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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 ATWO-HOLED TORUS, T²# T²ByPeter R. NewburyB.Sc.Hons., University of Manitoba, 1990A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF MATHEMATICSINSTITUTE OF APPLIED MATHEMATICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAMarch, 1993© Peter R. Newbury, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature) Department of  MathematicsThe University of British ColumbiaVancouver, Canada -Date ^12.^OVJA. 1993DE-6 (2/88)AbstractResearch into the relationships between General Relativity, topology, and gauge theoryhas, for the most part, produced abstract mathematical results. This thesis is an attemptto bring these powerful theories down to the level of explicit geometric examples. Muchprogress has recently been made in relating Chern-Simons gauge field theory to (2+1)-dimensional gravity over topologically non-trivial surfaces. Starting from the dreibeinformalism, we reduce the Einstein action, a functional of geometric quantities, down toa functional only of the holonomies over flat compact surfaces, subject to topologicalconstraints. We consider the specific examples of a torus T 2 , and then the two-holedtorus, T2#T2 . Previous studies of the torus are based on the fact that the torus, and onlythe torus, can support a continuous, non-vanishing tangent vector field. The results weproduce here, however, are applicable to all higher genus surfaces. We produce geometricmodels for both test surfaces and explicitly write down the holonomies, transformationsin 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 suggeststhe quantum mechanical version of the theory exists on curved spacetime.iiTable of ContentsAbstract^ iiList of Figures^ vAcknowledgements^ viiIntroduction^ 11 The Dreibein Formalism^ 51.1 The Dreibein Formalism of General Relativity  ^51.2 Chern-Simons Gauge Field Theory over the Poincare Group  ^92 The Einstein Action^ 132.1 Reduction of the Einstein Action  ^163 The Simple Torus^ 223.1 The Action over a Torus ^  223.2 Consistency of the Constraints  283.3 A Model of T 2 ^  313.4 An Alternative Approach to the Torus ^  354 A Two-Holed Torus^ 394.1 The Action over a Two-Holed Torus ^  394.2 Consistency of the Constraints  474.3 A Model of T2#T2 ^  51iii4.3.1 Tilings  ^524.3.2 Construction of an Octagonal Tile ^  534.3.3 Holonomies ^  584.3.4 The Fields Uab and pa  ^615 Canonically Conjugate Variables and Quantization^ 646 Conclusions^ 726.1 Generalization of the Result  ^726.2 Future Research ^  73Bibliography^ 77Appendices 78A Properties of Lorentz Transformations Uab^ 78B Setting C'Ji ab 0 and ez a = 0^ 80C Properties of {W}^ 86ivList of Figures0.1 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.   22.1 The difference -yi — 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.   153.1 (i) Labels for the sides and corners of the tile, and identifications of sidesA&C and B&D. (ii) The neighborhood of a vertex where four tiles meet. 233.2 The transformations W across the identified edges (i)C&A, (ii)D&B. . . 243.3 Values for the 'phase' U at the corners of the tile.   343.4 Tiles of the tangent space over the torus are parallelograms in boost-trans-lation space^  364.1 (i) Label for the sides and corners of the tile, and identifications of sides.(ii) The neighborhood of a vertex where eight tiles meet. ^ 404.2 The transformations W across the identified edges (i)C&A, (ii)D&B,(iii)G&E, and (iv)H&F ^  424.3 Circles of radius i- intersect the unit disk orthogonally. The arcs withinthe unit disk are parametrized in 0^  554.4 For various values of F, arcs (i) do not intersect, (ii) are tangent to oneanother, (iii) form an almost regular octagon about the origin, and (iv)intersect with angle 7/4^  57v^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 fromthe origin ^594.6 Values of the phase U on the corners of the r#T2 's octagonal tile. Theconstraints guarantee U(1) is well defined.  ^625.1 (i) A principle ISO(2,1)-bundle over the base space r#T2 . (ii) The 'phasespace' is like an IS0(2,1)-bundle over the space of configurations of thesurface.   675.2 J2 and J3 are generators of Lorentz boosts while Ji generates rotations Because the same eigenvalue 0 results from rotations by 0 + 2irn for allintegers n, the coordinate 7/ 1  0 is periodic and the Hilbert space of wavefunctions has a "cylindrical" geometry.   70BA An infinitesimal loop of generated by vectors T4 , Su with area ATAS. . . 82B.2 Two infinitesimally different paths to xri = y. . ^  83viAcknowledgementsI would like to thank my supervisor, Bill Unruh, for guiding me through this project. Histop-down insight met with my ground-up exploration to produce the results. I am alsograteful to David Austin for reading this thesis from the point of view of a mathematiciansympathetic with the pedagogical style of physics, and to Nathan Weiss for his criticalreading 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. Findingher was the most unexpected, but easily the greatest result of this whole endeavor.viiIntroductionUsing General Relativity (GR) to study of the dynamics of a spacetime generally followsa 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 thespatial slices and constraints that govern the splitting; vary this action with respect to thedynamical 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 canbe 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 thantackle 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 splitinto (2+1)-dimensions — 2 spatial and 1 temporal. While the results of these studies areclearly 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)-dimensionsis much simpler and the physics of the space is not (so) lost in mathematically difficultequations that much be examined.With 3-dimensional spacetime sliced into (2+1)-dimensions, a system is defined bythe choice of the 2 spatial dimensions. The simplest cases are those where the spaceis a plane or some similar infinite region. More interesting problems arise when the 2dimensions are wrapped up into compact surfaces, especially when the surfaces are notsimply (topological) spheres. In this thesis we will consider first the torus T 2 and thena genus 2 surface, the 2-holed torus T*T2 (See Figure 0.1), and the interplay betweenthe differential geometry of GR and the non-trivial topology of this surface.1Figure 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 becontinuously 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 ofGR on non-trivial spacetimes. Nothing helps to answer questions more than a workingmodel — where would mechanics be without the simple harmonic oscillator, or quantummechanics with the Stern-Gerlach experiment? These idealized models answer questionsabout physics without the clutter of experimental error or unsolvable mathematics. Realphysical 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 hopeto achieve here. By constructing a simple, though unphysical, model on which we cansee the roles of GR, topology, and ultimately quantum mechanics, more complex modelsmay be built. Hopefully this will contribute to one day producing a theory of quantumgravity, 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 formalismfor GR and see how this approach, together with gauge field theory, has recently renewed2interest 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: theholonomies associated with each non-trivial loop on the 2-dimensional surface. The resultcomes from carefully keeping the boundary terms in the action, rather than dismissingthem as irrelevant, as far as the action is concerned. In Chapter 3, we construct a modelfor T2 . Taking this model through the prescription of Chapter 2 endows it with a framefield and connection. The resulting action depends only on the holonomies of T 2 . Anexplicit, geometric model of the torus also shows that there are enough degrees of freedomthat many different collections of holonomies are possible, some possibly admitting closedtime-like curves. As a more complicated trial, Chapter 4 repeats the process over a genus2 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#T2 by folding up an octagonaland calculate the holonomies of this particular construction. In Chapter 5, we speculateon the classical conjugate variables. The action we produce is not canonical and wemake a qualitative interpretation of the phase space. While the results of Chapter 2 areclassical, the form of the "conjugate variables" strongly suggests the quantum mechanicalversion of the model lives on curved spacetime. Finally, in Chapter 6, we see how theresults may be generalized to higher genus surfaces, and where further research can bedone 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 aPhysics problem. It can equally be approached as an example of differential geometryover surfaces with non-trivial fundamental groups. The relations we encounter and chooseto interpret as topological constraints imposed on the space are none other than the3representation of the fundamental group of the surface in the gauge group. The particularcollection of holonomies of the surface, a subset of the gauge group, obey relations whichdemonstrate the structure of the fundamental group. So while this thesis emphasizes thePhysics interpretation of the results, the Mathematically-minded reader can translatethese same results into Theorems, Proofs, and Corollaries.Finally, a word on whom to attribute the results of this thesis. The ideas of Chapter 2are developed as a special case of the theory of GR, and not as an interesting observationof the manipulation of the dreibein and connection. The difference between these twoapproaches 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, Chapter2 is a reproduction of work done by my supervisor W.G. Unruh, and to him the resultsshould be attributed. The calculations in the rest of the thesis, the "easy part", werecarried out by both of us independently, in the sense that we ended up with two stacksof paper, although I was prodded in the right direction at several stages. Therefore, thisthesis should be viewed as a report of Unruh's exploration of (2+1)-dimensional gravityin the dreibein formalism, annotated and "demystified" to allow graduate students likemyself to understand and appreciate the results.4Chapter 1The Dreibein FormalismThe basis for studying spacetime dynamics in classical GR is the Einstein action. 