The geometry of dynamical triangulations

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Jan Ambj0rn Mauro Carfora Annalisa Marzuoli

The Geometry of Dynamical Triangulations

Springer

Authors Jan Ambj0rn Niels Bohr Institute, University of Copenhagen Blegdamsvej 17, DK-2100 Copenhagen, Denmark Mauro Carfora International School for Advanced Studies, SISSA-ISAS Via Beirut 2-4, 1-34013 Trieste, Italy Annalisa Marzuoli Department of Nuclear and Theoretical Physics University of Pavia Via Bassi 6, 1-27100 Pavia, Italy CIP data applied for.

Die Deutsche Bibliothek - CIP-Einheitsaufnahme

Ambjsrn, Jan:

The geometry of dynamical triangulations / Jan Ambjfl}rn ; Mauro Carfora ; Annalisa Marzuoli. - Berlin; Heidelberg ; New York ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris ; Santa Qara ; Singapore; Tokyo: Springer, 1997 . (Lecture notes in physics: N.s. M, Monographs; 50) ISBN 3-540-63330-8

ISSN 0940-7677 (Lecture Notes in Physics. New Series m: Monographs) ISBN 3-540-63330-8 Edition Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re- use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1997 Printed in Germany

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Preface

The express purpose of these lecture notes is to go through some aspects of the simplicial quantum gravity model known as the dynamical triangulations approach. Emphasis has been on laying the foundations of the theory and on illustrating its subtle and often unexplored connections with many distinct mathematical fields ranging from global Riemannian geometry, to moduli theory, number theory, and topology. Our exposition will concentrate on these points so that graduate students may find in these notes a useful exposition of some of the rigorous results one can -establish in this field and hopefully a source of inspiration for new exciting problems. We try as far as currently possible to expose the interplay between the analytical aspects of dynamical triangulations and the results of Monte Carlo simulations. The techniques described here are rather novel and allow us to address points of current interest in the subject of simplicial quantum gravity while requiring very little in the way of fancy field-theoretical arguments. As a consequence, these notes contain mostly original and until now unpublished material, which will hopefully be of interest both to the expert practitioner and to graduate students entering the field. Among the topics addressed here in considerable detail are the following. (i) An analytical discussion of the geometry of dynamical triangulations in dimensions n == 3 and n == 4. (ii) A constructive characterization of entropy estimates for dynamical triangulations in dimension n = 3, n = 4, and a comparision of the analytical results we obtain with the data coming from Monte Carlo simulations for the 3-sphere §3 and the 4-sphere §4. (iii) Indications (convincing, we feel) that our analytical model and the numerical simulation of the genuine model provide the same critical line k4(k 2 ) characterizing the infinite-volume limit of simplicial quantum gravity. (iv) An analytical characterization of the critical point k2rit in our analytical model, which is in good agreement with the location of the critical point obtained by Monte Carlo simulations. (v) A simple entropical understanding of the fact that the weak coupling phase of simplicial gravity seems to consist of manifolds that degenerate to branched polymer-like structures. (vi) We also show that in the 3-dimensional case the comparison between the analytical and numerical data is very satisfactory. Trieste, Italy, May 1997

J. Ambj0rn, M. Carfora, A. Marzuoli

VI

Contents

1.

Introduction.............................................. 1 3 1.1 The Model: Simplicial Quantum Gravity 1.1.1 Summing over Topologies 5 1.1.2 Simplicial Quantum Gravity. . . . . . . . . . . . . . . . . . . . . . . 7 10 1.2 Summary of Results

2.

Triangulations............................................ 2.1 Preliminaries: Simplicial Manifolds and Pseudo-manifolds. . .. 2.1.1 Piecewise-Linear Manifolds. . . . . . . . . . . . . . . . . . . . . . .. 2.1.2 Dehn-Sommerville Relations. . . . . . . . . . . . . . . . . . . . . .. 2.2 Distinct Thiangulations of the Same PL Manifold. . . . . . . . . ..

17 18 20 24 33

3.

Dynamical Triangulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.1 Dynamical Thiangulations as Length-Spaces. . . . . . . . . . . . . . .. 3.1.1 PL Connections and the Incidence Matrix. . . . . . . . . .. 3.1.2 Curvature Assignments 3.1.3 The Einstein-Hilbert Action for Dynamical Triangulations . . . . . . . . . . . . . . . . . . . . .. 3.2 Dynamical Thiangulations as Singular Metric Spaces . . . . . . .. 3.2.1 Geodesic Triangles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.3 Approximating Riemannian Manifolds Through Dynamical Thiangulations . . . . . . . . . . . . . . . . . . . . . .. 3.4 Topological Finiteness Theorems for Dynamical Thiangulations of Bounded Geometry

39 40 42 44

Moduli Spaces for Dynamically Triangulated Manifolds. .. 4.1 Romer-Zahringer Deformations of Dynamically Triangulated Manifolds. . . . . . . . . . . . . . . . . . .. 4.2 Dynamical Thiangulations and Locally Homogeneous Geometries. . . . . . . . . . . . . . . . . . . .. 4.2.1 Locally Homogeneous Geometries 4.2.2 Moduli of Locally Homogeneous Geometries 4.3 Moduli of Dynamical Triangulations . . . . . . . . . . . . . . . . . . . . .. 4.4 Gauge-Fixing of the Moduli of a Dynamical Triangulation. ..

69

4.

46 49

51 56 65

71 75 76 79 82 86

VIII

Contents

4.5

5.

6.

7.

A Measure on the Moduli Space. . . . . . . . . . . . . . . . . . . . . . . . .. 4.5.1 Moduli Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.5.2 Moduli Asymptotics for DT-Surfaces. . . . . . . . . . . . . . .. 4.5.3 A Compact Formula for Surfaces . . . . . . . . . . . . . . . . . ..

88 90 92 93

Curvature Assignments for Dynamical Triangulations. . . .. 5.1 Partitions of Integers and Curvature Assignments. . . . . . . . . .. 5.2 Distinct Dynamical Triangulations with Given Curvature Assignments 5.3 The Counting Principle 5.4 A Remark on SU(2) Holonomy

95 95 102 120 121

Entropy Estimates 6.1 The Asymptotic Generating Functions for the Enumeration of Dynamical Triangulation 6.2 Gauss Polynomials and Dynamical Triangulations 6.3 Dynamical Triangulations and co-Dimensional Operators 6.4 Asymptotics and Entropy Estimates 6.5 The 2-Dimensional Case 6.6 The n ~ 3-Dimensional Case 6.6.1 The Infinite Volume Limit 6.7 Distinct Asymptotic Regimes 6.7.1 Strong Coupling 6.7.2 Critical and Weak Coupling 6.7.3 Weak Coupling and Complete Polymerization

125

Analytical vs. Numerical Data 7.1 The 4-Dimensional Case 7.2 Polymerization 7.3 Summing over Simply-Connected 4-Dimensional Manifolds 7.4 The 3-Dimensional Case 7.5 Concluding Remarks: the Order of the Phase Transition

157 158 160

125 126 127 129 132 132 142 144 145 151 153

167 170 173

A. Appendix A.1 Pachner Moves A.l.l The 2-Dimensional Case A.1.2 The 3-Dimensional Case A.1.3 The 4-Dimensional Case A.2 The Tangent Space to the Representation Variety

