB){WDB-1 WCAWDB} • Chapter 3. The Simple Torus^ 28 An example of the {W} algebra is shown below in the 7 12#712 case. If we now impose the constraints on the system, the first term above vanishes, as {1} oc 1 = 0. The action is finally reduced to (147DB-111DB){WcA} — (WcA 1 ricA ) { a b ( WC A WD B 141D B WC A) a b a ((1 — WC A)11 D B (1 — WD B) 11 C ^ (3.6) where the constraints are included with the Lagrange multipliers e and C. The most important feature of this result in that it is written entirely in terms of the holonomies [W ITT 1C A I—CAJ and [W Ill 1DB i _DB J • All reference to the geometry of the tile has been removed. This is exactly the property that flat connections are completely specified by a collection of holonomies. Furthermore, we see that the action is very nearly in the canonical 1,4 form, except that {W} is not merely I- W. Before we attempt to extract the canonical variables, we will look at the r#T 2 case for more insight. 3.2 Consistency of the Constraints It appears from (3.6) that 6 degrees of freedom will be removed by imposing the con- straints on the system: 3 from the vector constraint, and just 3 from the matrix con- straint, due to the symmetry of Lorentz transformations. In fact, explicitly writing out the constraints shows that these six equations are not independent, and only four degrees of freedom are directly removed. Also, we will see that under a interesting identification of the H with translations in the solution space of W's, the relations (3.4) and (3.5) contain the same constraints. This association is not precise, but merely suggestive of the roles of W and H as canonically conjugate variables. Chapter 3. The Simple Torus^ 29 The first constraint of the T 2-system is the closure relation WCA abi/VDB b c TVDB abWCA b c = 0 . The W's are forced by this constraint to lie on a surface in W-space. Hence variations of the solution WCA^WDB must also lie on the this surface. That is, variations of the constraint with respect to the the coordinates Wab must vanish. We will see that this new condition is a copy of the second constraint on the system, subject to some interpretation of the variables. The variation of WcA WDB — WDBWcA is calculated by SWCA ab rTAT^6.WCAb a vvCAYTIVDB b^TVDBab TAT C SWCA sAac TVC A Xy^ V IVA Ty (5 WDB b  1LIT^C5WB ab r+ 147C A ab^xC "rDBxy^U "ixTDB Ty WC A b c 'TDB y^WDDB Xy where swa 6 — sa SWxy x Thus A ac^SWCAabWDBb WDB ab 0WCA b C 14/6 A abSWDB 6 SWD B ablVCA 6 or, in matrix form, A = 8WcAWDB — WDB6WcA WcASWDB — SWDBWCA • By varying the constant metric 77 ab , observe that (SW)W' is antisymmetric: (S (77 ab ) = s(wawbc) = swac wbc w a c swbc = swac wbc swb c w ac^ (3.7) Chapter 3. The Simple Torus^ 30 Insert factors of 1 = W'W where necessary into the expression for A to get all SW into this antisymmetric form: A = (SWcAWcA-1 )WcAWDB — WDB( 6WCAWCA-1 )WCA + WC A(CSWDBWDB-1 )WDB (SWD BWD B-1 )WD BWC A = (SWC AWC A-1 )WC A WD B WD B ( 6. WC AWC A-1 )WD B-1 WD B WC A + WC A (6. WD B 14713 B-1 )WC A-1 WC A IlVD B (8 WD BWD B-1 ) 1413 B (VC A where the last step puts each (SW W') into a similarity transformation. We right- multiply by IV and apply the relation (3.4) to find ^\ ^— 0^(6WCA WCA-1) WDB (8WCA WCA-1 ) r HT DB 1 WCA ( 8WDB WDB-1 )WCA-1 (SWDB WDB-1 ) Now as (SW W -1 ) is antisymmetric, we can replace this matrix by an equivalent vector A a defined as Aa fabc swbd wcd The similarity transformations U(SWW -4 )/4' become linear transformations U of the vector. "Rotate" the antisymmetric matrix (SW W') with a Lorentz transformation U: swbdvrd^ubxjwxdwyducy This new matrix is also antisymmetric in b, c: interchange b and c to find ucx (swsdwy d) uby = ucx (6w dwyd)uby ucx (5wyd wx.d) uby as SW W-1  antisymmetric. Now relabel the dummy indices, interchanging x and y: _= ucy swsd wylubx ub x ( (Swsd wy d) ucy cAWDB, or equivalently WDBWCA, without changing the vanishing variation Chapter 3. The Simple Torus^ 31 Define a new vector A'a from this antisymmetric matrix: ya fabc (11bxtiVW dWyd) . By (A.4), SO which is simply Eabc tibs ti c y = tia z exy , Ala = ti az Ezxy ST/VxdWYd A'a = tfazAz . Under this substitution of A a , the variation of the constraint on the W's is equivalent to the constraint 0 = ACA — WDBAC A + WCAADB — \DB = (1 — WDB)ACA — (1 — Wc,A)ADB • This is exactly the second constraint (3.5) under the identification Aa^ila . The Aa are infinitesimal translations in the space of W's, suggesting the holonomy com- ponents II are related to translation-generating momenta, conjugate to the configuration variables W. We will expand on this idea further, after considering the T2_//.T2 case, where we reproduce the almost canonical p4 action. 3.3 A Model of T2 To transform the action in {W} and H into something that can be written on the back of an envelope, we construct an explicit model for T2 and find the Poincare transformations Chapter 3. The Simple Torus^ 32 [Will] across its boundaries. We will see that even a very simple model reveals interesting details. It is easy to construct a region representing the torus because the plane R 2 can be tiled in unit squares. By identifying points on opposite sides of the square, the torus' topology is produced. To bring this tile into the arena where we can study the action, we must attach the holonomies [W ITT 1 andj [WDB IllD )3] between the identified sides. The constraints (3.4) and (3.5) make this a fairly simple procedure. Because the Lorentz transformations Wc A and WDB commute, they must be boosts in the same direction. Let us choose coordinates (x l , x 2 , x3) = (t, x, y) over the tiled R2 plane so that this direction is the x-direction, with the origin at corner 1 of the tile. Recall from §1.2 that boosts in the x-direction are generated by exponentiation of the matrix J3. We can choose cosh(y) sinh(,u) 0 \ ^WCA = eA.13^sinh(,u) cosh(,u) 0 0^0^1 j cosh(ay) sinh(a,u) 01 ^WDB = e la3^sinh(ay) cosh(au) 0 0^0^l/ where ,u is some boost parameter and au ensures WDB is parallel to WCA. It is interesting to look at the translation constraint (3.5) with respect to this choice of WCA and WDB. The constraint reads 1 — cosh(y) — sinh(y) 0 HDB 1 — sinh(u) 1 — cosh(y) 0 HDB 2 0 0 0 / HDB3 Chapter 3. The Simple Torus^ 33 / 1 — cosh(ap) — sinh(a,u) 0 \ TIC A l = — sinh(ap) 1 — cosh(ap) 0 TIC A 2 0 0 0 1 \ ITC A 3 I While IIDB 1 , HDB 2 , HcA 1 , HcA 2 are coupled by two equations, IIDB 3 and HcA 3 are com- pletely unconstrained. That is, for any translations Ir in the y-direction, the sequence of transformations that circumvents the tile vertex is still the Identity. While initially it appeared that the constraints would directly remove six degrees of freedom (df) from the system, in fact only four are removed. This is not merely an artifact of our particular choice of the x-direction for the boosts, for a rotation will not affect the indeterminacy of the matrices 1— WDB or 1 — WCA• We will see below in the more complicated r#T 2 case that this failure of the six constraints to remove six df is unique among the non-zero genus surfaces to the torus. Furthermore, whereas the Lorentz component of the holonomies confine the transformations to a surface of constant 0 2 — x 2 , the translation carries vectors off this hyperbolic plane. With some translation in the t- and x-directions and arbitrary y-translation, it seems possible that the sequence of transformations around the vertex can be a loop with time-like sections, carrying vectors along closed time-like curves. To attach these Lorentz transformations to the unit square recall that W is de- fined at the link between the values of the field Uab across the identification boundary: W21ti2 = U1. The field Uab which is consistent with this choice of W's is found by treating the transformations W as a sort of "phase difference" between identified points. Referring again to Figure 3.1, first consider just the corners labelled 1,2,3,4. As the vertex at cor- ner 1 on side D maps to the vertex at corner 2 on side B under WDB , U (2) = WDBU (1) . To find U(3), we see that U(3) maps onto U(2) under Wc A, so U(2) = WcAU(3) or Chapter 3. The Simple Torus^ 34 Figure 3.3: Values for the 'phase' U at the corners of the tile. U(3) = WcVU(2) = WcA-1 WDB U(1). Similarly, U(4) = WcZ1U(1). Now if we cal- culate U(3) from U(4), instead of U(2), we write U(3) = WDB U(2) = WDBMA 1 U(1). So that the result is independent of the choice of evaluation, it must be true that Wcit 1 WDB = WDBWCA 1 or WDBWCA = WCAWDB, exactly the constraint we encountered earlier. Since the transformations W are determined only by the phase difference be- tween the values of U at identified points, there is an over all arbitrary choice for U(1). Setting U(1) = Wc corner values are now interpolated smoothly over the tile, taking care to keep the correct W phase difference between identified points on the boundary. The particular choice of interpolation does not change the action, which depends only on the difference across the tile (or across the boundary). One simple example of an interpolation is ti(x, y) = e ( 1— f(Y) -Ectf(x))0J3 SA gives simple values to the corners of the tile (See Figure 3.3). These where f is linear between f (0) = 0 and f(1) = 1. Smoother interpolations (quadratic, Chapter 3. The Simple Torus^ 35 cubic, ... ) can be used where continuity of derivatives of U is required in further calcu- lations. The final step in the construction of our model is finding a set of H's. These trans- lations entered the calculation through W21p2 + 1121 = p1, where pa(x b) = Fa(x b) is the function chosen to eliminate the e i a. A simple choice is to give the translations only y-components, n-CA = (0, 0, a) and IlDB = (0, 0, b), and set pa (x b) = xa . The Lorentz components of the holonomies preserve 0 2 — x 2 while the translations shift the y-components of parallel transported vectors by a constant. The tangent space over the tile is a parallelogram on the surface of constant 02 - x 2 (See Figure 3.4). The relation W21P2 + 1121 = p1 tells us which points are identified in the tangent space. 