1 Thisaction is a complicated functional of the metric components g . An alternative approachis 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 alsoreview recent work that recasts the dreibein approach to GR in a Chern-Simons gaugefield theory.1.1 The Dreibein Formalism of General RelativityThe most common approach for finding the metric on a spacetime is the Lagrangianformulation with the Einstein actionI =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 motionare found by variations of the action with respect to the metric components, g Theseequations of motion are generally highly non-linear PDE's, owing in part to the squareroot VIM. The advantage of the dreibein formalism is that is supplies a kind of squareroot 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].5Chapter 1. The Dreibein Formalism^ 6the collection of three vectors^where indices ,a, v,^= 0,1, 2 are the tangent-spacecomponents. We will use i, j, = 1, 2 to indicate the spatial components. The indicesa, b, .= 1, 2, 3, sometimes known as the Lorentz indices, merely label the vectors in theframe. The metric on the spacetime is defined by egab, = epa evahence giving a kind of 'square root' to gov . The vectors must remain orthogonal in theLorentz space:a pcbe e = abwhere qab =diagf— + +} is the usual Minkowski metric in (2+1)-dimensions. All raisingand lowering of a, b indices is done with qab. A new connection co: b is required to accountfor curvature in this mixed space. Covariant differentiation is defined byDtheva = 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'sAvMinkowski space, which forces Witab to be antisymmetric:0 = Doabab=+WAacil cb + Witb criac= ab^baThe 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 Ricciscaler R is found by contracting the "internal" curvature tensor Rit, 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 repeatedindices, both latin and greek.Chapter 1. The Dreibein Formalism^ 7While 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 spacesvia the dreibein: = eacpbRtwab. A flat internal space, RiLvab = 0 implies a flatspacetime, Ritv P = 0. In this mixed spacetime-internal space, the Lagrangian becomesO g iR = ( Epvp a^c, \( ^dfCabC)k e deaf  -"Aa ) •By the associativity of addition, we can re-arrange the summed Lorentz indices:= ftwP Eadf epaevbepc ObeucRAu dfE""Eadf e tLa j pA d;RA, dfby the orthogonality (1.1), leaving simply= cpvp cab, e a R be .AThe 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,,,bVbWith the definition of covariant derivative (1.2), it is simple to calculateAlp ab^„ ab^, a , db-"pv^'11 ,v^"Iv d'p •Finally, with volume form fluiPd3x =^A dxv A dxP, we arrive at the expressiona(I = fabcep kWvbc ,pb dWpdc )ale A dxv A dxP (1.3)The great advantage to this formulation of the problem is that we will nowhere encounterbelow a raised tangent-space index ,a. This allows us to work in the more general casewhere 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^ 8variables 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, thefull 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, inthis particular case also implies the Riemann tensor vanishes: the solution spacetimesare truly FLAT. This is one of the main reasons why (2+1)-dimensions are studied — thegeometry does not contribute to the physics of the spacetime, thereby leaving topologicalconsiderations more apparent.In analogy with the ADM formulation [7] of (3+1)-dimensions, the action (1.3) isre-written in terms of the canonical variables and their conjugate momenta. Because ofthe 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. Withoutconjugate momenta, these variables are constant in time. In the language of variationalcalculus, eoa and c00% are Lagrange multipliers of constraints which govern the way the2-dimensional slices of 'space' evolve in 'time'. Explicitly, we findI = —2 fdt P2 x Eijej aci7j,•eoa cij(wi bc j^w j b dwi dc wi bdw j dc)idt fd2 X eabcf^wobc fij ( ci a^j^e^eidw j ad^eidwi ad ilThe constraints, proportional to e oa and wo bc , are recognized asRij bc^0 (1.4)= 0 ,^ (1.5)respectively. These two constraints, which tell us the spatial slices are flat and torsionfree, greatly simplify the system, as we will see in chapter 2.Chapter 1. The Dreibein Formalism^ 9We see that of all the ways to attempt to describe the geometry of 3-dimensionalspacetime, the result produced by the theory of GR is fairly simple. In fact, most of thegymnastics of differential geometry have disappeared! It is no wonder, then, that this"simple" result can be reproduced from a quite different abstract approach, Chern-Simonsgauge field theory.1.2 Chern-Simons Gauge Field Theory over the Poincare GroupIn recent work that rekindled interest in (2+1)-dimensional GR, Witten [1] recognizedthe Einstein action written as (1.3) as a Chern-Simons action of the the gauge field theoryfor 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 Vb 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 physicaltransformations. An ISO(2,1) representation (Inhomogeneous Special Orthogonal) of thePoincare group is the collection of 4x4 matrices\^0[PIT] =Tatiabwhere tiab is an SO(2,1) Lorentz transformation. Properties of these matrices liab that willbe used below are found in the Appendix A. Transformations of 3-dimensional vectorsChapter 1. The Dreibein Formalism^ 10Va are carried out by appending a fourth component, of fixed length 1, to the 3-vectorsforming VA = (Va, 1):/^\ /^\^/^\tiabVb + Ta=Uab Ta V'Ik 10The 3-dimensional component of these resulting 4-vectors reproduces the 3-dimensionalPoincare transformation on Va.The Poincare group has generators/ 0 0 0 1 /^0 0 —1 /0 1 00 0 —1 0 0 0 0 0 1 0 0 0J1 = J2 = J3 =0 1 0 —1 0 0 0 0 00 0/ 0 0 / \ 0 0/Under exponentiation, these generate, in Minkowski space where xa = {t, x, y}, rotationsin 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 3submatrices by themselves generate the pure Lorentz group SO(2,1). The generators oftranslation in the t-, x-, and y-directions are, respectively,/ 11 / 0 " / 0 10 0 0 1 0 0P1 = P2 = P3 =0 0 1\ 0 0 / \ 0 0 / 0 0 /The generators of this group obey the Lie algebra[fa , 'Id .= Cabcf cChapter 1. The Dreibein Formalism^ 11Val Pb] = EabcPc[Pal Pb] = 0In [1], Witten observes that in taking the 1-form A = A pdx 1` with componentsA, = coaP, c.o4aJaas a gauge field, the Chern-Simons action/cs = 2 f Tr(A A dA IA A A A A)Mexactly coincides with the Einstein action (1.3). We see that ISO(2,1) is the gauge groupby varying the gauge field A. The infinitesimal transformation of A, generated by aparameter u = pa Pa + TaJa with infinitesimal pa, Ta is defined as8A„ = 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 + (5.co:JawhereSe a =^_ fabcembrc — cabc„4bpc,^a —Ta — abc bTp,^c •Witten notes that by setting pa = Vaq, for an infinitesimal spacetime vector -PL, thedifference between these transformations of Ella  and cot:, and those generated by infinites-imal Lorentz transformations along with infinitesimal diffeomorphisms (translations!) inthe —V4L direction, is simply a Lorentz transformation. Since Lorentz transformationsChapter 1. The Dreibein Formalism^ 12are 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 thePoincare group is the same as the Einstein action of GR. Gravity can be re-expressed asa gauge field theory, which greatly increases the chances of finding a quantum mechanicalversion of the theory. Though this thesis is based on classical GR, and not quantum orgauge field theory, we will see the importance of the Poincare group in the results thatfollow.Chapter 2The Einstein ActionIn this chapter we derive the main result of this thesis: the Einstein action over a compactsurface can be written explicitly in terms of the holonomies on the surface. Before weproduce this result, let us briefly review holonomies and boundary terms.Intuitively, a holonomy is the failure of a vector to return to its original orientationafter being parallel transported around a closed path, or loop. We will define holonomyas the transformation which carries the initial vector onto the final. While in generala holonomy is an automorphism of the tensor fields over a manifold, the holonomiesencountered here are simple transformations on the tangent space. Curvature, also ameasure of the changes in parallel transported vectors, is closely related to the holonomyof trivial, or contractible, loops. When a surface is flat, the holonomy of all trivial loopsis simply the identity — no change occurs in a vector when it is parallel transportedabout a flat surface. When curvature is present, the holonomy depends both on thebase point of the loops and the shape of the loops itself: parallel transport around "longand twisting" loops will alter a vector more than parallel transport around "small andsimple" loops.As we saw in Chapter 1, the surfaces we deal with here are flat, and the presenceof interesting holonomies seems unlikely. However, the surfaces we deal with are alsotopologically non-trivial, and are covered in non-contractible loops, those paths whichget "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 is13Chapter 2. The Einstein Action^ 14everywhere 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 besmoothly deformed into each other differ by a contractible loop (See Figure 2.1). Thispath does not contribute to the holonomy as the surface is flat, so the holonomy of thetwo homotopic paths is the same. Therefore all holonomies on a surface will be knownonce those about a few representative non-trivial loops are known. Just as the genus 1torus 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 these2g loops, the geometry of the surface is completely determined, at least up to globalgauge transformations. These are the transformations which transform the entire surfacewhile leaving the action (or any other gauge-invariant function) invariant. It is these 2gholonomies which will play the role of the dynamical variables in the expression for theaction 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, dowe handle boundary terms that arise in the action? The answer comes by looking at theway 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 infiniteplane with these polygons, so that moving off one tile onto an adjacent one is the sameas 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 surfaceopen. This does not affect integration over the surface: A continuous function has thesame value at two identified points on the boundary. Since the outward-pointing normalsfrom the two identified boundary points have opposite orientations, any contribution toChapter 2. The Einstein Action^ 15Figure 2.1: The difference 'Y1-72 between two homotopic curves 7 1 and 72 is contractibleto a point. The holonomy of this trivial loop is the identity, so 7 1 and 72 have the sameholonomy.Chapter 2. The Einstein Action^ 16a boundary integral along one boundary component is exactly cancelled by the matchingboundary's