175 175 178 178 180 182

References

187

Index

193

1. Introduction

Recent. years witnessed a massive introduction of methods of statistical field theory in attempts to quantize gravity along the lines of a conventional field theory [7, 41, 40, 90], [42, 49, 51, 81, 77]. Such an emphasis is the direct outspring of the possibility of discretizing gravity in a way consistent with the underlying reparametrization invariance of general relativity [114, 74],[42, 49, 51, 81]. The basic idea is to use a variant of the standard Regge calculus [99], first suggested by Weingarten[113], in an attempt to make quantum gravity well defined as a lattice statistical field theory. This discretization is known as the theory of Dynamical 'friangulations (DT) or simplicial quantum gravity. It consists of a replacement of the (Euclidean) functional integral over all equivalence classes of metrics with a summation over all abstract triangulations of the given manifold. The fixed edge-length of the links of the triangulations plays here the role of lattice spacing of the more familiar lattice gauge theories. The basic idea is to search, in the parameter space of these discretized quantum gravity models, for critical points where the lattice spacing can be taken to zero and contact can be made to continuum physics. Credit to this picture is lend by noticing that within this framework, Euclidean 2-dimensional quantum gravity can be formulated as the scaling limit of a dynamically triangulated collection of surfaces thought of as an ordinary statistical system. Contact with the continuum, as described by Liouville theory[43, 47], can successfully be made, and the robust alliance of the theory of critical phenomena and Monte Carlo simulations even sheds light on many aspects of the theory not liable of an analytical approach. If we extend the theory to higher dimensions we have to face the fact that we have no continuum theory of Euclidean Quantum Gravity with which we can compare, nonetheless many aspects of the theory can be investigated by computer-assisted simulations with a good degree of precision. These numerical methods allow us to get a rather complete picture of the phase diagram of the discretized theory and they serve as an important inspiration for analytical studies of higher dimensional models of quantum gravity. These lecture notes attempt to partially fill the gap between such numerical studies and a fully- fledged analytical approach to higher dimensional dynamical triangulations. Our emphasis here is on the applications of elementary ideas of Piecewise-Linear (PL) geometry, global riemannian geometry, and number

2

1. Introduction

theory in order to understand dynamically triangulated models of quantum gravity in dimension n = 3, 4. The occasion for such an analytical approach comes by exploiting the techniques developed for the recent proof[16, 34, 35] of the conjecture concerning the existence of an exponential bound to the number of distinct triangulations on a PL-manifold Mn, (n ~ 3), of given volume and topology. This exponential bound is germane to the definition of dynamical triangulations as a lattice statistical field theory model. Analytical proofs are well known in two dimensions[23, 31, 22, 109, 17, 18]' (see[29] for an attempt in dimension n = 3), and computer simulations supported very strongly the existence of such bound also in higher dimensions[9, 7]. The actual proof for higher dimensions draws from techniques of controlled topology and geometry initiated by Cheeger, Gromov, Grove, and Petersen[65, 59, 52], [68, 70, 71, 69, 118], and it suggests that such mathematical methods may bear relevance to the whole program of dynamical triangulations. A check with Monte Carlo simulations (in dimension n = 4) shows a good agreement between the numerical data and the analytical results obtained by pursuing this geometrical approach to higher dimensional dynamically triangulated models. It is not yet possible to reach the level of sophistication of the 2dimensional case, but as we show in these notes, the analytical methods we develop in dimensions n = 3 and n = 4, provide a good start for an analytical understanding of dynamical triangulations in such dimensions. In particular, we characterize with greater precision the entropy estimates[16, 34, 35], and provide the corresponding generating functions. Among other properties common to both the 3- dimensional and 4-dimensional case, we prove that in 4 dimensions dynamically triangulated models it makes sense to sum over all simply connected manifolds. We also discuss the existence of a polymer phase in these models[10]. We give analytically arguments in favor of a scenario where the system admits a critical point k2 = k2rit corrisponding to a phase transition between a strongly coupled phase and a weakly coupled phase of simplicial quantum gravity. The arguments presented here should be amenable to considerable improvements, and hopefully these notes can be used as a working tool. For this reason, we have presented a detailed summary of the more important and useful results in Sect. 1.2. In Sect. 1.1 we review the definition of the model. In Chaps. 2 and 3 we recall few elementary aspects of the PL-geometry of dynamical triangulations. In particular in Sects. 3.2, 3.3, and 3.4, we establish the connection with Gromov's spaces of bounded geometry. This connection provides mathematical foundation to the heuristic arguments which motivate the approximation of distinct riemannian structures with distinct dynamical triangulations. In Chap. 4 we introduce the Moduli space associated with dynamical triangulations. These moduli spaces parametrize the set of inequivalent deformations of dynamical triangulations needed to approximate riemannian structures with large symmetries. In Chaps. 5, and 6 we provide all entropy estimates needed for the higher dimensional DT models.

1.1 The Model: Simplicial Quantum Gravity

3

Here we exploit elementary number theory and geometry in order to provide such estimates and construct the associated generating functions. We discuss also the 2-dimensional case which fully agrees with the known results. The 4-dimensional case, and the 3-dimensional case are discussed starting from in Sect. 6.6. It should be stressed that the analytical results we obtain are two- fold: we obtain exponential bounds on the number of triangulations. These bounds establish the existence of the model of dynamical triangulations in dimensions higher than two. In addition we can, making few (quite reasonable, we think) assumptions, shapen the bounds to actual estimates, which in turn can be used to deduce a number of properties of the model. In this way we can compare our "analytical predictions" with the result of actual Monte Carlo simulations. In particular, we find a critical line k4 (k 2 ) where the infinite volume limit has to be taken. It agrees well with the Monte Carlo simulations. Further, our analytical results predict a phase transition between a weak coupling and a strong coupling phase of gravity. Again the agreement with the numerical simulations are quite good. The fact that our "predictions" fit well the actual numerical simulations give us confidence in assumptions made.

1.1 The Model: Simplicial Quantum Gravity Let M be an n-dimensional, (n ~ 2), manifold of given topology, and with a finite number of fixed (n - I)-dimensional boundaries E k , k == 1,2, .... Let Riem(M) and Dif f(M) respectively denote the space of riemannian metrics 9 on M, and the group of diffeomorphisms on M. In the continuum formulation of quantum gravity the task is to perform a path integral over equivalence classes of metrics: Z(A,G,Ek,h)

=

L

Top(M)

1

V[g(M)]e-Sg[A,G,El,

(1.1)

Riem(M)/Diff(M)

weighted with the Einstein-Hilbert action associated with the riemannian manifold (M, g), viz.,

+Boundary Terms.

(1.2)

The boundary terms depend on the metric h and on the extrinsic curvature'induced on the boundaries 17k, and they are such that for the action obtained by glueing any two manifolds (M1,gl) and (M2,g2) along a common boundary (E,h), we get Sgt+g2[A,G] == Sgt[A,G,E] +Sg2[A,G,E]. In

4

1. Introduction

this way, the partition function (1.1) associated with a manifold M with two boundaries (E1 , hI) and (E2, h2), satisfies the basic composition law

L Z(E

Z(E1 , hI; E 2, h2) =

1,

hI; hi, hi)Z(hi, hi; h2, h2)

(1.3)

Ei,hi

which describes how M can interpolate between its fixed boundaries (hI, hI) and (E2, h2) by summing over all possible intermediate states (hi, hi)' In

Fig. 1.1. A manifold M interpolating between the fixed boundaries (EI , hI) and (E 2 , h 2 ) by passing through the intermediate state (E3 , h3 )

order to form the partition function we need to pick up the formal a priori measure V[g(M)]. Even without pretense of rigor, it is clear that the V[g(M)] characterizing such sort of path integration should satisfy some basic properties. In particular: (i) V[g(M)] should be defined on a suitably topologized space of riemannian structures

Riem(M)jDif f(M)

(1.4)

so to avoid counting as distinct any two riemannian metrics gl (M) and g2(M) which differ one from the other simply by the action of a diffeomorphism of ¢: M ~ M, viz., such that g2(M) = ¢*gl(M);

1.1 The Model: Simplicial Quantum Gravity

5

(ii) The measure V[g(M)] should playa kinematical role and not a dynamical one. More explicitly, for a given r E jR+, let [Riem(M)/Diff(M)]xl,X2;T denote the set of all riemannian structures on a manifold M with two marked points Xl and X2 with preassigned distance dg (M)(XI,X2) = r. When restricted to this set of riemannian structures, V[g(M)] should factorize: fluctuations in the geometry of M which are localized in widely separated regions, should be statistically independent. This implies that V[g(M)] should not describe the a priori existence of long range correlations on the set of riemannian manifolds considered. Such correlations should be generated by the spectrum of fluctuations of (1.1) by means of a statistical suppression-enhancement mechanism similar to the energy versus entropy argument familiar in statistical mechanics. In a field theoretic formalism, this property is translated in the familiar requirement of ultra-locality, namely in the absence of derivatives in the formal £2 norm on the space of metrics, (the De Witt supermetric). (iii) Finally, V[g(M)] should be so constructed as to allow for the introduction of a Di f f (M)- invariant short distance cut-off representing the shortest wavelength allowed in discussing fluctuations of the geometry of M. This cut-off should be removable under appropriate circumstances, in particular when, in a critical phase, long range correlations are generated. 1.1.1 Summing over Topologies Long before addressing the physical and mathematical characterizations of such requirements on V[g(M)], both in the continuum field-theoretic formalism and in its possible discretized versions, (in this connection see the recent work of Menotti and Peirano[90]), one must also discuss the proper meaning to attribute to the formal sum over topologies appearing in (1.1): are we summing over homotopy types, homeomorphism types, or over smooth types of manifolds?