3.4 An Alternative Approach to the Torus Another, more "standard", method for studying (2+1)-dimensional GR on a torus is based on the fact that of all the compact surfaces, only the genus 1 torus can support a continuous, non-vanishing tangent vector field. The approach, therefore, cannot be generalized to higher genus surfaces. Because the plane R2 can be tiled in unit squares, we break spacetime into R2 0 R. By identifying opposite sides of a unit square, or defining spatial coordinates x and y to be periodic with period 1, the spacetime T 2 R is generated. With this geometry, Carlip [4] proceeds by specifying the holonomies of this surface, two commuting Poincare transformations: A l : (t, x, y) —f (t cosh^x sinh^x cosh + t sinh^y a) A2 : (t, x, y) --+ (t cosh /2 x sinh^x cosh p + t sinh ,a, y b) A dreibein and connection which exhibit these holonomies under a path-ordered integral Chapter 3. The Simple Torus^ 36 Figure 3.4: Tiles of the tangent space over the torus are parallelograms in boost-transla- tion space. Chapter 3. The Simple Torus^ 37 or Wilson line [5] calculation of the holonomy are e1 = e2 = 0 , col = co2 = 0 and e3 = (0, a, b)^w3 = (0, A, ,a) This dreibein produces a singular metric, but we can gauge transform to a non-singular system: el = (— /3 , 0, 0 )^col = (0, 0, 0) e2 = (0, #A, OF) w2 = (0, 0, 0) e3 = (0, a,b)^co3 = ( 0, A, it) where 13(0 is a function only of the time on the slice. On each slice of constant time, a constant, continuous non-vanishing tangent vector field is realized. The metric arising from this choice of dreibein is ds2 = la 2 dt2 — (a2 + # 2 A2)dx 2 —2(ab + 13 2 A,a)dx dy — (b2 + /3 2 it 2 )dy2 . Now Carlip observes that the spatial part of the metric, the metric on the torus, is unchanged by the two coordinate transformations (x + (ab + 0 2 Aµ ^Nap — Ab) ,,,) ___+ a2 + 13 2A2 Y a2 + /3 2A2 Y (ab + 13 2A,a) ,,,, 0(a,tt — Ab) ,„)((x +1) + a2 + 0 2A2 '.9 a2 + 13 2A2 Y (x + (at. + 0 2 A,u) ,,,, 13(a,a — Ab) ,,,) a2 + 0 2x2 Y a2 + /32A2 Y (x + (ab + a2 + 0 2 A,a) (y +1), ,3(af 3 2A2 t — Ab)  (y + 1))0 2A2^a2 + / The geometry of the space, therefore, is characterized by these two coordinate translations in a, b, A, ,a, the parameters that fix the holonomies. Treating a, b, A, ,a as a new set of coordinates, the Hamiltonian produced is simply H = i3(a,u — Ab) . Chapter 3. The Simple Torus^ 38 One interpretation of the canonical variables is to take the boost parameters A, p as coordinates and the translations a, b as conjugate momenta. This result is based on the existence of a non-singular spatial metric on the surface. Its spatial periodicity is equivalent to the periodic tiling of the tangent space over the torus. The holonomies (A, a) and (pi, b) completely determine the action because of the flatness of the space. A model for the genus g = 1 torus is simple to construct because the plane R2 can be tiled in regular (4g = 4)-gons, or squares. This surface is also easier to study than other genus surfaces because it is the only one that can support a non-vanishing tangent vector field, immediately giving the surface a non-singular metric. We now turn a to more complicated surface, the two-holed torus, where the results for T2 are closely mimicked. Chapter 4 A Two-Holed Torus In this chapter we apply the results of Chapter 2 to the more complicated genus 2 two- holed torus, r#T2 . One would suspect that this surface is more difficult to study than T 2 , for it is impossible to put a continuous non-vanishing tangent vector field onto this surface whose non-zero Euler characteristic x = 2-2g = —2 is non-zero.[6] We will find, however, that T2#T2 is simply the connected sum of two tori T 2 , and the action is more complicated only because it is a functional of twice as many variables. This surface has 2g = 4 generating non-trivial loops. It can be constructed by identifying pairs of sides of a (4g = 8)-sided polygon. Or equivalently, the two-holed torus can be conceived by a tiling of a plane with octagonal tiles, with eight tiles meeting at each vertex. As we shall see, but intuitively understand already, the plane R 2 cannot be tiled in regular octagons without leaving gaps in the tiling. Hence we look to hyperbolic geometry where the condition that triangles have 180° no longer applies. Before we construct such a tile and its collection of holonomies W and II, we first consider the case of a general octagonal tiling, and translate the results of Chapter 2 into the r#T 2 variables. 4.1 The Action over a Two-Holed Torus We calculate the action over the two-holed torus exactly as we did for T 2 . Cover each surface of constant time with octagonal tiles. Label the sides of the octagon A, B , . . . , H and the corners 1,2, ... , 8 (See Figure 4.1(i)). The identification of sides is indicated by the arrows on each side. Gluing the matching sides together generates the 39 Chapter 4. A Two-Holed Torus^ 40 Figure 4.1: (i) Label for the sides and corners of the tile, and identifications of sides. (ii) The neighborhood of a vertex where eight tiles meet. boundary-less two-holed torus. Note the reversed orientation in the pairs of identified sides C&A, D&B, G&E, H&F. The neighborhood of a vertex where eight tiles meet also shows the identifications and labels (See Figure 4.1(ii)). As before, we bootstrap around the vertex using the identifications and label the sides and regions of the vertex. Figure 4.1(ii) is a truer representation of the tile because the neighborhood of a vertex is a patch of R2 , and the angular contribution of each tile is if. By drawing the whole tile on paper (R2 ) as opposed to the hyperbolic plane where the tile really sits, we are forced to stretch the angles out to ,ir-. Across the boundary between sides A and C we find from (2.8) WCA ab PC a + IICA a = PA a • Here WcA is the Lorentz transformation which relates WcA Uc = UA and HCA is the constant defined by (2.7). This relation holds all along the edge between sides A and C Chapter 4. A Two-Holed Torus^ 41 as Wab,, = 0 (2.5), so we push the result down to the 3-2 region of the vertex where WCA ab PC a ( 3 ) + TICA a = pAa(2) • The term pc a (3) is the limiting value Similar relations across each of the identified of pc a as the vertex is approached along edge C. edges are defined in Figure 4.2: mAab pc b (4) + HCAa^pAa( 1) Figure 4.2 (1) WCA ab PC b ( 3 )^HCA a = PA a (2) Figure 4.2 (ii) WDBab pDb (4) + HDB a^Wa(3) WDB ab PD b (5 )^IIDB a = PC a (2 ) (4.1) Figure 4.2 (iii) { WGE ab pGb (7) + HGE a = PE a (6) wGE ab pGb(8) IIGEa pEa(5) Figure 4.2 (iv) WHF ab PH b ( 8 )^HH Fa = PFa (7) TVHF ab PH b ( 1 ) + 111 F a^pFa(6) We must evaluate Ap i a by integrating along identified edges to produce the action. The return integrals along 2-sides are accounted for by the WW terms, so we need only consider the 1-sides, from corners 1 2, 2 -+ 3, 5 -+ 6, and 6 -4 7. The action becomes I =^(PAa(2) PA a ( 1 ))WCA bsWCA cx (PB a (3) PB a ( 2 ))WDB bxWDB cs (PE a ( 6) PE a ( 5 ))WGE b zWGE cx ^(PF a^PFa (6)0H Fb^Fcs fabc • ^ (4.2) The field pa is discontinuous across the boundaries, the discontinuities related to the holonomies W and H. But in region 4 of the vertex, for instance, pp (4) = pc (4), as this is the limiting value of a function pa, continuous on the tile, without any identification of points required. Call the common value just p 4 . An analogous relation holds in each Chapter 4. A Two-Holed Torus^ 42 Figure 4.2: The transformations W across the identified edges (i)C&A, (ii)D&B, (iii)G&E, and (iv)H&F. Chapter 4. A Two-Holed Torus^ 43 region of the vertex: pA(1) pi-1M= pi^pA(2)= pB(2) =--- p 2 pB(3) = pc(3) = p3^pc(4) = PD(4) = P4 pD(5) = pE(5) = P5^P E( 6 ) = PF ( 6) = P6 PF( 7) = PG( 7) = P7^PG(8) = PH(8) = P8 By bootstrapping around the vertex from region 1 with these relations and the W, H above (4.1) we can express all the pi in terms of p l . As vectors and matrices rather than in components, we see P2 = WCA P3 + HC A WCA(WDB P4 + HDB) + HCA WCA(WDB(WCA 1 (pi —HcA))+ HDB) + "CA Repeating the process for each corner at the vertex gives P2^WCA(WDB(WCA1 (pi —HcA))+HDB)+ 11 CA P3^WDB(WCA1 (pi — HcA)) HDB P4^111' ( p — "CA P5^WGEWH F l (WGE 1 (WHF P1 + HHF HGE) HHF) + HGE P6^WHFP1 HHF -GE 1 \ •^1•HFP -HF -GP7^W (W TT^TT E 1 P8^WHF1 (WGE 1 (WHF P1 + HHF HGE) - HHF) We arbitrarily chose to write ps = Ps(P7(06(P1))) in finding the last result in this list, but we equally could have bootstrapped the other way around the vertex, writ- ing ps = ps(p5(p2(p3(p4(p1)))))• For the pi to be well-defined, it must be true that a complete circuit of the vertex is the Identity: (4. 