contribution. The converse to this situation need not by true, though, as wewill see below. By starting with the tile and matching pairs of sides, we are no longerguaranteed that functions are continuous across the boundary. The discontinuities weallow are exactly the holonomies that appear in the action.2.1 Reduction of the Einstein ActionThe dynamical variables that appear in the Einstein action (1.3) are the dreibein e ma andthe connection co:b . Let us assume there exists a collection of e fia(xv) and woab (xv) whichare continuous over the surface, or equivalently, continuous on the tile, even across theidentified boundaries. Because only the torus T 2 can support a non vanishing tangentvector field [6], we must allow the possibility of the dreibein becoming singular on thetile.First consider the symmetries of the action. The action, as the CorrespondencePrinciple implies, is Lorentz invariant. Transform the variables under, aFiaWA bUab ettbco1:d audb uc a(ucb,m )^ (2.1)Some index manipulation along with the properties of U (Appendix A) show the action isthe same functional of e and L. Recall from the (2+1)-dimensional splitting of spacetimethat the space is subject to the constraint Ri j ab = 0. This implies that on any coordinatepatch, we can find a particular Lorentz transformation Uab (e) such that c"-;.5i ab = 0 (SeeAppendix B). Note that only the spatial components of (.7., can be made to vanish becauseonly the spatial Rid ab = 0.Chapter 2. The Einstein Action^ 17In 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 assumedthat eif is continuous across the boundary, it is not necessarily true that e lia is continuous.With this transformation, the action becomesI =^fabc (6Aa noa)fiupbc a A de A dx f)Mwhere M is the (2+1)-dimensional manifold and R is the curvature, also written in termsof a An integration-by-parts on the second term givesI = et: fcpbc axt.,, A de A dxPi fabcM+ I Cabc pa ft,bc dx"A deI— IEabc pa b[1Wc de A dxv A dxP .it4-The last term vanishes by the Bianchi Identity. A second constraint imposed on thesystem 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 ei a = 0 (See AppendixB). The first term of the above action vanishes leaving only the boundary integral. Theboundary here includes the spatial boundary where the identification takes place and alsothe initial and final hypersurfaces in time. The integrals over these temporal boundaries,however, come from the termbo(pa fiv \ ,x oEabc^pbc )a A de A dxP .imThis total time derivative can be removed form the Lagrangian as it adds only a constantto the action and has no effect on the equations of motion found by variational principles.= Eedges^gCabc (Pla^be^a^bc4.7610 — p2 632 o ) dxied e .Chapter 2. The Einstein Action^ 18This leaves onlyI = Eabc pa w bc• — be^ib dcjodc^ob i dc)dxiwhere 0 represents the spatial boundary. As the spatial components (Di ba vanish, theaction is further reduced toI =Eabc^bcdxp^.jOne more integration-by-parts givesI =PaCab Pa coobcdxi— JaThe 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 getthe topology. Thus each distinct edge of the tile contributes twice to this integral. Callthe two matching sides '1' and '2'. Summing over the 2g different edges givesaRecall 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 isbd tiice — p2 a i^bd^de dxiCabc (P1  edges edgeAs U1 and U2 are both Lorentz transformations, there is a Lorentz transformation betweenthem. Defineui ab = wac u2 cb^ (2.4)Chapter 2. The Einstein Action^ 19This maps U on one side of the boundary onto U on the other side, in some sense carryingU across the boundary. Observe that= (V V c ati2da )(Wcelk e= ( Wc dWce) 142 datt2 e b,ibe+ Wed /122 w'e7i u2 ebWith C3i ab vanishing in (2.1) and co lobc = co2obc,HenceOruida^= u2da u2 dbiwc 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 forU2 in terms of W and U1 . Property (A.4) then reduces this term of the action to simplyE f (pi ai wab p2^bd Ce wode dxedges^iedge(2.6)Before we calculate the second part of the action from the UU term in (2.3), notthe following. Recall Coo de does not appear in the action by the antisymmetry of thevolume form. Thus wo de is a Lagrange multiplier. The UU term contains no CA) o de sothe constraint associated with this Lagrange multiplier comes from the first term of theaction alone, namelyPic; — wa bb p2,, = 0 •We found in (2.5) that Wab is constant on each time slice, so this constraint is equivalentto(2.7)w bp2 19 + Ha = pi a^ (2.8)Chapter 2. The Einstein Action^ 20where Ha is constant. Under the identification of points that generates the topology of thesurface, p1 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 (see1.6)). Another way to compare vector P 1 with vector p2 is to parallel transport p 1 acrossthe tile between the two identified points on the boundary. The resulting vector is definedto be p2 . Now (2.8) shows the holonomy of this loop in a Poincare transformation. Wehave seen that the W are constant on each time-slice and that the holonomies are pathindependent. Thus p on the whole of side 1 maps onto p on the whole of side 2 underthis 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 442edges edgeFrom (2.7) and (2.4) we substitutedx .P2 a,i = Wb a Pl b/42 cd = wy c ul ydBecause Wab does not necessarily vanish (only the spatial derivatives do), we cannot yet•substitute for /42b d. These substitutions givecdE^cab, (pi a 141 b,i^d"1edges edge•Wea P1 e ,i U2b dWv c tOd ) dxi .Replace Eabc under (A.4) in the second term. Then (A.1) and (A.3) reduce the action toNowedges ledgeb^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 ydChapter 2. The Einstein Action^ 21givingE Eabcp1a,i wbx wcx chiedges edgeRecall the Wab are constant and hence can be pulled out of this integral along the edgesof the tile, leaving only • • fpi a 'i dxiedge This integral, simply the difference in p i between the ends of edge 1 of the tile, we denoteby Api a.Finally, we produce the action on the tile. By imposing the constraint (2.7), whichintroduces the constant Ha, the first term of the action (2.6) vanishes. All that remainsfor the action is the sum over representative edges= E Api awbx wcx cabc^ (2.9)edgesBy constructing a tile and assigning on the tile fields of W and II, we can explicitlycalculate this action for a torus T2 and later, the more complicated but interesting two-holed torus T2#T2.Chapter 3The Simple TorusAn 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, byits holonomies around the nontrivial loops..."Mathematically, this is a concrete and definitive corollary, distilled from a much largertheory. It is not an explicit statement of the physics of the system, though, for it stilldeals with abstract ISO(2,1) transformations across some surface or region with identifiedpoints. In this Chapter, we see how the result (2.9) of Chapter 2, which asserts that theaction in a function(al) of the holonomies (supporting Carlip's statement), is manifest ona simple 1-holed torus T2 . This is relatively easy to do, as the flat genus g = 1 torus hasonly two independent holonomies, and can be constructed by identifying pairs of sides ofa square in the Euclidean plane R 2 .3.1 The Action over a TorusEach 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 onthis tile, let us first label its components as in Figure 3.1(i). We call the sides A, B, C, Dand the corners 1,2,3,4. The identification of sides A&C and B&D is indicated by the1 [4], p. 2649.22Chapter 3. The Simple Torus^ 23Figure 3.1: (i) Labels for the sides and corners of the tile, and identifications of sidesA&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 withthe indicated orientation creates a torus, T2 . The same gluing information can also beexhibited 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 Figureby "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 identifiedas part of the topology generating gluing. Adjacent to side C is side D, and these twosides meet at corner 4. Next, side B is glued onto D, side C is adjacent to B, and corner3 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 , Cji onone side, and p2 , w2 on the other. The corresponding edge along which we calculate Ap i ais, in this case, the line that runs out from the 1-4 region of the vertex and returns to1 23 4DWDB12(i)^A3CWC A4Chapter 3. The Simple Torus^ 24Figure 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 findfrom (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 theconstant defined by (2.7). This relation holds all along the edge between sides A and Cso we push the result down to the 1-4 region of the vertex. Evaluated at the vertex wewriteWCAab 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 edgeC. There is a similar relation at the other end of this CA boundary, the end in the 3-2region of the vertex. The relations across each of the four edges of the tile can be writtendown by looking at Figure 3.2:Chapter 3. The Simple Torus^ 25WCA 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 identifiededges. Each edge contributes twice, recall, but the return integral along the 2-side istaken care of with the WW terms in the action. We need only consider the 1-sides, fromcorners 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  EabcNow it is not true that, say, pc (4) = pA (1), even though these two functions are evaluatedat the same point, under the identification of sides A and C. There is the discontinuityis pa defining 14/cA and licit . It is true, however, that at vertex 4, pD(4) = pc (4), as theseare evaluated at the same point on the tile, without any identification of points required.Call the corner value just p4 . 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) = p4With these relations, along with the W and II above, we can express p in each regionof the vertex in terms of, say, pl . 