The homotopy type of a manifold is a topological notion which (with the exception of dimension two) is too weak to appear of some immediate utility. The homeomorphism type of a manifold M is the notion which is more directly related to a natural summation over topologies, however it is difficult to handle in dimension larger than two, (see below). Finally, the smooth type yields for a summation over the possible distinct differentiable structures that a manifold M can carry. This latter summation appears more relevant to a field- theoretic formalism since in an expression such as (1.1) the diffeomorphism group of the underlying manifold Dif f(M) plays a basic role, but as we shall see momentarily, a summation over the smooth types arises more naturally in simplicial gravity. Thus, this latter interpretation for ETop(M) is perhaps the most appealing since it suggests an unexpected bearing of simplicial quantum gravity on the field-theoretic formalism, and in this sense the two approaches are not alternative to each other. Simplicial quantum grav~ty may be necessary in addressing the important issue of summation over distinct topologies, whereas the field-theoretic formalism is necessary in

6

1. Introduction

order to understand the nature of the continuum limit (if any exists) of the discretized models. In dimension two there is equivalence between smooth structures and the homeomorphism types, and the summation over topologies can be given the unambigous meaning of a summation over the Euler characteristic X(M) of the manifolds, since this invariant characterizes surfaces. However, even in such a simple case it is very difficult to· obtain clear-cut results for what concerns a reliable procedure for summing over topologies. The use of matrix models of 2D-gravity and the associated double scaling limit have shed some light on this issue, but we are still far from a clear understanding[7] . In dimension three topological manifolds are uniquely smoothable[92] , and the question of summing over topologies is again synonimous of summation over smooth structures. Thus, as long as we confine our attention to the topological category or to the smooth category, summation over topologies is reduced to the yet unsolved problem of enumerating the homeomorphism types of three-manifolds. This enumeration cannot be realized as long as the Poincare conjecture is not proved. For instance, if there were a fake three- sphere then one could prove[53, 54] that there cannot be finitely many homeomorphism types of three-manifolds even under bounds on curvatures of the manifold, (typically, under curvature, volume and diameter bounds one gets topological finiteness theorems for homeomorphism and diffeomorphism types[68, 70, 71] for dimension n =1= 3,4). In dimension four the situation is even more complex since smoothing theory is not yet completely known. In open contrast to the theory of 3manifolds, there is no equivalence between topological manifolds and smooth structures, and Donaldson-Freedman's theory shows that there are topological manifolds which are not smoothable as well as manifolds admitting uncountably many inequivalent smooth structures[55, 56, 48]. The situation is further worsened by the fact that the topological classification of all (compact, orientable) 4-manifolds is logically impossible, (again, the fundamental group should be blamed for this). As stressed by Frohlich[57], this circumstance has even be used to foster the credence that only simply-connected, spinable 4-manifolds should contribute to the gravitational path integral. But clearly this point of view, if not better substained, cannot be advocated. Suppression of a class of manifolds from path integration can be justified only on a dynamical ground, as a form of statistical suppression driven by the spectrum of fluctuations of the theory. This situation does not improve even if we confine our attention to a manifold M of fixed topology and with a given smooth structure, for, in that case we have the additional problems associated with: (i) The unboundedness of the Euclidean action; (ii) The mathematical difficulties in defining a proper path integration over the stratified manifold of riemannian structures; (iii) The Einstein-Hilbert action is not renormalizable.

1.1 The Model: Simplicial Quantum Gravity

7

Fig. 1.2. Fluctuations in the geometry of M which are localized in widely separated regions should be statistically independent

1.1.2 Simplicial Quantum Gravity The hope behind simplicial quantum gravity is that some of the above problems, concerning both the characterization of the measure V[g(M)] or the issues related to ETop(M), can be properly addressed, in a non-perturbative setting, by approximating the path integration over inequivalent riemannian structures with a summation over combinatorially equivalent piecewise linear manifolds. The first attempt of using PL geometry in relativity dates back to the pioneering work of Regge [99]. His proposal was to approximate Rieman-

nian (Lorentzian) structures by PL-manifolds in such a way as to obtain a coordinate.:.free formulation of general relativity. The basic observation in this approach is that parallel transport and the (integrated) scalar curvature have natural counterparts on PL manifolds once one gives consistently the lengths of the links of the triangulation defining the PL structure. The link length is the dynamical variable in Regge calculus, and classically the PL version of the Einstein field equations is obtained by fixing a suitable triangulation and by varying the length of the links so as to find the extremum of the Regge action. If the original triangulation is sufficiently fine, this procedure consistently provides a good approximation to the smooth spacetime manifold which is the corresponding smooth solution of the Einstein equations. This approach can also be successfully extended so as to provide a quantum analogue to Regge calculus[114, 74, 75], in which we replace the formal path integration over the space of Riemannian structures with an integration over the link variables.

8

1. Introduction

Dynamical triangulations are a variant of Regge calculus in the sense that in this formulation the summation over the length of the links is replaced by a direct summation over abstract triangulations where the length of the links is fixed to a given value a. In this way the elementary simplices of the triangulation provide a Diff-invariant cut-off and each triangulation is a representative of a whole equivalence class of metrics. Regge calculus still mantains its validity and it provides both the metric assignment for the PL manifold, obtained by glueing the simplices, and the corresponding action. Since all simplices now are identical, the action will only depend on the numbers, respectively N n and N n - 2 , of n- and (n - 2)-dimensional simplices of the n-dimensional PL manifold. In this way we get that the EinsteinHilbert action for n-dimensional (Euclidean) gravity formally goes into the combinatorial action 1 S[kn- 2, kn ] = knNn(T) - kn- 2 N n- 2(T) + "2kn- 2 N n- 2(8T), (1.5) where ~Nn-2(8T) is the boundary term, and kn , and kn- 2 are (bare) coupling constants related to the cosmological constant A and to the gravitational coupling G, respectively. In particular, we can view 1/kn - 2 as a bare gravitational coupling constant. The partition function associated with such discretized action is

Z[kn- 2 , knJ

=L

e-knNn+kn-2Nn-2-!kn-2Nn-2(8T) ,

(1.6)

TET

where the summation is over distinct triangulations T in a suitable class of triangulations T. Roughly speaking, we consider any two triangulations

(say with the some number of vertices) distinct if there is no map between the vertices which is compatible with the assignments of links, triangles, etc., while T restricts the class of triangulations to those satisfying suitable topological constraints. The proper choice of the class T is strictly connected to the difficult question of how to sum over topologies in quantum gravity mentioned above. The interpretation of (1.6) as providing also a sum over topologies is based on the observation that a PL manifold is uniquely smoothable in low dimension (in particular for n = 4, by Cerf theorem, see e.g.,[55, 56, 48, 94]), (it is also worth stressing that every smooth manifold admits a natural ai-smooth PL-structure). In particular, in dimension n = 4 there is a bijective correspondence between (isotopy) classes of smooth structures and PL-structures. Thus (1.6), if no restrictions are imposed on T, subsumes in a rather natural way a summation over topologies if we decide that ETop actually means summation over smooth structures. Even if not ambiguous in its definition, (1.6) blends summation over metric structures and summation over smooth structures only in a formal way. Already in dimension two, the sum (1.6) is divergent since if the topology is not fixed, the number of distinct triangulations grows factorially with the volume of the manifold (i. e., with