3) P1(p4(p3(P2(P5(P8(p7(p6(P1) • • •)) = P1 Chapter 4. A Two-Holed Torus^ 44 The Lorentz part of this Poincare transformation must be the Identity and the translation must vanish. The transformation carrying p i around the vertex back onto p i is given by P1 = WC AWD B 1 WC A-1 WD BWG E WH F 1 WG E 1 WHF P1 + WC AWD B 1 WCA4 WD BWG EWH F 1 WG E 1 H H F - WC AWD B 1 WC A-1 WD BWG E WH F 1 WG E 1 H G E - WC AWD B 1 WC A 1 WD BWGEWH F 1 H H F + WC AWD B 1 WC A 1 WD BHG E + WC AWD B 1 WC A 1 H D B TAT 11,17 -ln - WCAvvDB wvCA - WCAWDB-111DB • H C A • The Lorentz transformations WCA,WDB1WGE,WHF must obey the closure relation WCA WDB 1 WCA-1 WDB WGEWHF 1 WGE 1 WHF = 1 • Again we see the fundamental group 7r i (T 2#T 2 ) represented by the W in the gauge group. Define 1^WC A WD B-1 WC A-1 WDB = WH WG EWH FWG E 1 • The constraint (4.5) is equivalent to (4.6) f21 - f22 = . While S2 1 = h1(WcA, WDB) but I/2 = 11 2(WGE, WHF), they are the same transformation: hi is a transformation halfway around the vertex in one direction, C2 2 is a transformation halfway around in the other direction, and the two results coincide there. The decoupling (4.4) (4.5) Chapter 4. A Two-Holed Torus^ 45 of the transformations into ABCD terms and EFGH terms is an indication of the connected sum construction of r#T 2 . The constraint (4.5) shows that the two tori glue together smoothly. By inserting the factors missing from the cycle (4.5) and removing the resulting factors of 1, we can reduce the translation part of (4.4) to the constraint 0 = 11CA — WCAWDB 1 WCA 1 HCA — WC A WD B-1 H D B WC A WD 13-1 WC A 1 H D B — WHF 1 HGE WHF 1 WGEWHFWGE 1 HGE +WH F-111H F WH F-1 WG EH H F •^ (4.7) We can now write the action in terms of p i (or simply p), W, and H, plus the constraints (4.5) and (4.7) with the help of (4.3). Again we introduce {W}. = Ikxw'eab, to more easily write H a lkbsW'cabc H{W} We use the properties of {W} found in Appendix C to reduce the action to a simple form: I = — p{WCAWDB 1- WCA 1 WDB} p{WHF 1 WGEWH FWG (WD 13-1 HG A) {wDB 1 f2i } — (wDB 1 HDB) {wcA 1 121} HGE{WHFC12} (WGEWHF-1HHF) {WGEg2} • The term in HcA, for example, is found as follows: Upon substituting the relations (4.3) into the action (4.2), the terms containing 11CA are — (wcAwDBwc,VHcA) {14/CA} + HCA{WcA} — (wDBMA i ncA) {WDB} + (wcAwDBwcA 1 11cA) {wDB} —11cA{WDB} • Chapter 4. A Two-Holed Torus^ 46 Rewrite this using (C.1) to give the terms the form of the right-hand side of (C.3): (— WC A WD B WC A-1 H C A) { WC A WD B WC A-1 WC A WD B-1 WC A-1 WC Al —^ —^ — 11 CA { WcA. — HCA{WDB} — (WDB , v 7A7 cA 1 -Fr iicA) {WDB v TAT CA 1  ry TA7 CAvv TA7 DB 1  ro' Tv' DB + (WC A WD B WC A 1 ri c A) WC A WD B WC A-1 WC A WD 13-1 WC A 1 WD B Now apply (C.3) to remove the under-braced terms: = —11 C A {WcAWDB-1 } + HC A {WC AWD B-1 WC A 1 } +HCA {VI/6A} — HCA{WDB} — IICA {WCA} + HcA{ WCAWDB 1 } +11cA{WCA WD B-1 WC A-1 WD B} HC A {1476 AWD WcA 1 } • The only two remaining terms are grouped together with (C.3) to give = (wDB-1 11cA) {WDB-1 WcAwDB-1 W6A-1 WDB} . Substituting 52 1 from (4.6) produces the fIcA term in the action above. By imposing the constraint Q i = 522, the terms in p cancel in the action, leaving the constrained action I = ( WDB-1 HcA) {wDB-l f/i} — (WDB-1 HDB) {Wcii-1 121} —HGE{WBFQ2} + (WGEWBF 1 HBF) {WGE512} — Q2)ab +Ca C A — WC AWD B-4 IVC A-4 NC A + • • • — WH F-1 WG EH H Fr •^(4.8) The Lagrange multipliers e and have been included to account for the constraints imposed by (4.5) and (4.7), respectively. Observe again that the terms in ABCD are decoupled from those in EFGH. As in the T 2 case, this action is written entirely in terms of the holonomies W and H. The geometry of the tile has been removed. The action is very nearly in the canonical • Chapter 4. A Two-Holed Torus^ 47 p4 form, except that the term {W} is not merely do W . The only difference between the "simple" T 2 and the "difficult" r#T 2 is that more variables have appeared to account for the increased number of incontractible loops and holonomies. 4.2 Consistency of the Constraints Recall the first constraint on the system is that the Lorentz transformations W obey the closure condition (4.5) T^T^1WC AWDB 1 WC A-1 WDBWGE vvHF1 v1, vGE 1 EA 7rHF 1 or with (4.6), C21 -^= 0 . The second constraint forces the sequence of translations to vanish (4.7), closing the circuit of Poincare transformations around the two-holed torus: 1,17 -1 TAT 1 rr 0^HCA WC A v^vvC A "-CA —wcAwDB 1 11DB+ WC A WD 13-1 WC A-1 H D B - WHF 11IGE WHF 1 WGEWHFWGE 1 HGE +WH F 1 HHF WH F 1 WG Ell' F • The "configuration" variables W (they are not quite the configuration variables be- cause the action is of the form 11{W}, not merely 11W) are forced by the first constraint to lie on a surface defined by the relation (4.5). Variations, or nearby solutions, must also lie in this surface. We will see that this new condition is the second constraint (4.7), under an identification of the "conjugate" variables H (as above, the H are not quite conjugate to W) with infinitesimal Lorentz generators. We will work both in tensor and component form in showing this result. Chapter 4. A Two-Holed Torus^ 48 The variation of C2 1 — S22 is defined as • = s ( — 112 ) ( — f22) ^SWCA^SWDB ^ SWCA^SWDB ^+ S(f21 n2) SWGE ^— C1 2) SWGE^SWIIFSWGE^SWHF where Swab^= SW sy SW b a y ^y ^asw x Wp 14lb (5TS the latter coming from 5(W W') = 0. In component form, the constraint reads 0 = WC A a bWDB cb WC A dc WDB d e — WH F b a WGE b e l/VH F c d •WGE e d • The variation, with the 5-functions evaluated, is Aae = owcAaowDBcbwcAdcwDB de WCAabWDBc5WCApcWCAdq(SWCAPOWDBde _wcA ab wDBpbwDBcg(swDBpowcAdcwDB de wcAabwDBcbwcAdc(swDBde) _KFba(swGE bc)wHF c dwGEed wHFbawGE b cwHF cd uT vvGEpd WGE e q (SWGE Pq ) +WH F pa WH F b q (SWH FPOWGE b c WHF cd 14 E e d — 147HFb a WGE b c (SWHF C d)WGE e d This is merely the ( )ae element of the matrix • = ( 8WcA)WDB-1 WcA-4 WDB — TVC AWD B-1 WC A-1 (SWC A)WC A-1 WD B —WC AWD B-1 (SWD B)WD B-1 WC A-1 WD B WC AWD B-1 WC A-1 (SWDB) —WH F-1 (SWGE)WH FWGE-1 WH F-1 WG EWH FWGE-1 (SWGE)WGE-1 +WH F-1 (SWH F)WH F-1 WGEWH FWGE-1^F-1WGE (SWH F)WGE-1 which is clearly the variation of WcAWDB 1 WcA 1 WDB — WHF 1 WGEWHFWGE 1 under s(w -1) = Chapter 4. A Two-Holed Torus^ 49 Recall that (SW)W -1 is antisymmetric (3.7). Insert factors of 1 = W'W where necessary into the expression for A to get all SW into this antisymmetric form. As the variation A must vanish, we can right-multiply A by WDB vvTA,C AWDBWC A1 or equivalently /17^T r WGEWHF- 1  roGE VVHF, to find the following: 0 = SWC A 147C A-1 ) — 147C AWD B-1 WC A-1 (SWC A 147C A-1 )WC AWD B-1 WC A-1 — WC A WD B-1 (SWD B 147D B-1 )WD B .147C A-1 + WC AWD B-1 WC A-1 (SWD B WD B-1 )WC AWD B .147C A-1 — 147H F-1 (SWGE WG E-1 )WH F WH F-1 WG E 147H F 147G E-1 (SWGE 147G E-1 )WG E 147H F-1 WG E-1 WH F +WH F-1 (SWH F 147H F-1 )WH F WH F-1 WG E (SWH F WH F-1 )WG E-1 147H F As (SW)W' is antisymmetric, it has only three independent components and can be replaced by a vector Aa: Aa = cabc swbd wdc Again, the similarity transformations U(SW W')/1 -1 in the variation are simple linear transformations UA of the vector A. The variation of the constraint becomes = ACA — WC AWD B 1 WC A 1 ACA — WCAWDB-1 ADB WCAWDB-1 WCA-1 ADB — WH F 1 AGE + WH F 1 WG EWH F 1 WG E 1 GE +WHF 1 AHF WHF 1 WGEAHF • Under the identification Axy a^xya , this is exactly the second constraint (4.7). The same notion of the 11 being momenta conjugate to the configuration variables W is suggested. We offer an interpretation of these "canonical variables" in Chapter 5. Chapter 4. A Two-Holed Torus^ 50 We arrive at the question of how many degrees of freedom are directly removed by these constraints. The condition 52 1 5221 = 1 implies WC A WD B 1 WC A-1 = 11 2 WDB1 •^ (4.9) Taking the trace of this matrix equation shows TrWDB-1 = Tr (WC A WD B-1 WC A-1 ) = Tr (0. W- -2 -DB 1 ) Whatever form 122 takes, the components of WD B must satisfy this (scalar) equation, removing one df. Using the fact that the Lorentz transformations form a Lie group, we can write each element in exponential form U = enaJa, where the generators Ja introduced in §1.2 are a basis for this vector space. We re-write (4.9) as e -nDBawcA.lawcA1 = 122 e —971JBaJa Now — 7/DB a WC A a WC A-1 is a vector in the space spanned by the basis vectors WCAJaWCA-1) the original basis Ja rotated by WC A.  