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, asP3 = 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^ 26Note that just as we define Wc A to jump from side C to side A, WAC jumps from side Ato side C. Each of the W is invertible, though, so WAC WC A l • In this way we findP2^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 truethat 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 Lorentzpart is the Identity, while the translation vanishes. Jumping all the way around thevertex producesTAT -1 Tv- -1 1,,pi^WCA vvDB vvCA vvDBP1+ WC AWD B-1 WC A-1 H D BLT -1^-11-rWCA WDB vvCA 11 CA— WCAWDB HDB+ FICA •Hence the Lorentz transformations must obey the closure relationTAT -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 gaugegroup ISO(2,1). It is equivalent to the conditionWCAWDB WDBWCAChapter 3. The Simple Torus^ 27so that the two Lorentz transformations commute. Applying the relation (3.4), we reducethe 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 furtherconstrains 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 simplyp), W, and II, plus the constraints (3.4) and (3.5). We first introduce notation to removethe cumbersome termviib wcxcx^abeDefine{W}c, = Flibxwcscab, •We now adopt a 0-index notation so thatHa 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, readsI = (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^ 28An example of the {W} algebra is shown below in the 7 12#712 case. If we now imposethe constraints on the system, the first term above vanishes, as {1} oc 1 = 0. The actionis finally reduced to(147DB-111DB){WcA} — (WcA 1 ricA ) {a b ( WC A WD B 141D B WC A) a ba ((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 ofthe holonomies [W ITT 1C A I—CAJ and [W Ill 1DB i _DB J • All reference to the geometry of the tile hasbeen removed. This is exactly the property that flat connections are completely specifiedby a collection of holonomies. Furthermore, we see that the action is very nearly in thecanonical 1,4 form, except that {W} is not merely I- W. Before we attempt to extractthe canonical variables, we will look at the r#T 2 case for more insight.3.2 Consistency of the ConstraintsIt 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 outthe constraints shows that these six equations are not independent, and only four degreesof freedom are directly removed. Also, we will see that under a interesting identificationof 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 ofthe roles of W and H as canonically conjugate variables.Chapter 3. The Simple Torus^ 29The first constraint of the T 2-system is the closure relationWCA abi/VDB bc TVDB abWCA bc = 0 .The W's are forced by this constraint to lie on a surface in W-space. Hence variationsof the solution WCA^WDB must also lie on the this surface. That is, variations of theconstraint with respect to the the coordinates Wab must vanish. We will see that this newcondition is a copy of the second constraint on the system, subject to some interpretationof the variables. The variation of WcA WDB — WDBWcA is calculated bySWCA ab rTAT^6.WCAb a vvCAYTIVDB b^TVDBab TAT C SWCA sAac TVC A Xy^ V IVA Ty(5WDB b  1LIT^C5WB ab r+ 147C A ab^xC "rDB xy^U "ixTDB Ty WC A b c'TDB y^WDDB Xywhereswa6 — saSWxy xThusA ac^SWCAabWDBb WDB ab0WCA bC14/6 A abSWDB 6 SWD B ablVCA 6or, in matrix form,A = 8WcAWDB — WDB6WcAWcASWDB — SWDBWCA •By varying the constant metric 77ab , 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^ 30Insert factors of 1 = W'W where necessary into the expression for A to get all SW intothis 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 ) 14113 B (VC Awhere the last step puts each (SW W') into a similarity transformation. We right-multiply by IVand apply the relation (3.4) to find^\ ^—0^(6WCA WCA-1 ) WDB (8WCA WCA-1 ) rHTDB 1WCA ( 8WDB WDB-1 )WCA-1 (S WDB WDB-1 )Now as (SW W-1 ) is antisymmetric, we can replace this matrix by an equivalent vectorAa defined asAa fabcswbd wcdThe similarity transformations U(SWW -4 )/4' become linear transformations U of thevector. "Rotate" the antisymmetric matrix (SW W') with a Lorentz transformation U:swbdvrd^ubxjwxdwyducyThis new matrix is also antisymmetric in b, c: interchange b and c to finducx (swsdwy d) uby = ucx (6w d wyd)ubyucx (5wyd wx.d) ubyas SW W-1  antisymmetric. Now relabel the dummy indices, interchanging x and y:_= ucy swsd wylubxub x ((Swsd wy d) ucycAWDB, or equivalently WDBWCA, without changing the vanishing variationChapter 3. The Simple Torus^ 31Define a new vector A'a from this antisymmetric matrix:ya fabc (11bxtiVW dWyd) .By (A.4),SOwhich is simplyEabc tibs tic y = tia z exy ,Ala = tiaz EzxyST/VxdWYdA'a = tfazAz .Under this substitution of Aa , the variation of the constraint on the W's is equivalent tothe constraint0 = ACA — WDBAC A + WCAADB — \DB= (1 — WDB)ACA — (1 — Wc,A)ADB •This is exactly the second constraint (3.5) under the identificationAa^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 configurationvariables 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 T2To transform the action in {W} and H into something that can be written on the back ofan envelope, we construct an explicit model for T2 and find the Poincare transformationsChapter 3. The Simple Torus^ 32[Will] across its boundaries. We will see that even a very simple model reveals interestingdetails.It is easy to construct a region representing the torus because the plane R2 can betiled 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 anddj [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 boostsin the same direction. Let us choose coordinates (x l , x 2 , x3 ) = (t, x, y) over the tiled R2plane 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 thematrix J3. We can choosecosh(y) sinh(,u) 0 \^WCA = eA.13^sinh(,u) cosh(,u) 00^0^1 jcosh(ay) sinh(a,u) 01^WDB = e la3^sinh(ay) cosh(au) 00^0^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 choiceof WCA and WDB. The constraint reads1 — cosh(y) — sinh(y) 0 HDB 1— sinh(u) 1 — cosh(y) 0 HDB 20 0 0 / HDB3Chapter 3. The Simple Torus^ 33/ 1 — cosh(ap) — sinh(a,u) 0 \ TIC A l= — sinh(ap) 1 — cosh(ap) 0 TIC A 20 0 0 1 \ ITC A 3 IWhile IIDB 1 , HDB 2 , HcA 1 , HcA 2 are coupled by two equations, IIDB 3 and HcA3 are com-pletely unconstrained. That is, for any translations Ir in the y-direction, the sequenceof transformations that circumvents the tile vertex is still the Identity. While initially itappeared that the constraints would directly remove six degrees of freedom (df) from thesystem, in fact only four are removed. This is not merely an artifact of our particularchoice of the x-direction for the boosts, for a rotation will not affect the indeterminacy ofthe matrices 1— WDB or 1 — WCA• We will see below in the more complicated r#T2 casethat this failure of the six constraints to remove six df is unique among the non-zero genussurfaces to the torus. Furthermore, whereas the Lorentz component of the holonomiesconfine the transformations to a surface of constant 02 — x2 , the translation carriesvectors off this hyperbolic plane. With some translation in the t- and x-directions andarbitrary y-translation, it seems possible that the sequence of transformations aroundthe vertex can be a loop with time-like sections, carrying vectors along closed time-likecurves.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 treatingthe transformations W as a sort of "phase difference" between identified points. Referringagain 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) orChapter 3. The Simple Torus^ 34Figure 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 1U(1).So that the result is independent of the choice of evaluation, it must be true thatWcit 1 WDB = WDBWCA 1 or WDBWCA = WCAWDB, exactly the constraint we encounteredearlier. 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) = Wccorner values are now interpolated smoothly over the tile, taking care to keep the correctW phase difference between identified points on the boundary. The particular choice ofinterpolation does not change the action, which depends only on the difference across thetile (or across the boundary). One simple example of an interpolation isti(x, y) = e( 1— f(Y) -Ectf(x))0J3SA gives simple values to the corners of the tile (See Figure 3.3). Thesewhere f is linear between f (0) = 0 and f(1) = 1. Smoother interpolations (quadratic,Chapter 3. The Simple Torus^ 35cubic, ... ) 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 ) isthe function chosen to eliminate the ei a. A simple choice is to give the translationsonly y-components, n-CA = (0, 0, a) and IlDB = (0, 0, b), and set pa (x b ) = xa . TheLorentz components of the holonomies preserve 0 2 — x2 while the translations shift they-components of parallel transported vectors by a constant. The tangent space over thetile is a parallelogram on the surface of constant 02 - x2 (See Figure 3.4). The relationW21P2 + 1121 = p1 tells us which points are identified in the tangent space.3.4 An Alternative Approach to the TorusAnother, more "standard", method for studying (2+1)-dimensional GR on a torus isbased on the fact that of all the compact surfaces, only the genus 1 torus can supporta continuous, non-vanishing tangent vector field. The approach, therefore, cannot begeneralized 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 yto 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 Poincaretransformations:Al : (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 integralChapter 3. The Simple Torus^ 36Figure 3.4: Tiles of the tangent space over the torus are parallelograms in boost-transla-tion space.Chapter 3. The Simple Torus^ 37or Wilson line [5] calculation of the holonomy are e1 = e2 = 0, col = co2 = 0 ande3 = (0, a, b)^w3 = (0, A, ,a)This dreibein produces a singular metric, but we can gauge transform to a non-singularsystem: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 arisingfrom this choice of dreibein isds2 = la 2 dt2 — (a2 + # 2 A2)dx2—2(ab + 13 2 A,a)dx dy — (b2 + /3 2 it2 )dy2 .Now Carlip observes that the spatial part of the metric, the metric on the torus, isunchanged 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 2A2t — Ab) (y + 1))0 2A2^a2 + / The geometry of the space, therefore, is characterized by these two coordinate translationsin a, b, A, ,a, the parameters that fix the holonomies. Treating a, b, A, ,a as a new set ofcoordinates, the Hamiltonian produced is simplyH = i3(a,u — Ab) .Chapter 3. The Simple Torus^ 38One interpretation of the canonical variables is to take the boost parameters A, p ascoordinates 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 thetorus. The holonomies (A, a) and (pi, b) completely determine the action because of theflatness of the space.A model for the genus g = 1 torus is simple to construct because the plane R2 canbe tiled in regular (4g = 4)-gons, or squares. This surface is also easier to study thanother genus surfaces because it is the only one that can support a non-vanishing tangentvector field, immediately giving the surface a non-singular metric. We now turn a to morecomplicated surface, the two-holed torus, where the results for T2 are closely mimicked.Chapter 4A Two-Holed TorusIn 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 thanT2 , for it is impossible to put a continuous non-vanishing tangent vector field onto thissurface 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 T2 , and the action is morecomplicated only because it is a functional of twice as many variables. This surface has2g = 4 generating non-trivial loops. It can be constructed by