1.1 The Model: Simplicial Quantum Gravity

9

the number of simplices). However, if one fixes the topology, the number of distinct triangulations of a 2-dimensional PL manifold is exponentially bounded[23, 31, 22, 109, 17, 18] according to E

(A)N2 N (x(2 2

(1.7)

) )(l'str- 2 )-1,

where A is a suitable constant and !str, the string exponent, is a topological suscettivity generated by the quantum fluctuations of the metric. In this way, LTET e-S(T) , where T is a given PL surface, is well defined. Topology is then allowed to fluctuate (i. e., one attempts to extend the summation to all PL surfaces), by a delicate limiting procedure (Double Scaling Limit). In higher dimension it is not yet known how to perform the summation (1.6), but numerical simulation as well as the results of the 2-dimensional theory, suggests that (1.6) makes sense if one fixes the topology, [9],[36]. This issue is clearly related to a rigorous characterization of the discretized counterpart of the formal a priori measure D[g(M)]. As a matter of fact, a necessary condition for attributing a meaning to (1.6), for fixed topology, is to require that the number of distinct triangulations of a given PL manifolds is exponentially bounded as a function of the number of (top-dimensional) simplices[7, 41]. Recently[16, 34, 35], the existence of the exponential bound in all dimensions and for all (fixed) topologies has been proved. The existence of the exponential bound implies that for a fixed kn - 2 there is a critical lower value kc: it (k n_ 2) of kn such that the partition function is well defined for k n > k~rit(kn_2) and divergent for kn < k~rit(kn_2). To be more specific this implies that we can introduce the (canonical) partition function for fixed (lattice) volume: The effective Entropy W(N, kn- 2 )eff ~

L TET(Nn

ekn-2Nn-2(T) ,

(1.8)

)

where T(Nn ) denotes the class of distinct triangulations of fixed volume (Nn ), fixed topology and boundary conditions. This effective entropy will characterize the infinite (lattice) volume limit of the theory, defined by the approach to the critical line k n -7 k~rit(kn_2) in the (k n- 2, kn ) coupling constant plane. The existence of the infinite volume limit is a necessary but not sufficient condition for the existence of a physically significant continuum limit of the theory, and we only expect interesting critical behavior, i.e., the onset of long range correlations, at certain critical values of kn - 2 . In the rest of these notes we will be mainly interested in the analytical characterization of the canonical partition function (the effective entropy) W(N, kn - 2 )e//, and in comparing the obtained results with the existing numerical data coming from Monte Carlo simulations. It is not difficult to see the origin of the difficulties in dealing with the set of dynamically triangulated manifolds (of given volume and topology) considered as a statistical system described by (1.8). Roughly speaking we have a

10

1. Introduction

collection of identical simplices whose only interaction is basically associated with unpenetrability and glueings according to certain rules. The energetic term is very simple and controls volume and average curvature. Thus the free energy of the system is basically characterized by the entropic factor enumerating the number of distinct dynamical triangulations of given topology, volume, and average curvature. Such an estimate is deeply non-local, (e.g., see (1.7) for n == 2), and as such its characterization is conceptually different from the standard approaches used to enumerate distinct configurations, say of spins, on a rigid lattice. This situation is not new in statistical mechanics since it is reminiscent of what happens when dealing with the hard sphere gas: a set of identical spherical particles whose only interaction is associated with unpenetrability. Here too, the free energy is of entropic origin, and the characterization of such an entropy is a highly non local problem since it is related to the characterization of the densest packings in spheres: the insertion of a sphere may change the packing up to a distance proportional to the inverse of the average separation between the surfaces of the spheres. For dynamical triangulations, as we shall see, the source of non- locality is subtler since there is a delicate feedback with the topology, (e.g., see (1.7), where the critical exponent IS depends from the Euler characteristic of the surface obtained by glueing the simplices). But these topological difficulties would be present also in the hard sphere gas had we considered the spheres evolving in a topological non-trivial ambient space. Thus, it is perhaps not so surprising to note that the first proof of the existence of an exponential bound for the entropy of dynamical triangulations was carried out[16, 34, 35] by exploiting the geometry of sphere packings in riemannian manifolds. This interplay between the geometry of packings, topology, and the metric properties of riemannian manifolds, (or more general metric spaces), can be traced back to ideas of M. Gromov, J. Cheeger, K.Grove, and P.Petersen. These ideas have sprung a renaissance in recent developments in Riemannian geometry, and we will exploit them here by showing that they are a good source of inspiration also in simplicial quantum gravity and in the fascinating field of the statistical mechanics of extended objects.

1.2 Summary of Results For the convenience of the reader we present here a summary of the constructive results that we prove in these notes and that can be useful in applications to higher dimens~onal dynamical triangulations. If not otherwise stated, we refer to dynamically triangulated manifolds, M, in dimension n ~ 2. The dynamical triangulations in question have N n - 2 (T) ~ A + 1 bones a n - 2 , and Nn(T)~N top-dimensional simplices an. We denote by {q(k)}~=o, with q(a) ~ q, (typically q == 3), the string of integers providing the numbers of top-dimensional simplices incident on the A + 1 bones, and often refer collectively to such numbers as curvature assignements. Note that, for a given PL-

1.2 Summary of Results

11

manifold M, we consider the set of distinct triangulations {T(i)} with given number of bones and a given average number, b(n, n-2) ~ !n(n+1)(N/ A+1), of n-simplices incident on a bone. Sometimes we need to mark n of these bones and a top-dimensional simplex incident on one of them. We refer collectively to triangulations with such markings as rooted triangulations. The topology of the triangulations explicitly enters our results through the representation variety Hom( 1rb(M),G): the set of conjugacy classes of representations of the fundamental group of the manifold, 7rl (M) into a (compact) Lie group G. This representation variety parametrizes the set of inequivalent deformations of dynamical triangulations needed to approximate riemannian structures endowed with G-structures, (for instance, this parameter space tells us how to deform a given 3-dimensional dynamical triangulation, approximating a flat torus, in such a way as to describe unambiguosly also the inequivalent flat 3- tori infinitesimally near the given one). Our enumeration procedure, in establishing entropy estimates, is based on the observation that the number of distinct dynamical triangulations admitted by a manifold of given topology, (with given number of bones A + 1 and given average incidence b(n,n - 2)), is provided by

an

Card{T(i)} == p~urv < Card{T(i)}curv >,

(1.9)

where p>..urv is the number of distinct curvature assignements over the A + 1 bones, and < Card{T(i)}curv > is the average (with respect to p~urv) of the number of distinct triangulations sharing a common set of curvature assignments, (see Sec. 5.2). Since our enumeration procedure overcounts the number of distinct curvature assignements p~urv admitted by dynamical triangulations of given topology, in what follows we also introduce a normalizing factor Cn , of the form

(1.10) where the constants an, and a n-2 ::; 0 depend only on the dimension n, (and on the kinematical bounds b(n, n - 2)min and b(n, n - 2)max between which b(n, n - 2) varies). The structure of this normalizing factor follows from a subtle interplay between geometry and general subadditive arguments. Note that in dimension n == 2, Cn simply reduces to a constant, whereas in dimension n == 4, a n -2!n=4 == 0, and a n =4 == -~ In(cos-1(1/4)) and similarly in dimension n == 3, (where however a n -2In=3 =I 0). With these notational remarks along the way, we have: Transition Between Weak and Strong Coupling. If n 2:: 3, there is a critical value bo(n), of the average incidence b(n, n - 2), (sufficiently near to the lower kinematical bound b(n, n - 2)min), and to which we can associate a critical value k~~~ of the inverse gravitational coupling, such that if

b(n, n - 2)min ::; b(n, n - 2; {q(k)}) ::; bo(n),

(1.11)

then, as A ~ 00, the rate of growth of the average number < Card{T(i)}curv > of rooted triangulations is at most polynomial, viz., there are constants,

12

1. Introduction

J-t(b(n,n-2)) > 0, andT(b(n,n-2)) 2:: O,(possibly depending onb(n,n-2)), such that

< Card{T(i)}curv

>~ J-t(b(n, n -

2)) . N nr(b(n,n-2).