In the original, non-rotated basis, this is the vector — (WCAT/DB) a Ja . This is the same as the earlier result of §3.2 that showed A' UA when the matrices (SW W-1  rotated by U. Thus we have e -(wcAnDB)aJa =11 2 e—naDEja •^ (4.10) The right-hand side is WDB1 transformed by 112. The vector representing this new Lorentz transformation is —(WCA 71DB) a a rotation of 7/15B . The equation specifies the direction about which this rotation must occur (two equations) and its magnitude (one equation). We see, however, 147C A WD B-1 WC A-1 = WC A (WD BWD B-1 )WD B-1 WC A-1 = (WC A WD B)WD B-1 (WC A WD B) -1 Chapter 4. A Two-Holed Torus^ 51 so that rotations about the riDB-direction are inconsequential. Therefore, the rotation specified by (4.10) is determined by only one parameter. Together with the magnitude of the rotation, two df are removed. Coupled with the trace relation, a full three df are directly removed by the relation 52 1 1221 = 1. The translation constraint (4.7) likewise removes three df, directly reducing the dimension of the phase space by six. The failure of the constraints to remove a full six df from the T 2 system is due to the triviality of the trace relation: The condition WcAWDB-1 WcA-1 WDB = 1 shows WDB 1 = WC A WD B 1 WC A 1 SO that Tr WD B1 = Tr ( WC A WD B-1 WC A-1 ) = Tr WDB-1 • This equation put no conditions of the transformation WDB. This, together with the corresponding translation "non-constraint", supplies the unexpected extra df in the torus system. Each constraint also removes a gauge degree of freedom, that gauge transformation generated by the constraint. Thus the dimension of the phase space over all genus g > 1 surfaces is (2g holonomies © 6 df per holonomy) — (6 constraints + 6 gauge choices), giving dimension 12g-12, except for the genus 1 torus, which has an unexpected 12-8=4 degrees of freedom. 4.3 A Model of T 2#712 While the results we have found are quite explicit, they are still based on some unspecified octagonal tiling and an abstract collection of Lorentz transformations and translations. We now propose to build an actual physical model (as physical as (2+1) can be...) for the tiling and the holonomies. We will see, though, that to construct a model, we have Chapter 4. A Two-Holed Torus^ 52 to simplify the geometry with high symmetry, eliminating the translation components IP of the holonomies altogether. 4.3.1 Tilings To construct a closed surface without boundary, like T 2#T 2 , one abstractly thinks of a plane or similar infinite region modulo some identification. Concretely this can mean covering the infinite region with tiles and identifying sides of the tile in pairs. The plane R2 , however, cannot be tiled with all regular polygons. Suppose the tiles are regular p-sided polygons. The interior angle at each vertex of a p-gon is w (P-2) . If q such tiles meet at every vertex then each tile contributes 29 so that 7r(P-2) = —21r or (p-2)(q-2) = 4. The solutions to this condition are {p = 3, q = 6}, {p = 6, q = 3}, and {p = 4, q = 4} which correspond to covering the plane in triangles, hexagons and squares, respectively. The latter is the tiling we use to construct the torus T 2 . Clearly there is no integral value of q for which p = 8 is a solution, meaning the R2 cannot be tiled in octagons. Instead we look to a hyperbolic plane where the sum of the angles in a triangle is less than 7r, and the interior angle of regular p-gon is less than 74P-2) . The neighborhood of a vertex where q hyperbolic p-gons meet is a patch of R2 and still requires a full 27r radians. Each tile contributes an angle of aLr. Since the p-gon is hyperbolic , r < w(P )q P 7 or (p — 2)(q — 2) > 4. One of infinitely many solutions to this condition is {p = 8, q = 8}. While it is not clear how an octagon can regularly cover a hyperboloid, one must recall that the hypersurface of constant time is embedded in Minkowski, not Euclidean, space so that every point on the hypersurface is the same (proper) distance from the origin, much like the 2-dimensional surface of a sphere in Euclidean R3. Chapter 4. A Two-Holed Torus^ 53 4.3.2 Construction of an Octagonal Tile The constraints tell us R 23 a b = 0, so the tile we construct must be flat. Consider the simplest flat 3-dimensional space, Minkowski. In polar coordinates, the 3-dimensional Minkowski metric is ds2 = —dt 2 dR2 R2 d02 . Change coordinates (t, R, 0) to (T, p, 0) defined by t^T cosh p R^sinh p.^ (4.11) Inverting this transformation shows = +0 2 — R2 p = tanh -1 — R . Observe that r is invariant under Lorentz transformations, while p is the magnitude of the Lorentz boost which takes R 0 out to R sinh p. We exploit the invariance of r under Lorentz transformations. By building the tile on a surface of constant r, points on the tile will be connected with merely Lorentz, rather than full Poincare, transformations. Finally, define a new coordinate r = Ttanh( e )^ (4.12) and consider for simplicity the r = 1 hypersurface. This coordinate transformation projects the T = 1 hypersurface onto a unit disk with infinity at r = 1, much the same way the stereographic map projects R2 onto the 2-sphere. The metric on this surface is conformally flat: 4^r2do.2)do-2 = ^ (dr2 (1 — r2)2 Chapter 4. A Two-Holed Torus^ 54 Because this disk is conformally flat, angles are preserved between the unit disk and the (T, p, 0) coordinate system. To construct the tile, we piece together 8 identical curves chosen in the following way. In the original Minkowski space, consider the intersection of the T = 1 hyperboloid and the "vertical" plane y = 0. This curve lies on the hyperboloid "above" the y-axis and can be parametrized by -y(A) = (t(A) = A, R(A) = NA 2 - 1, 0(A) = for A > 1. Now Lorentz-boost every point in the y=0 plane in the x-direction by some magnitude p; the points which lie on the y-axis are unaffected by this transformation and the y = 0 plane is 'tilted' in the x-direction. With increasing boosts the intersection curve y(A) moves away from lying over the y-axis until finally with an infinite boost, the plane has tilted by 7r/4 and is just tangent to the hyperboloid at x = +oo. With arbitrary but finite boosts by p, the family of curves -y(A; p) is parametrized by t(A) = A cosh p R(A) = VA2 cosh 2 p — 1 0(A) = tan -1 (+\/A2 — 1 A sinh p ) One can check that y(A) = R(A) sin 0(A) remains unchanged under the p-boost, so that the plane is tilted without any stretching. On the unit disk with coordinates (r, 0) found by projecting down the T = 1 hyper- surface, consider the collection of circles centred outside the disk which intersect r = 1 orthogonally (See Figure 4.3). The arcs within the unit disk are the geodesics of this Poincare Disk model of hyperbolic geometry [8]. Parametrize in 0 the arc within the unit disk of a circle of radius "r: Chapter 4. A Two-Holed Torus^ 55 Figure 4.3: Circles of radius F intersect the unit disk orthogonally. The arcs within the unit disk are parametrized in O. Chapter 4. A Two-Holed Torus^ 56 r(0)^+ P2 COS 0 -^+ .7"2 ) COS 2 0 - 1 tan 0 E [ - , . This circle is centered a distance 1/1 +^> 1 from the origin. Transforming this family of curves r(0; F) with parameter F back to Minkowski coor- dinates, we find t(0) = 1 + r(0) 2 1 — r(0) 2 R(0) = 2r(0) 1 — r(0) 2 (^0 = e^) Comparing this with the family of intersection curves y(A; p) we see t (A) 2 — R(A) 2 = 1 = t(e ) 2 — R (19 ) 2 so that both families of curves lie on the T = 1 hypersurface. Furthermore, by comparing the 0 = 0 points of both families we find the correspondence between p and "7-. and finally that these two families of curves are identical. That is, the curve in Minkowski space where the y 0 plane, tilted by tanh(p), intersects the T 1 hypersurface becomes the arc of a circle of radius F(p) which orthogonally intersects the boundary of the unit Poincare Disk. By simply rotating these curves about the origin, we can piece together arcs in the unit disk to form an 8-sided figure. We must now find the value of the parameter F , or equivalently p, which produces the correct tile. The polygonal tile we are constructing is regular, so the 8-sides must be spaced at equal intervals of 7/4. Consider the figure produced by laying down 8 arcs of radius F centered at radius N/1 +^on the 8 7/4-'spokes' (See Figure 4.4). When i 0 the arcs belong to small circles centered just beyond r^1 and the arcs do not intersect (Figure 4.4(i)). At some larger F when each Chapter 4. A Two-Holed Torus^ 57 Figure 4.4: For various values of F, arcs (i) do not intersect, (ii) are tangent to one another, (iii) form an almost regular