identifying pairs of sidesof a (4g = 8)-sided polygon. Or equivalently, the two-holed torus can be conceived by atiling of a plane with octagonal tiles, with eight tiles meeting at each vertex. As we shallsee, but intuitively understand already, the plane R2 cannot be tiled in regular octagonswithout leaving gaps in the tiling. Hence we look to hyperbolic geometry where thecondition that triangles have 180° no longer applies. Before we construct such a tile andits collection of holonomies W and II, we first consider the case of a general octagonaltiling, and translate the results of Chapter 2 into the r#T 2 variables.4.1 The Action over a Two-Holed TorusWe calculate the action over the two-holed torus exactly as we did for T 2 . Covereach surface of constant time with octagonal tiles. Label the sides of the octagonA, B , . . . , H and the corners 1,2, ... , 8 (See Figure 4.1(i)). The identification of sidesis indicated by the arrows on each side. Gluing the matching sides together generates the39Chapter 4. A Two-Holed Torus^ 40Figure 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 identifiedsides C&A, D&B, G&E, H&F. The neighborhood of a vertex where eight tiles meet alsoshows the identifications and labels (See Figure 4.1(ii)). As before, we bootstrap aroundthe vertex using the identifications and label the sides and regions of the vertex. Figure4.1(ii) is a truer representation of the tile because the neighborhood of a vertex is a patchof 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 stretchthe 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 theconstant defined by (2.7). This relation holds all along the edge between sides A and CChapter 4. A Two-Holed Torus^ 41as Wab,, = 0 (2.5), so we push the result down to the 3-2 region of the vertex whereWCA ab PC a (3) + TICA a = pAa(2) •The term pc a (3) is the limiting valueSimilar relations across each of the identifiedof 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) IIGE a pEa(5)Figure 4.2 (iv) WHF ab PH b ( 8 )^HH Fa = PFa (7)TVHFab 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 onlyconsider the 1-sides, from corners 1 2, 2 -+ 3, 5 -+ 6, and 6 -4 7. The action becomesI =^(PAa(2) PA a ( 1 ))WCA bsWCA cx(PB a (3) PB a ( 2 ))WDBbxWDB cs(PEa (6) PE a ( 5 ))WGE b zWGE cx^(PFa^PFa ( 6 )0H Fb^Fcs fabc •^ (4.2)The field pa is discontinuous across the boundaries, the discontinuities related to theholonomies W and H. But in region 4 of the vertex, for instance, pp (4) = pc (4), as thisis the limiting value of a function pa, continuous on the tile, without any identificationof points required. Call the common value just p 4 . An analogous relation holds in eachChapter 4. A Two-Holed Torus^ 42Figure 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^ 43region of the vertex:pA(1) pi-1M= pi^pA(2)= pB(2) =--- p 2pB(3) = pc(3) = p3^pc(4) = PD(4) = P4pD(5) = pE(5) = P5^P E( 6 ) = PF ( 6 ) = P6PF( 7) = PG( 7) = P7^PG(8) = PH(8) = P8By bootstrapping around the vertex from region 1 with these relations and the W, Habove (4.1) we can express all the pi in terms of pl . As vectors and matrices rather thanin components, we seeP2 = WCA P3 + HC AWCA(WDB P4 + HDB) + HCAWCA(WDB(WCA 1 (pi —HcA))+ HDB) + "CARepeating the process for each corner at the vertex givesP2^WCA(WDB(WCA1 (pi —HcA))+HDB)+ 11 CAP3^WDB(WCA1 (pi — HcA)) HDBP4^111' ( p — "CAP5^WGEWH F l (WGE 1 (WHF P1 + HHF HGE) HHF) + HGEP6^WHFP1 HHF- GE 1 \ •^1•HFP - HF - GP7^W (W TT^TT E 1P8^WHF1 (WGE 1 (WHF P1 + HHF HGE) - HHF)We arbitrarily chose to write ps = Ps(P7(06(P1))) in finding the last result in thislist, 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 acomplete circuit of the vertex is the Identity:(4.3)P1(p4(p3(P2(P5(P8(p7(p6(P1) • • •)) = P1Chapter 4. A Two-Holed Torus^ 44The Lorentz part of this Poincare transformation must be the Identity and the translationmust vanish. The transformation carrying pi around the vertex back onto p i is given byP1 = 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 BTAT 11,17 -ln- WCAvvDB wvCA- WCAWDB-111DB• H C A •The Lorentz transformations WCA,WDB1WGE,WHF must obey the closure relationWCA WDB 1 WCA-1 WDB WGEWHF 1 WGE 1 WHF = 1 •Again we see the fundamental group 7ri (T2#T2) represented by the W in the gauge group.Define1^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 = 112(WGE, WHF), they are the same transformation:hi is a transformation halfway around the vertex in one direction, C2 2 is a transformationhalfway around in the other direction, and the two results coincide there. The decoupling(4.4)(4.5)Chapter 4. A Two-Holed Torus^ 45of the transformations into ABCD terms and EFGH terms is an indication of theconnected sum construction of r#T2 . The constraint (4.5) shows that the two tori gluetogether smoothly. By inserting the factors missing from the cycle (4.5) and removingthe resulting factors of 1, we can reduce the translation part of (4.4) to the constraint0 = 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 pi (or simply p), W, and H, plus theconstraints (4.5) and (4.7) with the help of (4.3). Again we introduce{W}. = Ikxw'eab,to more easily writeHa lkbs W'cabc H{W}We use the properties of {W} found in Appendix C to reduce the action to a simpleform: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^ 46Rewrite 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—^ —^ —11CA { WcA. — HCA{WDB} — ( WDB , v7A7cA 1 -FriicA) {WDB vTATCA 1 ryTA7CAvvTA7DB 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 BNow apply (C.3) to remove the under-braced terms:= —11C 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 theconstrained actionI = ( 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 constraintsimposed by (4.5) and (4.7), respectively. Observe again that the terms in ABCD aredecoupled from those in EFGH.As in the T2 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^ 47p4 form, except that the term {W} is not merely do W . The only difference between the"simple" T2 and the "difficult" r#T2 is that more variables have appeared to accountfor the increased number of incontractible loops and holonomies.4.2 Consistency of the ConstraintsRecall the first constraint on the system is that the Lorentz transformations W obey theclosure condition (4.5)T^T^1WC AWDB 1 WC A-1 WDBWGE vvHF1 v1, vGE1 EA 7rHF 1or with (4.6),C21 -^= 0 .The second constraint forces the sequence of translations to vanish (4.7), closing thecircuit of Poincare transformations around the two-holed torus:1,17 -1 TAT 1 rr0^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 constraintto lie on a surface defined by the relation (4.5). Variations, or nearby solutions, mustalso 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 quiteconjugate to W) with infinitesimal Lorentz generators. We will work both in tensor andcomponent form in showing this result.Chapter 4. A Two-Holed Torus^ 48The variation of C21 — S22 is defined as• = s ( — 112 ) ( — f22) ^SWCA^SWDB^SWCA^SWDB^+ S(f21 n2) SWGE ^— C12) ^SWGE^SWIIFSWGE^SWHFwhereSwab^=SW sySW b a y^y ^asw x^Wp 14lb (5TSthe latter coming from 5(W W') = 0. In component form, the constraint reads0 = WC A a bWDB cb WC A dc WDB d e— WH F b a WGE bel/VH Fc d •WGE ed •The variation, with the 5-functions evaluated, isAae = owcAaowDBcbwcAdcwDBdeWCAabWDBc5WCApcWCAdq(SWCAPOWDBde_wcA ab wDBpbwDBcg(swDBpowcAdcwDB de wcAabwDBcbwcAdc(swDBde)_KFba(swGE bc )wHFcd wGEe d wHFbawGEbcwHFcduTvvGEpd WGE eq (SWGE Pq )+WH F pa WH F b q (SWH FPOWGE bc WHF cd 14 E e d — 147HFb a WGE bc (SWHFC d)WGE edThis 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-1which is clearly the variation of WcAWDB 1 WcA 1 WDB — WHF 1 WGEWHFWGE 1 unders( w-1) =Chapter 4. A Two-Holed Torus^ 49Recall that (SW)W-1 is antisymmetric (3.7). Insert factors of 1 = W'W wherenecessary into the expression for A to get all SW into this antisymmetric form. As thevariation A must vanish, we can right-multiply A by WDB vvTA,C AWDBWC A1 or equivalently/17^T rWGEWHF-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 (SWGEE 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 FAs (SW)W' is antisymmetric, it has only three independent components and can bereplaced by a vector Aa:Aa = cabc swbd wdcAgain, the similarity transformations U(SW W')/1-1 in the variation are simple lineartransformations 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 identificationAxy a^xya ,this is exactly the second constraint (4.7). The same notion of the 11 being momentaconjugate to the configuration variables W is suggested. We offer an interpretation ofthese "canonical variables" in Chapter 5.Chapter 4. A Two-Holed Torus^ 50We arrive at the question of how many degrees of freedom are directly removed bythese constraints. The condition 52 1 5221 = 1 impliesWC A WD B 1 WC A-1 = 112 WDB1 •^ (4.9)Taking the trace of this matrix equation showsTrWDB-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 eachelement in exponential form U = enaJa, where the generators Ja introduced in §1.2 are abasis for this vector space. We re-write (4.9) ase-nDBawcA.lawcA1 = 122 e —971JBaJaNow — 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 whenthe matrices (SW W-1  rotated by U. Thus we havee-(wcAnDB)aJa =11 2 e —naDEja •^ (4.10)The right-hand side is WDB1 transformed by 112. The vector representing this new Lorentztransformation is —(WCA 71DB) a a rotation of 7/15B . The equation specifies the directionabout 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) -1Chapter 4. A Two-Holed Torus^ 51so that rotations about the riDB-direction are inconsequential. Therefore, the rotationspecified by (4.10) is determined by only one parameter. Together with the magnitudeof the rotation, two df are removed. Coupled with the trace relation, a full three df aredirectly removed by the relation 52 1 1221 = 1. The translation constraint (4.7) likewiseremoves three df, directly reducing the dimension of the phase space by six. The failureof the constraints to remove a full six df from the T 2 system is due to the triviality of thetrace relation: The condition WcAWDB-1 WcA-1 WDB = 1 shows WDB 1 = WC A WD B 1 WC A 1 SOthatTr 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 thecorresponding translation "non-constraint", supplies the unexpected extra df in the torussystem.Each constraint also removes a gauge degree of freedom, that gauge transformationgenerated by the constraint. Thus the dimension of the phase space over all genus g > 1surfaces 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=4degrees of freedom.4.3 A Model of T2#712While the results we have found are quite explicit, they are still based on some unspecifiedoctagonal 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...) forthe tiling