(1.12)

Note that this polynomial rate of growth also holds in the two- dimensional case for < Card{T(i)}curvln=2 >. Conversely, if bo(n) < b(n, n - 2) :::; b(n, n - 2)max,

(1.13)

then the asymptotics of < Card{T(i)}curv > is exponential. Namely there is a constant m(b(n, n - 2)) > 0, possibly depending on the average incidence b(n, n - 2), and an nH 2:: n such that

< Card{T(i)}curv

>~

J-t(b(n, n - 2)) . exp[-m(b(n, n - 2))N~/nH]N~(b(n,n-2)),

(1.14)

as N n goes to infinity.

The Generating Function. Let 0 :::; t :::; 1 be a generic indeterminate, and let p)..(h) denote the number of partitions of the generic integer h into (at most) ,x + n - 1 parts, each:::; (b(n, n - 2) - q)(,x + 1). In a given holonomy represention B:1rl(M;ao) ~ G, and for a given value of the parameterb ==

b(n, n - 2), the asymptotic generating function for the number of distinct rooted dynamical triangulations with Nn_2(T~i») == ,x + 1 bones and given number (ex h) of n-dimensional simplices incident on the n marked bones is given by Q[W(8,,x, b; t)] == Cn· < Card{T(i)}curv > . LP)..(h)t h == h~O

Cn

. [(b(n,n-2)-q)(,x+l)+(,x+l)-n] (,x+l)-n '

(1.15)

where Cn is the above-mentioned normalizing factor, and

is the Gauss polynomial in the variable t. The Rooted Entropy. In a given represention B: 1rl(M; ao) ~ G, and for

a given value of the parameter b == b(n, n - 2), the number W(B,,x, b) of distinct rooted dynamical triangulations with Nn_2(T~i») == ,x + 1 bones, is given, for large ,x, by W(8,'x, b) == Cn· < Card{T (i) }curv>

X

1.2 Summary of Results

X

(

(b(n, n - 2) - q)(,\ + 1) + (,\ + 1) - n ) (,\+1)-n .

13

(1.17)

Asymptotics for the Entropy. The number of distinct dynamical triangulations, with ,\ + 1 bones, and with an average number, b == b(n, n - 2), of n-simplices incident on a bone, on an n-dimensional, (n ~ 4), PL- manifold M of given fundamental group 1rl (M), can be asymptotically estimated according to

W('\, b) ~ W1r

•

en

. /(C.

y21r

< Card{T (') }curv > 1,

(b - q + 1)1-2n (b _ q)3 X

(b - q+ 1)b- +l]..\+1(b(n,n - 2) >..)D/2>.._2n2+3 [ (b _ q)b- Q n(n + 1) , Q

X

(1.18)

where W 1r is a topology dependent parameter, and D~dim[Hom(1rl(M), G)]. The Canonical Partition Function. Let us consider the set of all simplyconnected n-dimensional, (n == 3,4), dynamically triangulated manifolds. Let us set

(1.19) and let

k~~2

1J*(kn- 2)

denote the unique solution of the equation

~ ~(1 - A(k~-2)) = 1Jmax,

(1.20)

where TJmax == 1/4, (for n == 4) and 1Jmax == 2/9, (for n == 3). Let 0 < € < 1 small enough, then for all values of the inverse gravitational coupling k n - 2 such that k:-~~ -

f

< kn - 2 < +00,

(1.21)

the large N -behavior of the canonical partition function (effective entropy) is given by the uniform asymptotics W(N , kn-2 ) ell ==.

"L..J TET(Nn

ekn-2Nn-2(T) )

== x

1. Introduction

14

x [

JNwo(tPmax(kn-2)~) + ~W-1(tPmax(kn-2)~)] ,

(1.22)

where wr(z) ~ T(l- r)ez2/4Dr_1(Z), (r < 1), D r - 1(z) and T(l- r) respectively denote the parabolic cylinder functions and the Gamma function, and and are given by where the constants

eo

~o ~ _

6 ~

e1

17~~2(1 - 217max)-n tPmax(kn- 2) , J(l - 3"7max) (1 - 2"7max) f1]("7max)

(1.23)

~o

1Pmax (k n- 2)

(1.24) Where, for notational convenience we have set f("7) ~ -"7 In "7 + (1 - 2"7) In(l - 2"7) - (1 - 3"7) In(l - 3"7)

+ kn - 2"7

(1.25)

2

with f 11 ~ df /d"7, and f1]1]~d2 f /d"7 ,and

(1.26) (Note that for the 3-dimensional case the above expression for W(N, kn - 2 ) is slightly more general if one uses the variables (No, N 3 )-see below for details). The Infinite Volume Limit. The critical value of the coupling k~rit corresponding to the infinite-volume limit for simply-connected n-dimensional

dynamically triangulated manifolds is given by kncrit(kn-2 ) == ~ n ( n 2

+ 1)

[1n A(k 3 + 2 + 2)

which, for 0 < f < 1 small enough, holds for all Whereas, for k:-~2 < kn - 2 < +00 we get

X

(1 - 2"7max) (1- 21]ntax ) ] '1/Tnaz ( )(1-3) { In [ "7max 1 - 3"7max 1]ntax

en] '

l' In 1m N n

Nn-+oo

k:-~2

-

f

.

(1.27)

< kn - 2 <

In en }

+ kn-217max + Nn-+oo hm -N n

k:-~2'

. (1.28)

The 4-Dimensional Case. The critical value of the coupling K4rit corresponding to the infinite-volume limit for a simply-connected 4-manifold, (e. g., the 4-sphere §4), is given by

1.2 Summary of Results

(1 - 2'T]max)(1-2'lJrnax)

10 { In [ 1/~';;x (1 _

31/max)(1-31Jmax )

]

+ k2 1/max -In

(

e )} 2.066

'

15

(1.29)

(with kr ax < k 2 < +00), and'T]max == 1/4, and k;:ax == In4

~

1.387,

(1.30)

(note that this is a theoretical upper bound; for instance for 'T]max == 1/4.1 one would get k 2rit ~ 1.2), whereas one obtains

kcrit == 10 In A(k2 ) + 2 when k'2 ax k max _ € 2

(1.31 )

3e/2.066

4

< k2 < k'2 ax . There exists a critical value k2rit crit < k max -< k 2 2 €

(1.32)

rit ~

with € sufficiently small, (i.e., k 2 1.387 - E), corresponding to which the system undergoes a phase transition between a strongly coupled phase and a weakly coupled phase of simplicial quantum gravity. The 3-Dimensional Case. Let us consider the set of all simply-connected 3-dimensional dynamically triangulated manifolds described in terms of the variables N 3 , No. Let k~nf, and k'[)ax respectively denote the unique solutions of the equations 1

1

1

"3(1 - A(ko)) + 6 =

1

1/min =

6'

1/max =

g'

111

"3(1- A(ko)) +

6=

2

(1.33)

(1.34)

Let 0 < € < 1 small enough, then for all values of the inverse gravitational coupling ko such that k~nf + € < ko < k ax + €, (1.35)

o

the large N -behavior of the canonical partition function W(N, kO)eff for 3dimensional simplicial quantum gravity on a simply connected manifold is given by the uniform asymptotics

16

1. Introduction

(1.36) where the constants ao, aI, ·and

~o, ~I

are given by

(1.37)

11~~~2(1 - 211max)-n 'l/Jmax(ko) J(l - 311max) (1 - 211max) f 1J (l1max) , .

6 =

~o

rJ *-1/2 (1 - 2rJ*)-n [ 1 ] 'l/Jmax(ko) - J(l- 37/*)(1- 27]*) 'l/Jmax(ko)J-f'1'1(7]*) .