octagon about the origin, and (iv) intersect with angle 7r/4. Chapter 4. A Two-Holed Torus^ 58 arc just intersects the two neighboring arcs, the angle at the intersection of two adjacent arcs is 0 because all arcs intersect orthogonally with r = 1 (Figure 4.4(ii)). When r oo, the arcs become diameters of the unit circle, and at large r, the 8 arcs intersect to form an 8-sided figure about the origin which is very nearly a regular octagon (Figure 4.4(iii)). The angle between adjacent arcs of this figure is almost - 147-r, the interior angle of a regular plane octagon. For each value of r we amputate the legs of the 8-sided figure about the origin and call the result an octagon. We must choose the value of r which generates an octagon whose adjacent sides intersect at an angle ((Figure 4.4(iv)) so that 8 such tiles will supply the 27r radians about the vertex. On the conformally flat Poincare Disk, we can use plane geometry to find r ( 4 ) = 1 2 + 2V-2- • (4.13) This r corresponds to a boost magnitude of  ri^) p(i) 1ln [ 1 + 1 r^ ln  1 — vl F(D 2 F(i) • (4.14) The magnitude p(i) generates a tile on the T = 1 hypersurface. The construction can be repeated for hypersurfaces at arbitrary 7, but the result is the same. This magnitude is actually independent of 7 and generates curves on all hypersurfaces of constant 7 from which these octagonal tiles can be constructed. 4.3.3 Holonomies The 8-sided figure on the hypersurface of constant time T is the tile we represent schemat- ically in Figure 4.1(i). For simplicity, suppose the centre of the octagon lies over the origin and that the 0 0 ray bisects side A. Corner 1 lies at 0 = -1, corner 2 at 0^corner 3 at 0 =^and so on (See Figure 4.5). The transformations W between the identified sides can be easily found by recalling the procedure used to construct the curves which Chapter 4. A Two-Holed Torus^ 59 Figure 4.5: The octagonal tile lying on a hyperbolic hypersurface of constant time r. In Minkowski space, every point on the tile is the same distance 7 from the origin. Chapter 4. A Two-Holed Torus^ 60 became the sides of the octagon. Hereon, p will refer to the value p(i) produced from i'(i). Just as a boost in the y-direction by p drops the curve over the x-axis down to form side C of the tile, a boost in the (—y)-direction by p (or equivalently a boost in the y-direction by —p) will lift side C back up to a curve lying over the x-axis. Rotate this curve by -Fi (+ to generate the right orientation), and boost it by p in the x-direction to drop it back down onto side A. This sequence of SO(2,1) transformations maps side C onto side A. As SO(2,1) is a group, the composition of the three is a single Lorentz transformation, which we call WCA• A general Lorentz boost of magnitude 1u in the 0-direction is i -+ A(,u, 0)Y where cosh Lt^sinh ,u cos 0^sinh ,u sin 0^N sinh ,u cos 0 (cosh ,u — 1) cost + 1 (cosh — 1) cos 0 sin 0 sinh ,u cos 0 (cosh ,u — 1) cos 0 sin 0 (cosh ft — 1) sin e c + 1 / Rotations about the origin in the xy-plane by angle 0 are produced under the transfor- mation I 1^0^0 R(0) = 0 cos 0 — sin 0 \0 - sin zi)^cost' The transformations WCA , then, is given by : maps side C onto side A 7r^ 7r = A(p, 0)R(-2-)A(p, 3 ) cosh 2p 0 — sinh 2p sinh 2p 0 — cosh 2p 0^1^0 This method of lift-rotate-drop gives each of the W transformations: WCA WCA Chapter 4. A Two-Holed Torus^ 61 maps side D onto side B —4 )R(-2 )A(P, —4 ) 1^cosh 2p^sinh 2p v 2 sinh 2p \v sinh 2p^sinh 2 p^— cosh 2 p sinh 2p^cosh2 p^— sinh 2 p maps side G onto side E ^ A(p,7r)R(i^ 2 ) cosh 2p 0 sinh 2p — sinh 2p 0 — cosh 2p ■ 0^1^0 maps side H onto side F (0, ) ( 7r ) (0 ,^)A., , 4 , R. 2 , A , 4 , cosh 2p^—  sinh 2p v,-2 sinh 2p \ sinh 2psinh 2 p^— cosh 2 p sinh 2p^cosh 2 p^— sinh2 p 4.3.4 The Fields tiab and pa The symmetry of the model for T 2#T 2 allows us to find a single transformation WcA that maps side C onto side A. Because WcA is independent of the pair of identified points on the matching sides, we see Wcki = 0 as required by (2.5). The same applies to WDB, WGE, and WHF. The W transformations, recall, are the link between the values of the field U on opposites sides of a boundary. We attach the W onto the octagon as we did for the square tile of T 2 , by regarding the holonomy as a "phase difference" between identified points. WDB WDB WGE WGE WHF WHF WH F 1 WG EWH FWGE or WDB WC AWD B 1 WC A-1 WDB Chapter 4. A Two-Holed Torus^ 62 Figure 4.6: Values of the phase U on the corners of the r#T 2 's octagonal tile. The constraints guarantee U(1) is well defined. Again we start on the corners of the tile (Refer to Figure 4.1(i)). Since only the phase difference between identified points matters, there is an arbitrary constant phase. If we set U(3) WcA-1WDB, the U field nicely decouples into ABCD and EFGH, as we found before (See Figure 4.6). By starting on corner 3 and transforming to each corner under the W's, we find on corner 1 both U(1) = WH F-1 WGEWH FWG E-1 and U(1) = WC A WD B-1 WC A 1 WD B . The constraint (4.5) ensures U(1) is well defined. It is now a simple matter of interpolating these vertex values over the whole tile, while preserving the phase difference between identified points. A simple choice is a linear interpolation between the corners along the boundary, coupled with a radial interpolation under which the U field, now defined on the boundary, decays down to the identity at the center of the tile. All the XA \ YA. / = WCA Chapter 4. A Two-Holed Torus^ 63 transformations W are in SO(2,1) and connected continuously to the identity. These two interpolations together give a continuous field over the tile with points on the boundary differing by the appropriate transformations W. Smoother interpolations can be used, if necessary. The last component of this model is the field pa, introduced to write the dreibein as eA a = DA pa . The choice we make is a very simple one; too simple, perhaps, for it sets the translations H to 0. Consider the choice pa = xa = (t, x, y) in Minkowski space, giving ex = (0, 1, 0) and ey = (0, 0, 1). Recall the solutions to the equations of motion are those for which Rii, ab = 0, so in the solution space we can take e t = (1, 0, 0) as well. The two fields U and p completely determine the holonomies. We found in (2.8) W21 ab P2 a -I- 1121 a = /h a where p 1 and p2 are the values of p on either side of the identified boundary. The transformations WCA, for instance, maps (t, x, y) on side C onto (t, x, y) on side A: But this is exactly a = WCA ab PCPA ^7 showing the translation fI cAa is not needed to reproduce the discontinuity in p across the CA-boundary. This choice of pa is "elegant" because it so easily exhibits the discontinu- ities required for the holonomies to be non-trivial. The solution H = 0 just means the configuration is (momentarily) stationary. This is consistent with the geometry of the tile being independent of the time 7 on the spatial slice, a property we found in (4.14) when we built the tile. Chapter 5 Canonically Conjugate Variables and Quantization We have seen from the two surfaces studied here that the Einstein action, on the solutions to the equations of motion and with the constraints imposed, is of the form 17'2 = (wDB-1 11DB){1VcA} — (wcA-1 1-1cA){14/DB} iT2#T2 = (WDB 1 11cA) {WDB 1 1/1} — (WDB 1 11DB) {wcief21} —HGE{WHFI12} + ( WGEwHF 1 11HF) {WGEC22} dThis result is not the canonical p4 form of the action because {W} is not —dt W but instead, recall, {W}. = wbxwexcabc While this is not the vector ciW tangent to the space of W's, {W} is still tangent to some W-space, related in a one-to-one way with the tangents of the space of Lorentz transformations. The translations H are not (quite) the momentum conjugate to the Lorentz transformations W, although the association or the H with infinitessimal trans- lations of the W is very suggestive. It is possible to bring the result even closer to the canonical form to reveal more about the phase space. Every Lorentz transformation can be written in the form W = ei r"1" where the Ja are the 3 x3 Lorentz generators introduced in §1.2. The components 11a can 64 Chapter 5. Canonically Conjugate Variables and Quantization^ 65 be determined as follows. Write W = eA for some A. As BeAB -1 = e l3A13 -1 , eA^W = W W W -1 = WeAW -1 = ewAw-1 so that A = WAW- 1 or [A, W] = 0. If we write A = 7rJa , then 0 = qa[Ja , W] There are only three distinct components in this matrix equation due to the symmetry of the Ja . The three equations are not linearly independent, though (the right-hand-side has vanishing determinant, as det(Ja )=0). Another relation is needed to solve for the components rya. Determinants will not suffice: As det(W)=1 1 = det(W) = eTrA Thus TrA=0 and any constant times the matrix A will not change the determinant of W. Instead, we consider NTr W = Tr(eA ) = eA l + eA2 + eA3 where A i , A2, )t3 , the eigenvalues of A, are functions of ,.. This comes