and the holonomies. We will see, though, that to construct a model, we haveChapter 4. A Two-Holed Torus^ 52to simplify the geometry with high symmetry, eliminating the translation componentsIP of the holonomies altogether.4.3.1 TilingsTo construct a closed surface without boundary, like T2#T2 , one abstractly thinks of aplane or similar infinite region modulo some identification. Concretely this can meancovering the infinite region with tiles and identifying sides of the tile in pairs. The planeR2 , however, cannot be tiled with all regular polygons. Suppose the tiles are regularp-sided polygons. The interior angle at each vertex of a p-gon is w (P-2) . If q such tilesmeet 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 valueof 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 isless than 7r, and the interior angle of regular p-gon is less than 74P-2) . The neighborhoodof a vertex where q hyperbolic p-gons meet is a patch of R2 and still requires a full 27rradians. Each tile contributes an angle of aLr. Since the p-gon is hyperbolic , r < w(P )q P 7or (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 recallthat the hypersurface of constant time is embedded in Minkowski, not Euclidean, spaceso 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^ 534.3.2 Construction of an Octagonal TileThe constraints tell us R23 ab = 0, so the tile we construct must be flat. Consider thesimplest flat 3-dimensional space, Minkowski. In polar coordinates, the 3-dimensionalMinkowski metric isds2 = —dt2 dR2 R2 d02 .Change coordinates (t, R, 0) to (T, p, 0) defined byt^T cosh pR^sinh p.^ (4.11)Inverting this transformation shows= +02 — R2p = tanh -1 —R .Observe that r is invariant under Lorentz transformations, while p is the magnitude ofthe Lorentz boost which takes R 0 out to R sinh p. We exploit the invariance of runder Lorentz transformations. By building the tile on a surface of constant r, points onthe tile will be connected with merely Lorentz, rather than full Poincare, transformations.Finally, define a new coordinater = Ttanh( e)^ (4.12)and consider for simplicity the r = 1 hypersurface. This coordinate transformationprojects the T = 1 hypersurface onto a unit disk with infinity at r = 1, much the sameway the stereographic map projects R2 onto the 2-sphere. The metric on this surface isconformally flat:4^r2do.2)do-2 = ^ (dr2(1 — r2)2Chapter 4. A Two-Holed Torus^ 54Because 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 andthe "vertical" plane y = 0. This curve lies on the hyperboloid "above" the y-axis andcan be parametrized by-y(A) = (t(A) = A, R(A) = NA2 - 1, 0(A) =for A > 1. Now Lorentz-boost every point in the y=0 plane in the x-direction by somemagnitude p; the points which lie on the y-axis are unaffected by this transformationand the y = 0 plane is 'tilted' in the x-direction. With increasing boosts the intersectioncurve 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. Witharbitrary but finite boosts by p, the family of curves -y(A; p) is parametrized byt(A) = A cosh pR(A) = VA2 cosh 2 p — 10(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 thatthe 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 = 1orthogonally (See Figure 4.3). The arcs within the unit disk are the geodesics of thisPoincare Disk model of hyperbolic geometry [8]. Parametrize in 0 the arc within the unitdisk of a circle of radius "r:Chapter 4. A Two-Holed Torus^ 55Figure 4.3: Circles of radius F intersect the unit disk orthogonally. The arcs within theunit disk are parametrized in O.Chapter 4. A Two-Holed Torus^ 56r(0)^+ P2 COS 0 -^+ .7"2 ) COS2 0 - 1tan 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 findt(0) = 1 + r(0) 21 — r(0)2R(0) = 2r(0)1 — r(0) 2(^0 = e^)Comparing this with the family of intersection curves y(A; p) we seet (A) 2 — R(A) 2 = 1 = t(e ) 2 — R (19 ) 2so that both families of curves lie on the T = 1 hypersurface. Furthermore, by comparingthe 0 = 0 points of both families we find the correspondence between p and "7-. and finallythat these two families of curves are identical. That is, the curve in Minkowski spacewhere the y 0 plane, tilted by tanh(p), intersects the T 1 hypersurface becomesthe arc of a circle of radius F(p) which orthogonally intersects the boundary of the unitPoincare Disk.By simply rotating these curves about the origin, we can piece together arcs in theunit disk to form an 8-sided figure. We must now find the value of the parameter F , orequivalently p, which produces the correct tile. The polygonal tile we are constructingis regular, so the 8-sides must be spaced at equal intervals of 7/4. Consider the figureproduced by laying down 8 arcs of radius F centered at radius N/1 +^on the 87/4-'spokes' (See Figure 4.4). When i 0 the arcs belong to small circles centered justbeyond r^1 and the arcs do not intersect (Figure 4.4(i)). At some larger F when eachChapter 4. A Two-Holed Torus^ 57Figure 4.4: For various values of F, arcs (i) do not intersect, (ii) are tangent to oneanother, (iii) form an almost regular octagon about the origin, and (iv) intersect withangle 7r/4.Chapter 4. A Two-Holed Torus^ 58arc just intersects the two neighboring arcs, the angle at the intersection of two adjacentarcs 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 forman 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 regularplane octagon. For each value of r we amputate the legs of the 8-sided figure about theorigin and call the result an octagon. We must choose the value of r which generates anoctagon whose adjacent sides intersect at an angle ((Figure 4.4(iv)) so that 8 such tileswill supply the 27r radians about the vertex. On the conformally flat Poincare Disk, wecan use plane geometry to findr ( 4 ) =1 2 + 2V-2- • (4.13)This r corresponds to a boost magnitude of ri^) p(i) 1ln [ 1 + 1r^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 berepeated for hypersurfaces at arbitrary 7, but the result is the same. This magnitude isactually independent of 7 and generates curves on all hypersurfaces of constant 7 fromwhich these octagonal tiles can be constructed.4.3.3 HolonomiesThe 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 originand that the 0 0 ray bisects side A. Corner 1 lies at 0 = -1, corner 2 at 0^corner3 at 0 =^and so on (See Figure 4.5). The transformations W between the identifiedsides can be easily found by recalling the procedure used to construct the curves whichChapter 4. A Two-Holed Torus^ 59Figure 4.5: The octagonal tile lying on a hyperbolic hypersurface of constant time r. InMinkowski space, every point on the tile is the same distance 7 from the origin.Chapter 4. A Two-Holed Torus^ 60became the sides of the octagon. Hereon, p will refer to the value p(i) produced fromi'(i). Just as a boost in the y-direction by p drops the curve over the x-axis down toform side C of the tile, a boost in the (—y)-direction by p (or equivalently a boost in they-direction by —p) will lift side C back up to a curve lying over the x-axis. Rotate thiscurve by -Fi (+ to generate the right orientation), and boost it by p in the x-directionto drop it back down onto side A. This sequence of SO(2,1) transformations maps sideC onto side A. As SO(2,1) is a group, the composition of the three is a single Lorentztransformation, which we call WCA•A general Lorentz boost of magnitude 1u in the 0-direction is i -+ A(,u, 0)Y wherecosh Lt^sinh ,u cos 0^sinh ,u sin 0^Nsinh ,u cos 0 (cosh ,u — 1) cost + 1 (cosh — 1) cos 0 sin 0sinh ,u cos 0 (cosh ,u — 1) cos 0 sin 0 (cosh ft — 1) sine c + 1 /Rotations about the origin in the xy-plane by angle 0 are produced under the transfor-mationI 1^0^0R(0) = 0 cos 0 — sin 0\0 - sin zi)^cost'The transformations WCA , then, is given by: maps side C onto side A7r^ 7r= A(p, 0)R(-2-)A(p, 3)cosh 2p 0 — sinh 2psinh 2p 0 — cosh 2p0^1^0This method of lift-rotate-drop gives each of the W transformations:WCAWCAChapter 4. A Two-Holed Torus^ 61maps side D onto side B—4 )R( -2 )A(P, —4 )1^cosh 2p^sinh 2pv 2 sinh 2p \v 2 sinh 2p^sinh2 p^— cosh2 psinh 2p^cosh2 p^— sinh 2 pmaps side G onto side E^A(p,7r)R(i^ 2 )cosh 2p 0 sinh 2p— sinh 2p 0 — cosh 2p■ 0^1^0maps side H onto side F(0, ) ( 7r ) (0,^)A., , 4 , R. 2 , A , 4 ,cosh 2p^—  sinh 2p v,-2 sinh 2p \sinh 2psinh2 p^— cosh2 psinh 2p^cosh 2 p^— sinh2 p4.3.4 The Fields tiab and paThe symmetry of the model for T2#T2 allows us to find a single transformation WcA thatmaps side C onto side A. Because WcA is independent of the pair of identified points onthe 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 Uon opposites sides of a boundary. We attach the W onto the octagon as we did for thesquare tile of T 2 , by regarding the holonomy as a "phase difference" between identifiedpoints.WDBWDBWGEWGEWHFWHFWH F 1 WG EWH FWGE or WDBWC AWD B 1 WC A-1 WDBChapter 4. A Two-Holed Torus^ 62Figure 4.6: Values of the phase U on the corners of the r#T 2 's octagonal tile. Theconstraints guarantee U(1) is well defined.Again we start on the corners of the tile (Refer to Figure 4.1(i)). Since only thephase 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, aswe found before (See Figure 4.6). By starting on corner 3 and transforming to eachcorner 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 simplematter of interpolating these vertex values over the whole tile, while preserving the phasedifference between identified points. A simple choice is a linear interpolation between thecorners 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 theXA\ YA. /= WCAChapter 4. A Two-Holed Torus^ 63transformations W are in SO(2,1) and connected continuously to the identity. These twointerpolations together give a continuous field over the tile with points on the boundarydiffering by the appropriate transformations W. Smoother interpolations can be used, ifnecessary.The last component of this model is the field pa, introduced to write the dreibein aseA a = DA pa . The choice we make is a very simple one; too simple, perhaps, for it sets thetranslations H to 0. Consider the choice pa = xa = (t, x, y) in Minkowski space, givingex = (0, 1, 0) and ey = (0, 0, 1). Recall the solutions to the equations of motion are thosefor which Rii, ab = 0, so in the solution space we can take e t = (1, 0, 0) as well. The twofields U and p completely determine the holonomies. We found in (2.8)W21 ab P2 a -I- 1121 a = /h awhere p1 and p2 are the values of p on either side of the identified boundary. Thetransformations WCA, for instance, maps (t, x, y) on side C onto (t, x, y) on side A:But this is exactlya = WCA ab PCPA ^7showing the translation fI cA a is not needed to reproduce the discontinuity in p across theCA-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 