(1.39)

(1.40)

The critical value of the coupling k 3rit corresponding to the infinite-volume limit for the 3-sphere §3 is given by

crit k 3 (ko)

A(O.16ko) + 2 = 61n 3[cos- 1( 1/ 3)]1/5

There exists a critical value k rit k omax _ f -< k 0crit -< k 0max ,

- 0.16ko·

(1.41)

o

(1.42)

with f sufficiently small, (i.e., k rit ~ 3.845 - f), corresponding to which the system described by (1.36) undergoes a phase transition between a strongly coupled phase and a weakly coupled phase of simplicial quantum gravity.

o

2. Triangulations

In this section we recall few basic notions of Piecewise-Linear (PL) topology in order to provide a self- contained set up for the some of the mathematical problems we wish to discuss in simplicial quantum gravity. Together with standard material, (for which we refer freely to [103, 108]), we also collect few definitions and results which are not readily accessible in textbooks.

Fig. 2.1. A 4-dimensional simplex 0- 4 , as seen by Diirers frau Agnes, 1495. It has 53-dimensional faces, (each in the shape of a regular tetrahedron), 10 plane faces, 10 edges and 5 vertices

18

2. Triangulations

2.1 Preliminaries: Simplicial Manifolds and Pseudo-manifolds We first list some well known, but necessary, preliminaries. By an n-simplex an == (xo, . .. ,xn ) with vertices Xo, . .. ,Xn we mean the following subspace of R d , (with d > n), n

an ~ LAiXi,

(2.1)

i=O

where Xo, ... ,Xn are n + 1 points in general position in R d, and L.:~=o Ai == 1 with Ai 2 o. A face of an n-simplex an is any simplex whose vertices are a subset of those of an, and a simplicial complex K is a finite collection of simplices in R d such that if al E K then so are all of its faces, and if aI, a2 E K then a 1 n a2 is either a face of a 1 or is empty. The h- skeleton of K is the subcomplex Kh c K consisting of all simplices of K of dimension:::; h. Let

Fig. 2.2. The star of a vertex for n == 2

K be a (finite) simplicial complex. Consider the set theoretic union IKI

C

Rd

of all simplices from K

(2.2) Introduce on the set IKI a topology that is the strongest of all topologies in which the embedding of each simplex into IKI is continuos, (the set A c IKI is closed iff A n k is closed in a k for any a k E K). The space IKI is the underlying polyhedron, geometric carrier of the simplicial complex K, it provides the topological space underlying the simplicial complex. The topology of IKI can be more conveniently described in terms of the star of a simplex a, star( a), the union of all simplices of which a is a face. The open subset of the underlying polyhedron IKI provided by the interior of the carrier of the

a

2.1 Preliminaries: Simplicial Manifolds and Pseudo-manifolds

19

star of a is the open star of a. Notice that the open star is a subset of the polyhedron IKI, while the star is a sub-collection of simplices in the simplicial complex K. It is immediate to verify that the open stars can be used to define the topology of IKI. The polyhedron IKI is said to be triangulated by the simplicial complex K. More generally, a triangulation of a topological space M is a simplicial complex K together with a homeomorphism IKI ~ M. Note that every simplicial complex with N vertices admits a canonical embedding into the (N -I)-dimensional simplex and consequently into Euclidean space.

Fig. 2.3. The link of a vertex for n = 2

Simplicial maps. A simplicial map f: K ~ L between two simplicial complexes K and L is a continuous map f: IKI ~ ILl between the corresponding underlying polyhedrons which takes n-simplices to n- simplices for all n, (piecewise straight-line segments are mapped to piecewise straight-line segments). The map f is a simplicial isomorphism if f-l: L ~ K is also a simplicial map. Such maps preserve the natural combinatorial structure of jRn. Note that a simplicial map is determined by its values on vertices. In other words, if f: KO ~ LO carries the vertices of each simplex of K into some simplex of L, then f is the restriction of a unique simplicial map. Combinatorial properties. Sometimes, one refers to a simplicial complex K as a simplicial division of IKI. A subdivision K' of K is a simplicial complex such that IK'I == IKI and each n-simplex of K' is contained in an n- simplex of K, for every n. A property of simplicial complex K which is invariant under subdivision is a combinatorial property or Piecewise-Linear (PL) property of K, and a Piecewise-Linear homeomorphism f: K ~ L between two simplicial

20

2. Thiangulations

complexes is a map which is a simplicial isomorphism for some subdivisions K' and L' of K and L. 7

3

4

... 7

3

4

7

~--.-01!~---....~--

N=7 2

Fig. 2.4. If we identify vertices and edges according to the indicated pattern we get a triangulation of the torus 'f2 with few vertices

2.1.1 Piecewise-Linear Manifolds

A PL manifold of dimension n is a polyhedron M = IKI each point of which has a neighborhood, in M, PL homeomorphic to an open set in Rn. PL manifolds are realized by simplicial manifolds under the equivalence relation generated by PL homeomorphism. Any piecewise linear manifold can be triangulated, however notice that any particular triangulation of a piecewise linear manifold is not well defined up to piecewise linear homeomorphism, (because such homeomorphisms involve subdivisions of the given triangulations). A triangulated space can be characterized as a PL-manifold according to the[108]

Theorem 2.1.1. A simplicial complex K is a simplicial manifold of dimension n if for all r-simplices aT E K, the link of aT, link(aT) has the topology of the boundary of the standard (n - r)-simplex, viz. if link(a T) ~ Sn-T-l. Recall that the link of a simplex a in a simplicial complex K is the union of all faces a f of all simplices in star(a) satisfying a f n a = 0, also recall that the Cone on the link link(a T), C(link(a T)), is the product link(a T) x [0,1] with link(a T) x {I} identified to a point. The above theorem follows[108] by noticing that a point in the interior of an r-simplex aT has a neighborhood homeomorphic to BT x C(link(a T)), where BT denotes the ball in jRn. Since link(a T) ~ sn-T-l, and c(sn-T-l) ~ Bn-T, we get that IKI is covered by neighborhoods homeomorphic to BT X Bn-T ~ Bn and thus it is a manifold. Note that the theorem holds whenever the links of vertices are (n-l)-spheres. As long as the dimension 4, the converse of this theorem is also true.

n:: ;

2.1 Prelim inari

F ig . 2 '\ ') . 't h e li

es: Simpli cial

Manifolds

a n d pseud

o-manifold

n k of a n e fold dge in a 3 _dimensio nal p L -M a ni

s

21

22

2. Triangulations

But this is not the case in larger dimensions, and there are examples of triangulated manifolds where the link of a simplex is not a sphere. In general, necessary and sufficient conditions for having a manifolds out of a simplicial complex require that the link of each cell has the homology of a sphere, and that the link of every vertex is simply-connected, (see e.g., Thurston's notes[108]). Since we are interested in dimension 2 ::s n ::s 4, we can simply disregard such subtler characterizations. However (see the next comments on glueings), this issue is actually bypassed by the particular way triangulations are generated in simplicial quantum gravity.

.. .',

T

N2»

1

2

Fig. 2.6. A triangulation of the torus y2 with a large number of triangles

Glueings. In dynamical triangulations theory one is not given from the outset a triangulation of a manifold, rather one generates an n-dimensional PLmanifold by glueing a finite set of n-simplices {an}. A rather detailed analysis of such glueing procedures is given in Thurston's notes[108] , (clearly for different purposes), and here we simply recall the most relevant facts. Given a finite set of simplices and the associated collection of faces, a glueing is a choice of pairs of faces together with simplicial identifications maps between faces such that each face appears in exactly one of the pairs. The identification space, T, resulting from the quotient of the union of the simplices by the equivalence relation generated by the identification maps, is homeomorphic to the polyhedron of a simplicial complex. The glueing maps are linear, and consequently the simplicial complex T obtained by glueing face-by-face n-simplices has the structure of a PL- manifold in the complement of the (n - 2)-skeleton. Since the link of an (n - 2)-simplex is a circle, it is not difficult to prove that the PL-structure actually extend to the complement of the (n - 3)- skeleton, and that the identification space of a glueing

2.1 Preliminaries: Simplicial Manifolds and Pseudo-manifolds

23

among finite n-simplices is a PL-manifold if and only if the link of each cell is PL-homeomorphic to the standard PL-sphere.