from the property that A can be written DAD' for some D and diagonal A, so that Tr eA = Tr( eDAD-1 ) = Tr( DeAD') = Tr e' by the cyclic nature of the trace. Since A is diagonal, Tr e A = e'' -+ + eA3. This relation, along with the two equations arising from [A, W] = 0, suffice to completely determine rf. Now we can write Ili = cii (ena.1“) Chapter 5. Canonically Conjugate Variables and Quantization^ 66 ^ow a ^ ^ = ( th a ) q ^ Wbx = ( aWirt )bx ir . > Therefore the action can be re-written as E^(ll a a_aw \ b x wcs cabc) 7y‘ re i holonomies [Will] This action is now in canonical /*form. This expression, however, is not as well behaved as one would like. In a neighborhood of each W in the space of Lorentz transformations, we can attach 77 coordinates and consider (OW107(). Writing W = e aJa is only shorthand + if ja + [qua, q bfb i 4_ . . .for the power series 1^ , so derivatives with respect to qz may not even exist. Furthermore, there is no guarantee that the coordinate patches around each W are part of some global coordinates over which we can compare the values of (OW/Oqz) for two different holonomies. With these problems in mind, let us speculate on the canonical coordinates. The configuration space variables are qa, the coordinates in the space of Lorentz transfor- mations. This space is now represented as a vector space with basis Ja . The relations [Ja7 Jb] = Cabc Jc show the structure constants are c ab', suggesting a non-trivial geometry. The momenta conjugate to these coordinates are related to the translations Ha. Under this association, we would expect the Ha, as momenta, to generate translations in the configuration space. And this is what we find in looking at the equivalence of the con- straints: we make the association Aa --> pa for Aa = fabc 8W bd Wdc • It is easy to verify that cab , = (Ja ) b, — these Aa are (infinitesimal) translations about the vector space of W's. While the interpretation is by no means rigorous, it is very suggestive: The Lorentz components W of the holonomies are the configuration coordinates. Translations in this space, generated by the momenta Ha ,s, OW W)Ja, are infinitesimal Lorentz transforma- tions. This is just what we expect for translation in a space of Lorentz transformations. fibre over * fibre over * global ISO(2,1) gauge transformat configurations of r#T 2^the surface holonomy (i ) Chapter 5. Canonically Conjugate Variables and Quantization^ 67 Figure 5.1: (i) A principle ISO(2,1)-bundle over the base space T 2#T2 . (ii) The 'phase space' is like an ISO(2,1)-bundle over the space of configurations of the surface. There is another, more mathematical description of this system. Vectors in the tan- gent space over each point of the surface, T 2 or r#T 2 , are subject to the action of the Poincare group ISO(2,1). This group leaves the Einstein action invariant, and its generators obey a Lie algebra. These are the ingredients needed to define a principle ISO(2,1)-bundle over the surface. The group action moves us along each fibre over the surface without changing the Einstein action. The holonomy at a point on one of the in- contractible loops is the element [Win] of the gauge group ISO(2,1) relating two distinct points in the fibre over this base point (See Figure 5.1(i)). In the language of princi- ple bundles, the phase space is also like an ISO(2,1)-bundle, this one over a base space consisting of configurations of the 2-dimensional surface (See Figure 5.1(ii)). The one point in the base space we have found represents the regular octagonal tile constructed above. Other points represent asymmetric tiles and their corresponding collections of holonomies. The momenta H are infinitesimal generators of the group action in each Chapter 5. Canonically Conjugate Variables and Quantization^ 68 fibre. The W and II vary as we move about in each fibre from one horizontal lift to the next under global gauge transformations. Yet the projections down to the base space of holonomies remains unchanged. If we rotate the tile, or globally boost it to a new location, a similarly transformed collection of holonomies is produced. The tile representing this new surface, though, is essentially unchanged, merely displaced. The canonical variables W and H are phase space coordinates in this bundle: The W are configuration coordinates in the fibre over a basepoint, a particular configuration of the tile representing the surface. The momenta H generate translations along this fibre un- der the infinitesimal group action. Different horizontal lifts, all differing by global gauge transformations, have different coordinates (W, II), but project down to the same model of the surface, perhaps displaced but leaving the Einstein action invariant. In the full ISO(2,1) representation, the Poincare holonomies can be written as [w(701 11 (0-a)] = enaJa+, apa where Pa , introduced with Ja in §1.2, generates translation in the xa direction. In the case of the torus T 2 , we can see T/CA a = (0, 0, p), 0-cA a = (0, 0, a) and 77DB a = (0, 0, au), 0DB a = (0, 0, b). For the two-holed torus r#T 2 , the transformations WCA, • • • , WHF are compositions of boosts and rotations and the corresponding 71 a must be evaluated by the method outlined above. The conjugate translations IIc A , ... , IIHF all vanish so ocA " = (0, 0, 0),..., UHF" = (0,0,0). This exponential ISO(2,1) representation of the holonomies matches the Chern-Simons approach. There, the gauge field is the collec- tion of flat connections A t, that transform under the action of an infinitesimal parameter u -,--- raJa +paPa . That is, the difference between two horizontal lifts along the same fibre, (W, II) and (W', II'), which project down to the same model of the surface, is generated by an infinitesimal Poincare transformation u. The coordinates W and H determine the point on the fibre over the base space. Chapter 5. Canonically Conjugate Variables and Quantization^ 69 When the canonically conjugate variables of a classical Hamiltonian system are known, quantization in the Schrodinger picture involves expressing the variables as operators on a Hilbert space of wave functions W. The observed values of the coordinates and mo- menta are eigenvalues of coordinate and momentum operators. Unfortunately our choice of canonical variables W and H have ill-defined operators, for the following reason. We suggest that the configuration space has coordinates 7/a, the components of the vector representing W in the vector space spanned by {J1, J2, J3} . The quantum mechanical 7/a are the spectrum of a position operator Na: Na T = qa kif While J3 and J2 generate boosts in the x- and (—y)-directions, respectively, and have spectra 7/3 , 7/ 2 E ( — oo, oo), recall that J1 generates rotations in the xy-plane. The eigen- value of this operator is the angle of rotation, 7/ 1 ti 0: N 1 (0)P =eW. Because rotations differing by 27r give the same reading 0, we require both N1 (0) 'P = 6411 1V 1 (0 + 27) = . Thus N 1 is no longer a linear operator on the Hilbert space of T's. This tells us that 0, or 7/ 1 , by itself cannot be an observable. Instead, some function of the operator N 1 , elN1 for instance, is needed for a well-defined operator. At the same time, the Hilbert space is no longer a vector space, but has a "cylindrical" shape (See Figure 5.2). Because the rotations are a subgroup of SO(2,1), which is itself a subgroups of the gauge group ISO(2,1), perhaps this problem can be circumvented by re-defining the configuration space modulo S 1 : R3181 0 R3/si 0 • • • ® R3is1, one term for each of the Chapter 5. Canonically Conjugate Variables and Quantization^ 70 Figure 5.2: J2 and J3 are generators of Lorentz boosts while J i generates rotations. Because the same eigenvalue 0 results from rotations by 0 + 2irn for all integers n, the coordinate 7/ 1 0 is periodic and the Hilbert space of wave functions has a "cylindrical" geometry. Chapter 5. Canonically Conjugate Variables and Quantization^ 71 2g holonomies of the flat genus g surface. In any event, this complex structure of the phase space arises from the choice of W and H as canonically conjugate variables, and this interpretation is only speculative, based on qualitative observations. Chapter 6 Conclusions 6.1 Generalization of the Result The ease with which we jumped from the torus to the two-holed torus suggests this formulation of the Einstein action can be applied to all higher genus surfaces. By choosing an appropriate r on the Poincare Disk to produce a regular (4g)-sided polygon with an interior angle of alr- tilings of the hyperbolic plane are produced. Algebraic topology4g describes a genus g surface, a connected sum of g tori, in terms of 2g cycles [9]: A -1 A-1 A -1A-1^-1A-1u i a 2 u2 a2 u2 • • • a l/ 1_4 11gg g g We use this expression to read off the identification of sides of the tile. In the r#T 2 case, a l b 1 adbTl a 2 b2 aZ1 b2l H ABCDEFGH tells us to identify sides A&C, B&D, E&G, F&H, each with the orientation of the 2 identified sides reversed. That is, choose a representation of the fundamental group of the surface in the gauge group ISO(2,1). Each additional "hole" formed by the surface simply adds two more holonomies, [W 29+1 ,IH2g+1 ] and 1W M29+21 --2g+2] to the collection that determines the geometry of the surface. The constraint that the transformation producing a complete circuit of the vertex is the Identity ensures each additional torus glues smoothly to the rest: W1 W2 -1 W1 -1 W2 • • • W2g-}-1W2g+2 1 W2g+0 W2g+2 = 1 W2g+2 1 W22+1 W2g+2^=^W2-1 W1 -1 W2 • • • W2g -1 W2g 1 W2g -1 1 W2g T92+ 1 # 7-112 #^# 29 ) 72 Chapter 6. Conclusions^ 73 These transformations can be attached to the tile with the "phase difference" approach. The closure relation on the W's guarantees a well-defined U field. Analogously, two more terms, (IT)\ ---- / 2g+1{W}2g+1 and (n)__,2g+2{W}2g +2, are added to the action to account for these new degrees of freedom. The problems of conjugate variables and quantization are still present, but not further obscured by the increase in genus. 