theconfiguration is (momentarily) stationary. This is consistent with the geometry of thetile being independent of the time 7 on the spatial slice, a property we found in (4.14)when we built the tile.Chapter 5Canonically Conjugate Variables and QuantizationWe have seen from the two surfaces studied here that the Einstein action, on the solutionsto the equations of motion and with the constraints imposed, is of the form17'2 = (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}. = wbxwexcabcWhile this is not the vector ciW tangent to the space of W's, {W} is still tangent tosome W-space, related in a one-to-one way with the tangents of the space of Lorentztransformations. The translations H are not (quite) the momentum conjugate to theLorentz 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 thecanonical form to reveal more about the phase space.Every Lorentz transformation can be written in the formW = eir"1"where the Ja are the 3 x3 Lorentz generators introduced in §1.2. The components 11a can64Chapter 5. Canonically Conjugate Variables and Quantization ^ 65be determined as follows. Write W = eA for some A. As BeAB -1 = el3A13-1 ,eA^W= W W W-1= WeA W-1= ewAw-1so that A = WAW - 1 or [A, W] = 0. If we write A = 7rJa , then0 = qa[Ja , W]There are only three distinct components in this matrix equation due to the symmetryof the Ja . The three equations are not linearly independent, though (the right-hand-sidehas vanishing determinant, as det(Ja )=0). Another relation is needed to solve for thecomponents rya. Determinants will not suffice: As det(W)=11 = det(W) = eTrAThus TrA=0 and any constant times the matrix A will not change the determinant ofW. Instead, we considerNTr W = Tr(eA ) = eA l + eA2 + eA3where A i , A2, )t3 , the eigenvalues of A, are functions of ,.. This comes from the propertythat 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 writeIli = 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 asE^(lla a_aw \ bx wcs cabc) 7y‘ re iholonomies [Will]This action is now in canonical /*form. This expression, however, is not as well behavedas 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, qbfbi 4_ . . .for the power series 1^ , so derivatives with respect to qz maynot even exist. Furthermore, there is no guarantee that the coordinate patches aroundeach 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. Theconfiguration 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. Underthis association, we would expect the Ha, as momenta, to generate translations in theconfiguration 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 8Wbd Wdc • It is easy to verifythat cab , = (Ja) b, — these Aa are (infinitesimal) translations about the vector space ofW's. While the interpretation is by no means rigorous, it is very suggestive: The Lorentzcomponents W of the holonomies are the configuration coordinates. Translations in thisspace, 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 transformatconfigurations ofr#T2^the surfaceholonomy(i )Chapter 5. Canonically Conjugate Variables and Quantization^ 67Figure 5.1: (i) A principle ISO(2,1)-bundle over the base space T 2#T2 . (ii) The 'phasespace' 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#T2 , are subject to the action ofthe Poincare group ISO(2,1). This group leaves the Einstein action invariant, and itsgenerators obey a Lie algebra. These are the ingredients needed to define a principleISO(2,1)-bundle over the surface. The group action moves us along each fibre over thesurface 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 distinctpoints 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 spaceconsisting of configurations of the 2-dimensional surface (See Figure 5.1(ii)). The onepoint in the base space we have found represents the regular octagonal tile constructedabove. Other points represent asymmetric tiles and their corresponding collections ofholonomies. The momenta H are infinitesimal generators of the group action in eachChapter 5. Canonically Conjugate Variables and Quantization ^ 68fibre. The W and II vary as we move about in each fibre from one horizontal lift tothe next under global gauge transformations. Yet the projections down to the basespace of holonomies remains unchanged. If we rotate the tile, or globally boost it toa new location, a similarly transformed collection of holonomies is produced. The tilerepresenting this new surface, though, is essentially unchanged, merely displaced. Thecanonical variables W and H are phase space coordinates in this bundle: The W areconfiguration coordinates in the fibre over a basepoint, a particular configuration of thetile representing the surface. The momenta H generate translations along this fibre un-der the infinitesimal group action. Different horizontal lifts, all differing by global gaugetransformations, have different coordinates (W, II), but project down to the same modelof 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(70111(0-a)] = enaJa+ , apawhere Pa , introduced with Ja in §1.2, generates translation in the xa direction. In thecase of the torus T2 , 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 arecompositions of boosts and rotations and the corresponding 71a must be evaluated bythe method outlined above. The conjugate translations IIc A , ... , IIHF all vanish soocA " = (0, 0, 0),..., UHF" = (0,0,0). This exponential ISO(2,1) representation of theholonomies matches the Chern-Simons approach. There, the gauge field is the collec-tion of flat connections At, that transform under the action of an infinitesimal parameteru --,--- 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 generatedby an infinitesimal Poincare transformation u. The coordinates W and H determine thepoint on the fibre over the base space.Chapter 5. Canonically Conjugate Variables and Quantization ^ 69When the canonically conjugate variables of a classical Hamiltonian system are known,quantization in the Schrodinger picture involves expressing the variables as operators ona Hilbert space of wave functions W. The observed values of the coordinates and mo-menta are eigenvalues of coordinate and momentum operators. Unfortunately our choiceof canonical variables W and H have ill-defined operators, for the following reason. Wesuggest that the configuration space has coordinates 7/a, the components of the vectorrepresenting W in the vector space spanned by {J1, J2, J3} . The quantum mechanical 7/aare the spectrum of a position operator Na:Na T = qa kifWhile J3 and J2 generate boosts in the x- and (—y)-directions, respectively, and havespectra 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:N1 (0)P =eW.Because rotations differing by 27r give the same reading 0, we require bothN1 (0) 'P = 64111V1 (0 + 27) = .Thus N1 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 N1 , elN1for instance, is needed for a well-defined operator. At the same time, the Hilbert spaceis 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 thegauge group ISO(2,1), perhaps this problem can be circumvented by re-defining theconfiguration space modulo S 1 : R3181 0 R3/si 0 • • • ® R3is1, one term for each of theChapter 5. Canonically Conjugate Variables and Quantization ^ 70Figure 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, thecoordinate 7/ 1 0 is periodic and the Hilbert space of wave functions has a "cylindrical"geometry.Chapter 5. Canonically Conjugate Variables and Quantization ^ 712g holonomies of the flat genus g surface. In any event, this complex structure of thephase space arises from the choice of W and H as canonically conjugate variables, andthis interpretation is only speculative, based on qualitative observations.Chapter 6Conclusions6.1 Generalization of the ResultThe ease with which we jumped from the torus to the two-holed torus suggests thisformulation of the Einstein action can be applied to all higher genus surfaces. By choosingan appropriate r on the Poincare Disk to produce a regular (4g)-sided polygon with aninterior angle of alr- tilings of the hyperbolic plane are produced. Algebraic topology4gdescribes a genus g surface, a connected sum of g tori, in terms of 2g cycles [9]:A -1 A-1 A -1A-1^-1A-1ui a2u2 a2 u2 • • • a l/ 1_4 11gg g gWe use this expression to read off the identification of sides of the tile. In the r#T 2case, al b1 adbTl a2 b2aZ1 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 arepresentation of the fundamental group of the surface in the gauge group ISO(2,1). Eachadditional "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. Theconstraint that the transformation producing a complete circuit of the vertex is theIdentity ensures each additional torus glues smoothly to the rest:W1 W2 -1 W1 -1 W2 • • • W2g-}-1W2g+2 1 W2g+0 W2g+2 = 1W2g+2 1 W22+1 W2g+2^=^W2-1 W1 -1 W2 • • • W2g -1 W2g 1 W2g -1 1 W2gT92+ 1 # 7-112 #^# 29 )72Chapter 6. Conclusions^ 73These 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 moreterms, (IT)\ ---- / 2g+1{W}2g+1 and (n)__,2g+2{W}2g +2, are added to the action to account for thesenew degrees of freedom. The problems of conjugate variables and quantization are stillpresent, but not further obscured by the increase in genus.6.2 Future ResearchThe Chern-Simons action based on gauge field theory and the usual Einstein actionof GR are two different representations of the same system. The former deals withthe gauge group theory of a principle ISO(2,1)-bundle over a (compact) surface, whilethe latter looks at invariants of the Einstein action over a (2+1)-dimensional splittingof 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 dreibeine: and connection w,,a, Witten [1] shows that the 2 representations differ only by atransformation 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 specificcase of (2+1)-dimensional spacetime. There is reason to suspect, however, that the twoapproaches are equivalent only under special circumstances.Recall from §1.2 that the variation of At, is given by SA A = Se: Pa + Sw:Ja wherea abcse a abcE Cilbrc — E WgbPcscot: = —T a — abc Lc)br c •Under the substitution pa = ViL etta these transformations coincide, on the solutions tothe 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^ 74this gauge transformation is a physical coordinate transformation generated by the vectorfield 174 = paella . Note that Va = Va(eaa ), where eac, = g4Lve„•Now on all but the torus T 2 , the tangent vector field to the spatial surface musthave at least one singularity. Suppose the spatial ei a fail to span the tangent space atthe point V'. Where does this point "slide" to under the diffeomorphism generated bypa = Vaeir We cannot say, as the vector field Vµ(e) cannot be determined! Theperfectly acceptable gauge transformation is no longer a coordinate transformation andhence 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 pointin flat spacetime where the dreibein is not singular. At this point, we perform the gaugetransformation with parameter pa chosen such that pa4 = e4a, so that Sem(' = — e4a andthe dreibein becomes singular. This simple gauge transformation does not have a corre-sponding coordinate transformation. Furthermore, we cannot perform a gauge invariantcoordinate transformation to get away from this singularity, for the diffeomorphism mustbe generated by VI-Lelia which vanishes for all V. This suggests we can take two verydifferent spacetime (Euclidean and Minkowski R3, for instance) and glue them togetherat this point e. Moreover, there are gauge transformations that allow us to pass throughthis 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 changessignature. 