Fig. 2.7. A glueing pattern, (8. la B. Fuller), of triangles for generating a triangulation of the 2-sphere §2

Pseudo-manifolds. It is important to stress that in dimension n > 2, not every simplicial complex T obtained by glueing simplices along faces is a simplicial manifold, and in general one speaks of pseudo- manifolds. ~ M is an n-dimensional pseudomanifold if: (i) every simplex ofT is either an n-simplex or a face of an n- simplex; (ii) each (n-l)simplex is a face of at most two n- simplices; (iii) for any two simplices an, in of T, there exists a finite sequence of n-simplices an == aD' af, ... ,aj == in

Definition 2.1.1. T

24

2. Triangulations

such that ai and ai+l have an (n - I)-face in common, (Le., there is a simplicial path connecting an and Tn).

Recall that a regular point p of a polyhedron ITI is a point having a neighborhood in ITI homeomorphic to an n-dimensional simplex, otherwise p is called a singular point. Absence of singular points in a pseudo-manifolds characterizes triangulated manifolds. Moreover, an n-dimensional polyhedron ITI is a pseudo- manifold if and only if the set of regular point in ITI is dense and connected and the set of all singular points is of dimension less than n - 1. Thus, in the applications to gravitational physics, the occurrence of pseudomanifolds is associated with the presence of local irregularities that should not have a sensible effect in the continuum limit of the theory, since the set of regular points has the density properties mentioned above.

Fig. 2.8. A pseudo-manifold as seen by a triangulated man (this latter from Villard de Honnecourt, MsJr.19093 Paris Bibliotheque Nationale)

2.1.2 Dehn-Sommerville Relations

In order to construct a simplicial manifold T by glueing simplices an, through their n - I-dimensional faces, the following constraints must be satisfied

2.1 Preliminaries: Simplicial Manifolds and Pseudo-manifolds

25

n

L(-l)iNi (T) == X(T),

(2.3)

i=O

.L n

i

'I,=2k-l

(i+1)!

(-1) (i _ 2k + 2)!(2k _1)!Ni (T)

=

0,

(2.4)

if n is even, and 1 ::; k ::; n/2. Whereas if n is odd n

i

(i+1)!

(2.5)

;L(-1) (i-2k+1)!2kI Ni (T) =0,

'I,=2k

with 1 ::; k ::; (n -1)/2. These relations are known as the Dehn-Sommerville equations. The first, (2.3), is just the Euler-Poincare equation for the triangulation T of which N i (T) denotes the number of i-dimensional simplices, the f-vector of the triangulation T. The conditions (2.4) or (2.5), are a consequence of the fact that in a simplicial manifold, constructed by glueings, the link of every (2k - I)-simplex (if n is odd) or 2k-simplex (if n is even), is an odd-dimensional sphere, and hence it has Euler number zero. Note also that in order to generate an n-dimensional polyhedron there is a minimum number q(n) of simplices an that must join together at a bone, (for instance, in dimension 2, we have q(2) == 3, otherwise we have no polyhedral surface). In general, since we will usually consider only the class of regular simplicial manifolds we assume that the number, q(a i ), of n-dimensional simplices an which share the subsimplex a i is such that

(2.6) with i ::; n - 2. There is a simple but important relation which must be satisfied by the numbers, q( a n - 2 ), of n-dimensional simplices sharing the (n - 2)-dimensional subsimplices, the bones of T, a n - 2 , viz.,

L

q(O'n-2) = n(n 2+ 1) N n

(2.7)

un - 2

which follows by noticing that the number of (n - 2)-dimensional sub- simplices in an n- dimensional simplex is !n(n + 1). We shall exploit this relation quite intensively, and for later convenience, let us introduce the average number of simplices an, in T, incident on the (n - 2)-dimensional subsimplices, a n - 2 , namely

~ (n-2) . 1 b(n, n - 2) = Nn - 2 (T) 0'~2 q 0'

1 (

) ( Nn(T) )

= '2 n n + 1

Nn - 2 (T)

.

(2.8)

It is easily verified that the relations (2.3), (2.4), (2.5) leave !n -1, (n even) or !(n - 1) unknown quantities among the n ratios N1/No, . .. ,Nn/No, [77].

26

2. Triangulations

Thus, in dimension n = 2,3,4, the datum of b(n, n-2), (trivial for n = 2), and of the number of bones N n - 2 , fixes, through the Dehn-Sommerville relations all the f-vectors Ni(T) of the dynamical triangulation considered. Together with the Dehn- Sommerville relations, equation (2.7) implies the following bounds for the average incidence number b(n, n - 2), when N n - 2 »1: (i) For n = 2:

(2.9)

b(2,0) = 6;

(ii) for n

= 3:

3 ::; b(3, 1) ::; 6;

(2.10)

(iii) for n = 4:

3 ::; b( 4, 2) ::; 5.

(2.11)

Notice however that the lower bounds in (2.10) and (2.11) are not optimal. This fact was noticed during computer simulations (see e.g. [45]). The actual characterization of the optimal lower bounds for b(n, n - 2) in dimension n = 3 and n = 4 it is rather important since it bears relevance to the location of the critical point in simplicial quantum gravity. It is not trivial to characterize such bounds since they are a consequence of the following result from the combinatorial topology of 3- dimensional PL-manifolds proved by D. Walkup [111]:

Theorem 2.1.2. For any combinatorial 3-manifold the inequality Nl

~

4No -10

(2.12)

holds with equality if and only if it is a stacked sphere

(a stacked sphere is by definition a triangulation of a sphere which can be constructed from the boundary of a simplex by successive adding of pyramids over some facets). By exploiting this result we can prove the following

Lemma 2.1.1. For any triangulation T

~ Mn of a closed n-dimensional PL- manifold M, the actual lower bound for the average incidence number b(n, n - 2) is given for n = 3 by

and for n

9

2'

(2.13)

= 4.

(2.14)

b(n, n - 2)ln=3 =

= 4 by

b(n, n - 2)ln=4

Proof. Let us consider the 3-dimensional case first. The Dehn-Sommerville relations for 3-manifolds are

2.1 Preliminaries: Simplicial Manifolds and Pseudo-manifolds

27

Fig. 2.9. A stacked sphere, (according to P. Halt: Perspectivische Reiss Kunst, Augsburg 1525)

28

2. Triangulations

(2.15) (2.16) from which we immediately get N 3 == N 1 - No. On the other hand, Walkup's theorem implies No ~ ~ N 1 + ~. Thus, N3 ~

3

5

4N1 - 2'

(2.17)

which yields the stated result for n == 3. In dimension n == 4, the Dehn-Sommerville relations read (2.18) (2.19) (2.20) Let us apply Walkup's theorem to to the link of every vertex v in our 4dimensional triangulation. In this way we get N1[link(v)] ~ 4No[link(v)] - 10,

(2.21 )

and by summing over all vertices one obtains 3N2

~

8N1

-

(2.22)

10No,

which is the basic inequality that we are going to exploit. From it, (multiplying by 3), we obtain (2.23) There is another independent relation of this sort which can be obtained if we multiply (2.18) by 30, and exploit (2.20), viz., 30No - 30N1

+ 30N2 -

18N3

== 30X.

(2.24)

From (2.19) one gets 2N1 - 3N2 + 2N3 == 0 which, (upon multiplication by 9), yields 18N3 == 27N2 - 18N1 . Inserting this latter expression in (2.24) we eventually get 3N2

== 30X + 12N1 - 30No,

(2.25)

which together with (2.23) provides a direct proof of the bound for b(n, n 2)\n=4. Explicitly, let us subtract (2.25) to (2.23). We get 6N2 ~ 12N1 ~ 30X,

(2.26)

namely (2.27) Since 2N1 - 3N2 + 2N3 == 0, we get 2N3 to (2.27), we have

-

2N2 == N 2

-

2N1 , thus, according

2.1 Preliminaries: Simplicial Manifolds and Pseudo-manifolds

2N3

-

2N2

~

-5X.