6.2 Future Research The Chern-Simons action based on gauge field theory and the usual Einstein action of GR are two different representations of the same system. The former deals with the gauge group theory of a principle ISO(2,1)-bundle over a (compact) surface, while the latter looks at invariants of the Einstein action over a (2+1)-dimensional splitting of spacetime. By comparing the ISO(2,1) gauge invariance of the Chern-Simons field = eµ Pa c.umaJa with the usual Lorentz + diffeomorphism invariance of the dreibein e: and connection w,,a, Witten [1] shows that the 2 representations differ only by a transformation that is part of the gauge group, and thus is inconsequential. Therefore, the general results of principle bundles and gauge theory can be used to study the specific case of (2+1)-dimensional spacetime. There is reason to suspect, however, that the two approaches are equivalent only under special circumstances. Recall from §1.2 that the variation of A t, is given by SA A = Se: Pa + Sw:Ja where a abcse a abcE C ilbrc — E WgbPc scot: = —Ta — abc Lc)br c • Under the substitution pa = V iL e tta these transformations coincide, on the solutions to the equations of motion, to infinitesimal diffeomorphisms and Lorentz transformations, the invariants of the Einstein action. In the case where e t," is everywhere non-vanishing, Chapter 6. Conclusions^ 74 this gauge transformation is a physical coordinate transformation generated by the vector field 174 = paella . Note that Va = Va(eaa ), where eac, = g 4Lve„• Now on all but the torus T 2 , the tangent vector field to the spatial surface must have at least one singularity. Suppose the spatial ei a fail to span the tangent space at the point V'. Where does this point "slide" to under the diffeomorphism generated by pa = Vae ir We cannot say, as the vector field Vµ(e) cannot be determined! The perfectly acceptable gauge transformation is no longer a coordinate transformation and hence is no longer physical. When we drop the requirement that the results be physical, we are left with an exercise in mathematics, not a theory of spacetime dynamics. There is another problem related to this singularity. Suppose we are at a point in flat spacetime where the dreibein is not singular. At this point, we perform the gauge transformation with parameter pa chosen such that pa4 = e4a, so that Sem(' = — e4a and the dreibein becomes singular. This simple gauge transformation does not have a corre- sponding coordinate transformation. Furthermore, we cannot perform a gauge invariant coordinate transformation to get away from this singularity, for the diffeomorphism must be generated by VI-Le lia which vanishes for all V. This suggests we can take two very different spacetime (Euclidean and Minkowski R3 , for instance) and glue them together at this point e. Moreover, there are gauge transformations that allow us to pass through this point from one spacetime to the other. It may even be possible to extend this "phe- nomenon" to a whole region, allowing us to construct a manifold whose metric changes signature. Clearly the equivalence of the Chern-Simons and Einstein actions has inter- esting details as yet unexplored. One more question raised by the results deals with a subtlety of the dreibein approach that suggests this formalism is somehow "larger" than GR. On all but the torus, the vector field tangent to the spatial surfaces must vanish at one or more points. This means the full spacetime metric must either be singular, or at least time-like. One of the Chapter 6. Conclusions^ 75 gauge transformations allowed by the dreibein representation of the Einstein action is the internal transformation coa --> e lf + D„pa. In general, this transformation will change the metric components la,tiv = epaeva• At the point(s) where the metric g, is time-like, a gauge equivalent metric .g,,,, may be space-like or null — clearly this is not a coordinate transformation. This phenomenon is prohibited in the usual form of GR written in terms of the metric and its derivatives. There is a freedom allowed by the dreibein formalism not allowed in GR. It may be possible to construct an explicit model and study it in analogy with the gauge theory explanation for the Bohm-Aharonov effect.[10] Overlapping coordinate patches may be related by a gauge transformation, but not a coordinate transformation. General Relativity is simpler in (2+1)-dimensions in the dreibein formalism because the conditions that describe the slicing of 3-dimensional spacetime force the spatial hy- persurfaces to be flat. We have considered the cases where these 2-dimensional slices are folded up into compact surfaces, the torus T 2 and the two-holed torus T 2#T 2 . While they remain flat, removing the geometric degrees of freedom, the topology of these higher genus surfaces becomes important. The Einstein action becomes a functional not of the geo- metric quantities g,„ but the topological quantities [WIIn, the ISO (2,1) holonomies over the surface. The flatness of the genus g surface removes all but 2g distinct holonomies, and the action is written entirely in terms of these Poincare transformations. From the form of the reduced action, we can speculate of the dynamical variables and the phase space. It appears that the configuration space is the space of Lorentz transfor- mations while the conjugate momenta lie in the collection of spacetime translations. We make this interpretation because the translations H a are related to infinitesimal Lorentz transformations, just as classically, momenta generate translation in configuration space. While the action we have produced does not truly reveal the dynamical variables of Chapter 6. Conclusions^ 76 this spacetime, and quantization of the phase space is not obvious, the method employed to reduce the action is quite revealing. It is apparent that the correct phase space of the classical conjugate variables is not simply R.2" . It is most likely curved, and quantum mechanics on curved space is a problem that will not be tackled here. The result raised interesting questions about the gauge structure of spacetime, and also, therefore, about (2+1)-dimensional gravity over compact surfaces. Bibliography [1] Witten, E. 1988, "2+1 Dimensional Gravity as an Exactly Soluble System," Nucl. Phys., B311, 46-78. [2] Misner, C.W., Thorne, K.S., and Wheeler, J.A. 1973, Gravitation (San Francisco: Freeman). [3] Wald, R.M. 1984, General Relativity (Chicago: The University of Chicago Press). [4] Carlip, S. 1990, "Observables, Gauge Invariance, and Time in (2+1)-Dimensional Quantum Gravity," Phys. Rev. D42, 2647-2654. [5] Carlip, S. 1989, "Exact Quantum Scattering in 2+1 Dimensional Gravity," Nucl. Phys., B324, 106-122. [6] Steenrod, N. 1951, The Topology of Fibre Bundles (Princeton: Princeton University Press). [7] Arnowitt, R., Deser, S., and Misner, C.W. 1962, "The Dynamics of General Rel- ativity" in Gravitation: An Introduction to Current Research, ed. L. Witten (New York: Wiley). [8] Kelly, P., and Matthews, G. 1981, The Non-Euclidean, Hyperbolic Plane (New York: Springer-Verlag). [9] Massey, W.S. 1967, Algebraic Topology: An Introduction (New York: Harcourt, Brace & World). [10] Ryder, L.H. 1985, Quantum Field Theory (Cambridge: Cambridge University Press). [11] Cornwell, J.F. 1984, Group Theory in Physics, Vol.2 (London: Academic Press). [12] Green, M.B., Schwarz, J.H., and Witten, E. 1987, Superstring Theory, Vol. 2:Loop Amplitudes, Anomolies and Phenomenology (Cambridge: Cambridge University Press). Chapters 4 and 12 contain an introduction to the dreibein formalism. 