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 approachthat suggests this formalism is somehow "larger" than GR. On all but the torus, thevector field tangent to the spatial surfaces must vanish at one or more points. Thismeans the full spacetime metric must either be singular, or at least time-like. One of theChapter 6. Conclusions^ 75gauge transformations allowed by the dreibein representation of the Einstein action is theinternal transformation coa --> elf + D„pa. In general, this transformation will changethe metric components la,tiv = epaeva• At the point(s) where the metric g, is time-like, agauge equivalent metric .g,,,, may be space-like or null — clearly this is not a coordinatetransformation. This phenomenon is prohibited in the usual form of GR written interms of the metric and its derivatives. There is a freedom allowed by the dreibeinformalism not allowed in GR. It may be possible to construct an explicit model andstudy 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 acoordinate transformation.General Relativity is simpler in (2+1)-dimensions in the dreibein formalism becausethe 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 arefolded up into compact surfaces, the torus T 2 and the two-holed torus T 2#T2 . While theyremain flat, removing the geometric degrees of freedom, the topology of these higher genussurfaces 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 overthe 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 andthe 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. Wemake this interpretation because the translations Ha are related to infinitesimal Lorentztransformations, just as classically, momenta generate translation in configuration space.While the action we have produced does not truly reveal the dynamical variables ofChapter 6. Conclusions^ 76this spacetime, and quantization of the phase space is not obvious, the method employedto reduce the action is quite revealing. It is apparent that the correct phase space of theclassical conjugate variables is not simply R.2" . It is most likely curved, and quantummechanics on curved space is a problem that will not be tackled here. The result raisedinteresting 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)-DimensionalQuantum 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 UniversityPress).[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 (NewYork: 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:LoopAmplitudes, Anomolies and Phenomenology (Cambridge: Cambridge UniversityPress). Chapters 4 and 12 contain an introduction to the dreibein formalism.77Appendix AProperties of Lorentz Transformations UabBy definition, a SO(2,1) Lorentz transformation Uab must keep the metric lab invariant:uac ubcocd = TiabFrom this we seeuctc ubc = labOr uac ?lb c _ Sg, (A.1)A similar relation can be derived can be derived from this one. Re-write (A.1) asud cridaubc = riabso thatud c 7idauboab = 1 .^ (A.2)Suppose, in all generality, that the transformation U has both different left and rightinverses: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) R78Appendix A. Properties of Lorentz Transformations tiab^79so that the left and right inverses are the same. From (A.2) we see ridaubc7lab is the rightinverse of lid c . Left and right inverses coincide, soridaubf 77ab /id =which is equivalent to tra b sab (A.3)A third property of Lorentz transformations is also used in producing the results ofChapter 2. We come across terms of the formCabc Uas Uby Ucz •The indices x, y, z are still Lorentz indices, named from the end of the alphabet to clarifythe index maniupulations. Relabel the dummy indices b, c:Ea bc Uax tiby ttc, = Cacb teix U ey Ubz= —6abctiaxUcyU 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 thereforemust be proportional to the only totally antisymmetric 3-tensor, c xyz : WriteCabcUaxilbyticz = A 6syzSolve for A:= exYz fabc Uax Uby ticz= det(Uab ) .These Special Orthogonal matrices have unit determinant, so A = 1, andCabc Uas Uby IA% = 6xYz (A.4)Appendix BSetting CJi ab = 0 and e a = 0We have writtenw„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, thereis a Uab for whichab =^pcb,,)orUa a cb,i = ticWib • (B.1)Formally we write the solution of this matrix equation= ef wp dsi`This is not well-defined as /4(x4 ) will in general have a different value for each path -y(A)integrated over, asde = dyµ(A) dAdAwill 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 under80Appendix B. Setting (:Di ab = 0 and ei a = 0^ 810 = Ao < Al < • • • < An = 1 with -y(A k ) = xk. Then the integral (B.2) is the limitingvalueurn ewP(„,),Axr,, • • • ewp(s 1 )Axii' ewp(x0 )Axt;n—).00Each term ewP(xk) °4 is a transformation acting on the term to the right, so that (B.2) isan 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 thendx(x k ) = d-y(Ak)dA dA Ax kandd ^f x0Pe -Y P d = Pef-Y wPdx1A 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 overa 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 bothdxv Pe f- Y wPdx.P = Pef-Y w"dx1A co,, ,p^=^wpdyPtiv 7dyvit is not true in general that, say,pef wjA cist`^coudxt`i^= Pe' ^co,, ( xk) ,dyvfor in generald-y(An )^d-y(-Ani)dx(x,i ) = dA dA ^ dA = dy(yrn )•dAAppendix B. Setting cVi ab = 0 and et a = 0^ 82Figure 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: Paralleltransporting a vector Va about an infinitesimal loop (See Figure B.1) changes Va in-finitesimally bySVa = ATASVbTt'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 directionsTA, 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 becauseRij ab = 0, the result holds for all paths on the surface:  i(xn ) li(x0wi(xn)dY2 (yn)for some other ya = yµ ( a) passing through xn.Appendix B. Setting iDi ab = 0 and ei a = 0^ 83Figure B.2: Two infinitesimally different paths to xn =Consider an infinitesimal change in the path 7 (See Figure B.2). Along coincidentpaths, U(xo ) = 14(y0 ),14(x i ) ,14(xn_1) =-14(y„_ 1 ). Whereas U(xn ) is calcu-lated in the limitllm ewp(Xn-1)AXt7:-1U(Xn-1)n—).coU(yn ) is calculated by traversing the loop of area —Axn_ i Ayn_i. The two paths fromXo = yo to xn = yn areXo, xi • • ' Xn-27 Xn-17 XnYo, Y1 • • • Yn-2) Yn-11 Y1) Y21 Yn •Their difference is the loopXn Yn-11 yl, Y27 Yn •If U(xn ) had been calculated by path-ordered integration along the varied path, then theAppendix B. Setting c.7.) i ab = 0 and "ei a = 0^ 84difference in its value would beSUab = xn- Agn-i X2 Y3 VbRuc a — UacRub 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 holdingat each stage. The result tells us that the value of U ab at xiL is independent of the pathalong which the integration occurs. Therefore, in checking that (B.1) is true at anypoint, we can assume that the path used to calculate WI, in (B.2) is the one for whichdx = (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 thetime components undetermined.In the coordinate system where CJi ab = 0, we transformed the dreibein undernwoaIn this coordinate sytem, Di pa = io`:i as the spatial C3i ab = 0. We claim that becauseD[iej Cil = 0, or E[iaji = 0, we can find a particular pa(xa) such that ema = 0. The solutionto the PDEe a^na (B.3)can formally be written aspa = PI e4adxii-ywhere again P indicates path-ordering along the path -y. The result (B.3) holds alongthe path where dxa = (d-yi DOA but the result need not be true for different paths.Like the co4ab case above, we interpret the path-ordered integral as a limit. Discretizethe path -y(A) so that xk = ry(Ak) and Axk is tangent to the path at, say, xk Axk/2.Appendix B. Setting C.3 i ab = 0 and ei a = 0^ 85The point where we evaluate does not matter in the limit. The path-ordered integral isthe limiting valueAx,2^Ax^ Axlim^ 1 e a(x0 + ^ )A4^2+ a(x i +^),A4 +^a(x„  ^A^2 nn)Ax .1-i To show that the value of pa(e) is in fact independent of the path to xa, again considerthe two paths between x, = yo and xn = y9., used above (See Figure B.2). The differencebetween the two paths, recall, is the loopXn Yn-11 yl, y2, Yn.This loop will contribute to the sum (or 'integral') an amountAxnetia(xn^_t^ AYn-1^Aepa(xn_ i2 2),A4^e a02 AYn-1 ^e a 01 + ^Axn_i2^-1^A^2These paths, and this loop, lie in a surface of constant time, so the displacements Ax kand Ayk have no time-, or 0-, components. Furthermore, we can assume that thesedisplacements are in the spatial 1- and 2-directions, for any loop can be approximated toan 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 contributesAye2a (xn-1 2 Ax) — e2a (xn-1 AxAy . .AxIn the limit where n -4 oo, this is simply(6172 - 6271 ) AxAy .The constraint tells us e[a3] = 0, so variations in the path do not affect the value ofthe path-ordered integral. Because of this path independence, we can always assumedx2 (d-yi /dA)clA so that el a = Apa . Therefore, et a must vanish.e1 'L — t + AY) — ei a (xn — t) (Appendix CProperties of {W}We define the symbol{ W}a = Wbx wcx cab,so thatHa -r -i- rbsW Wcs Eabc = 11{W} .There are three properties of this symbol which will simply the algebra of reducing theaction to a more canonical form.By definition,(un){uW1 = proaubyti/Yxuczwzx cabc= fictuad . ; -, ,u ucz IksWzx cabcRecalling property (A.4) of Lorentz transformations U,= rid Wyx wzx cabcRelabelling dyz^abc gives=_ Ha Wbs wcs cabcThat is,(un){uW} = H{W}^ (C.1)86Appendix C. Properties of {W}^ 87Next, in full component form,H { w1w2 } = Ha (wi b dw2 d s)(wi b dw d2 ) 6abc=^dw2c1 s wi c e w2 ex eabc+Ha W1 b^d0472 x Wl c e W2 ex EabcNow the first term can be rewritten by first raising and lowering the e:• b^ b^cdHa Wi d Wi ce dW2 x W2 x Eabc = II ^dW1 Eabcbe= 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 roWhile (W2 -1 11) {W2 -1 W1 } H11141 by (C.1),(w2_11-)^_ 0472-1472_11-0 {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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