77 Appendix A Properties of Lorentz Transformations Uab By definition, a SO(2,1) Lorentz transformation Uab must keep the metric lab invariant: uac ubcocd = Ti ab From this we see uctc ubc = lab Or  uac ?lb c _ Sg, (A.1) A similar relation can be derived can be derived from this one. Re-write (A.1) as ud c ridaubc = riab so that ud c 7idauboab = 1 .^ (A.2) Suppose, in all generality, that the transformation U has both different left and right inverses: u(u -1 ) R = 1^(u- ')Lu = 1 . Together, these give (u-1)L u (u-1 ) R (u — 1 ) 1, (u-1 ) 1, = (u-1) R 78 Appendix A. Properties of Lorentz Transformations tiab^79 so that the left and right inverses are the same. From (A.2) we see ridaubc7lab is the right inverse of lid c . Left and right inverses coincide, so ridau bf 77ab /id = which is equivalent to  tra b sab (A.3) A third property of Lorentz transformations is also used in producing the results of Chapter 2. We come across terms of the form Cabc Uas U by U cz • The indices x, y, z are still Lorentz indices, named from the end of the alphabet to clarify the index maniupulations. Relabel the dummy indices b, c: Ea bc Uax ti by tt c, = Cacb teix U ey U bz = —6abctiaxU cyU bz • Comparing the first and last expression, we see Eabcuaxuby IA% is antisymmetric in b and c. Analogous relabelling shows that EabcUaxUbyticz is totally antisymmetric, and therefore must be proportional to the only totally antisymmetric 3-tensor, c xyz : Write CabcUaxilbyticz = A 6syz Solve for A: = exYz f abc Uax U by ticz = det(Uab ) . These Special Orthogonal matrices have unit determinant, so A = 1, and Cabc Uas U by IA% = 6xYz (A.4) Appendix B Setting CJi ab = 0 and e a = 0 We have written w„ab = 73„cdtica udb + tica(ticb,,i) and claimed that because Rij ab = 0, we can find a Uab such that 473, 6 = 0. That is, there is a Uab for which ab =^pcb,,) or Ua a cb,i = ticWib • (B.1) Formally we write the solution of this matrix equation = ef wpdsi` This is not well-defined as /4(x 4 ) will in general have a different value for each path -y(A) integrated over, as de = dyµ (A) dA dA will be different for each path. Instead we write = pef,^ (B.2) where P stands for path-ordering the integral along the path -y. The meaning of a path- ordered integral can be seen as follows. Discretize the path -y(A) over the surface under 80 Appendix B. Setting (:Di ab = 0 and e i a = 0^ 81 0 = Ao < A l < • • • < An = 1 with -y(A k ) = xk. Then the integral (B.2) is the limiting value urn ewP(„,),Axr,, • • • ewp(s 1 )Axii' ewp(x0 )Axt; n—).00 Each term ewP(xk)°4 is a transformation acting on the term to the right, so that (B.2) is an infinite succession of infinitesimal transformations in a direction tangent to the path, parallel transporting along -y(A). If Ax k is tangent to the path at x k then dx(x k ) = d-y(Ak) dA dA Ax k and d ^f x0Pe -Y P d = Pef-Y wP dx1A Wv (Xk) dX 1' ( X1c) or, 14,,, =14w, as required in (B.1). There is no reason, however, for this result to hold for nearby paths. Integrated over a different path "--y between x, and x l , so that -y (5m ) = yin = xn = -y(An ), ti(yrn ) and /4(x,2 ) may have different values. While both dxv Pe f- Y wP dx.P = Pef-Y w" dx1A co,, , p^=^wpdyP tiv 7dyv it is not true in general that, say, pef w jA cist`^coudxt`i^= Pe'^co,, (xk) ,dyv for in general d-y(An )^d-y(-Ani) dx(x,i ) = dA dA ^dA = dy(yrn )•dA Appendix B. Setting cVi ab = 0 and et a = 0^ 82 Figure B.1: An infinitesimal loop of generated by vectors Tµ, S" with area ATAS. The constraints, however, tell us that the spatial components of the Reimann cur- vature tensor vanish, R2 j ab = 0. Recall what the Reimann tensor represents: Parallel transporting a vector Va about an infinitesimal loop (See Figure B.1) changes Va in- finitesimally by SVa = ATASV b Tt'S" 'RAvt a • The change in the vector Va is proportional to Rt,,,b a and to the loop's area ATAS. Notice that the indices ,a, v link the components of the tensor to the paths in the directions TA, S". Returning to our problem, suppose /1(xµ) is calculated along the path -y(A) as in (B.2). When dxil = (d-ya / dA)dA, equation (B.1) holds. We want to show that because Rij ab = 0, the result holds for all paths on the surface:   i(xn ) li(x0wi(xn) dY 2 (yn) for some other ya = yµ ( a) passing through xn. Appendix B. Setting iD i ab = 0 and e i a = 0^ 83 Figure B.2: Two infinitesimally different paths to xn = Consider an infinitesimal change in the path 7 (See Figure B.2). Along coincident paths, U(xo) = 14(y0 ),14(x i) ,14(xn_1) =-14(y„_ 1 ). Whereas U(xn) is calcu- lated in the limit llm ewp(Xn-1)AXt7:-1U(Xn-1) n—).co U(yn) is calculated by traversing the loop of area —Axn_ i Ayn_i. The two paths from Xo = yo to xn = yn are Xo, xi • • ' Xn-27 Xn-17 Xn Yo, Y1 • • • Yn-2) Yn-11 Y1) Y21 Yn • Their difference is the loop Xn Yn-11 yl, Y27 Yn • If U(xn ) had been calculated by path-ordered integration along the varied path, then the Appendix B. Setting c.7.) i ab = 0 and "e i a = 0^ 84 difference in its value would be SUab = xn- Agn-i X 2 Y3 VbRuc a — UacRu b c ) • Only the spatial components are present as the paths lie in a surface of constant time. Since R23 ab = 0, there is no change in Uab because of this infinitesimal change in the path. Macroscopic changes in -y can be built from many infinitesimal ones, the result holding at each stage. The result tells us that the value of U ab at xiL is independent of the path along which the integration occurs. Therefore, in checking that (B.1) is true at any point, we can assume that the path used to calculate WI, in (B.2) is the one for which dx = (d-yi I dA)dA, so that (B.1) holds. Finally, with U,1 = Ucoi at all points on the tile,Co' i ab = 0 must vanish, leaving only the time components undetermined. In the coordinate system where CJi ab = 0, we transformed the dreibein under nwo a In this coordinate sytem, D i pa = io`:i as the spatial C3i ab = 0. We claim that because D[iej Cil = 0, or E [iaji = 0, we can find a particular pa(xa) such that e ma = 0. The solution to the PDE e a^na (B.3) can formally be written as pa = PI e4adxii -y where again P indicates path-ordering along the path -y. The result (B.3) holds along the path where dxa = (d-yi DOA but the result need not be true for different paths. Like the co4a b case above, we interpret the path-ordered integral as a limit. Discretize the path -y(A) so that xk = ry(Ak) and Axk is tangent to the path at, say, x k Axk/2. Appendix B. Setting C.3 i ab = 0 and e i a = 0^ 85 The point where we evaluate does not matter in the limit. The path-ordered integral is the limiting value Ax, 2^ Ax^ Axlim^ 1 e a(x 0 + ^ )A4^ 2 + a(x i +^),A4 +^a(x„  ^A ^2 n n)Ax . 1-i To show that the value of pa(e) is in fact independent of the path to xa, again consider the two paths between x, = yo and x n = y 9., used above (See Figure B.2). The difference between the two paths, recall, is the loop Xn Yn-11 yl, y2, Yn. This loop will contribute to the sum (or 'integral') an amount Axn etia(xn^_t^ AYn-1^Aepa(xn_ i 2 2 ),A4^e a02 AYn-1 ^ e a 01 +^ Axn_i 2^-1^A^2 These paths, and this loop, lie in a surface of constant time, so the displacements Ax k and Ayk have no time-, or 0-, components. Furthermore, we can assume that these displacements are in the spatial 1- and 2-directions, for any loop can be approximated to an arbitrary degree by a tiling of parallelograms whose sides are in the 1- and 2-directions. Thus with Axn_ i = (0, Ax, 0) and Ayn_ i = (0, 0, ay), the loop contributes Ay e2a (xn-1 2 Ax) — e2a (xn-1 AxAy . . Ax In the limit where n -4 oo, this is simply (6 1 72 - 6271 ) AxAy . The constraint tells us e[a3] = 0, so variations in the path do not affect the value of the path-ordered integral. Because of this path independence, we can always assume dx 2 (d-y i /dA)clA so that e l a = Apa . Therefore, e t a must vanish. e1 'L — t + AY) — ei a (xn — t) ( Appendix C Properties of {W} We define the symbol { W} a = Wbx wcx cab, so that H a -r -i- rbsW Wcs Eabc = 11{W} . There are three properties of this symbol which will simply the algebra of reducing the action to a more canonical form. By definition, (un){uW1 = proaubyti/Yxuczwzx cabc = fictuad . ; -, ,u ucz IksWzx cabc Recalling property (A.4) of Lorentz transformations U, = ri d Wyx wzx cabc Relabelling dyz^abc gives =_ Ha Wbs wcs cabc That is, (un){uW} = H{W}^ (C.1) 86 Appendix C. Properties of {W}^ 87 Next, in full component form, H { w1w2} = Ha (wi b dw2 d s)(wi b dw d2 ) 6abc =^dw2c1 s wi c ew2 ex eabc +H a W1 b^d0472 x Wl c e W2 ex Eabc Now the first term can be rewritten by first raising and lowering the e: • b^ b^cdH a Wi d Wi ce dW2 x W2 x Eabc = II ^dW1 Eabc be = H{W1} The second term is simply 11{W1 W2 } so that  II{Wi W2} = 11{ -1/4}+ H{W1l4l2} (C.2) Lastly, using (C.2) just derived, we can write ( vv2 _11-) {w2 -1 wi} (w2 _11-T) {v2 _1 1471} (w2 _1 ro While (W2 -1 11) {W2 -1 W1 } H11141 by (C.1), (w2 _11-)^_ 0472 -1472 _1 1-0 {w214;-1 } = n ({W214/2-1}— = — H{W} as^a (S! = 0. Therefore (W2 -41) { 1472 -1 mo = HA} - {C) (C.3)


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