29

(2.28)

And from (2.20) we obtain 5N4

~

2N2

which, for N 2

-

5X,

» x,

(2.29)

yields for b(n,n - 2)ln=4 ~ 4.

K2

K

2

Fig. 2.10. A particular triangulation of the 2-sphere §2 with q( aO) small and just two vertices supporting most of the incident triangles is constructed by considering two distinct vertices VI and V2 on each one of which N /2 triangles are incident. The link of both vertices is an N /2-gon, and by glueing the stars of VI and V2 through such links we get a triangulation of the sphere §2 with N triangles, with q( VI) == q( V2) == if. For large N, this triangulation minimizes the incidence on the unmarked vertices {a?}i=3,4, ...

It is also interesting to note that according to relation (2.7) the incidence numbers {q(a n - 2 )} of a simplicial manifold cannot be provided by a monotonically increasing sequence of integers, for otherwise E un-2q(a n - 2) would be O(N~) in plain contrast with (2.7). The actual distribution of the incidence numbers q(a n - 2) of a triangulation has typically a peak around a rather small value of q( a n - 2 ) (~ 3) and only a few of the bones a n - 2 can support a large number (i.e., O(Nn )) of incident n-simplices. In particular, from (2.7) it is immediate to prove that if we let qmax ~ sUPT(N,B) sUPun-2{q(an-2)} ,denote the maximum value of q(a n - 2), as (1n-2 varies in the set of triangulations with N simplices an and given value of b(n, n - 2), then, for N n - 2 » 1, _ n(n+1) (b(n,n-2)-ii)N qmax - 2F(n) b(n, n - 2) n,

(2.30)

30

2. Thiangulations

where F(n) is the number of bones over which qmax can be attained, and ij is the average incidence over the remaining bones, i.e., Nn -

2

_. 1 " (n-2) . q = ~ L..-t qaa

n-2 F(n)+1

(2.31 )

An elementary example of such a configuration is easily constructed in dimension n = 2 by considering two distinct vertices VI and V2 on each one of which N /2 triangles are incident. The link of both vertices is an N /2-gon, and by glueing the stars of VI and V2 through such links we get a triangulation of the sphere §2 with N triangles, with q(Vl) = q(V2) = J¥-. Actually, the existence of this sort of triangulation is a rather general fact of some relevance to us. Indeed it provides, for large N n , a triangulation minimizing the average ij, (ij = ij + 1 = 4 in the above example), with few vertices carrying a large incidence. In general, the existence of such triangulations follows from the construction of a triangulation Tn+l ~ Bn+l of the (n + I)-dimensional ball Bn+l inducing on §n ~ 8Bn+l a triangulation Tn ~ §n of the n-dimensional sphere §n with N n simplices and N n- 2 = i!(n - I)(n + I)Nn + n bones. ~ of the n-dimensional simplices are incident on n distinguished bones {a*~-2}i=I, ... ,n spanning an embedded §n-2 in Tn. Thus, for n = 2, {a*~-2}i=I, ... ,n consists of two vertices (viz., §o which is indeed topologically identified with a double point). For n = 3, {a*~-2}i=I, ... ,n consists of 3 edges bounding a triangle (~ §1), whereas for n = 4 the supporting bones {a*~-2}i=I, ... ,n yields for 4 triangles bounding a tetrahedron (viz., an §2). In order to construct such triangulation of §n consider the (n + 1)dimensional ball Bn+ 1 triangulated by a sequence of Nn+ 1, (n+ 1)-dimensional simplices {a n+1} all sharing a common (n - I)-dimensional simplex a~-1 ~ 0'*, i.e., ) n+ 1(a*, ) ) 0'1n+ 1 ( 0'*,0'2 ... ,aNn++11(a*) =0'1n+ 1 ( 0'*,

(2.32)

and such that

n F[ajtl(a*), ajti(a*)] = 0'*, F[aj+l(a*),ajtl(a*)] ~ aj+l(a*) n ajtl(a*) is the

F[aj+l(a*), ajtl(a*)]

(2.33)

where n- dimensional face through which aj+l(a*) and ajtl(a*) are glued together. The boundary of such T n+1 ~ Bn+l is a triangulation of the sphere Tn ~ §n with N n = nNn+1 top-dimensional simplices an, and N n- 2 = ~(n-I)(n+I)Nn+n bones. The intersection 0'* n Tn defines an embedded §n-2 in Tn generated by n bones, {a*~-2}i=I,...,n, on each of which ~n simplices an are incident. These results can be naturally extended to PL-manifolds of arbitrary topology by glueing, the above triangulation of the n-sphere into a small triangulation of the given PL- manifod M, (say a triangulation of M with a minimal number of vertices). Explicitly, let TSmall(M) be a triangulation with few simplices, Nn(Tsmall) , of an n-dimensional PI-manifold M of given topology,

2.1 Preliminaries: Simplicial Manifolds and Pseudo-manifolds

31

Fig. 2.11. By glueing a large number of 3-simplices all incident on a commong edge we get a triangulation of a 3-dimensional ball B 3 • The boundary of such triangulated B 3 yields for a triangulation of the 2-sphere where all the top-dimensional simplices are incident on the vertices {0"1, O'2} spanning an §o 0"1

32

2. Thiangulations

(for a characterization of such type of triangulations and for examples see the paper by W. Kiihnel [87]). Consider the large N n triangulation T(§n) of the n-sphere §n defined above, let af(M) be a simplex in M and a~(§n) a simplex in §n. Set (2.34)

where a(·) denotes the interior of the given a's, and f is an homeomorphism from the boundary of af(M) to the boundary of a~(§n). If Nn(§n) » Nn(Tsmall), such T(M) provides a large N n triangulation of M whose distribution of incidence numbers {q(a n - 2 )} has the properties of the collection of incidence numbers of T(§n). Anticipating a little bit, when this sort of

Fig. 2.12. By glueing a large number of 4-simplices all incident on a commong triangle 0'2 we get a triangulation of a 4- dimensional ball B 4 • The boundary of such triangulated B 4 yields for a triangulation of the 3-sphere where all the top1 dimensional simplices are incident on the bones {aI, 0'2, O'3} spanning an §

triangulations entropically dominate in the canonical partition function (1.8) we should expect that the resulting statistical system as described by (1.8) exhibits a very large effective dimensionality as N n ~ 00. A useful clue to this is to note that an effective dimensionality on a triangulated manifold can be defined by counting the number of vertices visited in a walk of k steps, starting from a typical vertex. If the triangulation is sufficiently regular then, for k large, the number of visited vertices grows as kn , n being the topological dimension of the underlying PL-manifold. On the other hand, triangulations

2.2 Distinct Triangulations of the Same PL Manifold

33

such as the ones described above are such that all vertices can be visited in a few steps: as N n increases the system effectively behaves as if we were dealing with a regular infinite-dimensional lattice. Such remarks are elementary, nonetheless they play an important role in what follows since the incidence numbers {q(a n - 2 } are directly related with curvature when discussing dynamical triangulations. Thus there is an intimate relations between the distributions of the incidence numbers {q(a n - 2 }, metric properties, (curvature), and effective dimension of the resulting statistical system.

Fig. 2.13. Triangulating an arbitrary 2-dimensional closed PL- manifold with a triangulation where most of the triangles are incident on the marked §o

2.2 Distinct Triangulations of the Same PL Manifold By the very definition of PL manifolds, it follows that there exist distinct triangulations, T(i), of the some PL manifold M. For later convenience, it is better to formalize this remark, and recall the following standard characterization by W. Tutte [109]:

34

2. Triangulations

Definition 2.2.1. Two triangulations,

T(l) and T(2) of the some underlying PL manifold Mare identified if there is a one-to-one mapping of vertices, edges, faces, and higher dimensional simplices of T(l) onto vertices, edges, faces, and higher dimensional simplices of T(2) which preserves incidence relations. If no such mapping exists the corresponding triangulations are said to be distinct.

Notice that sometimes, (e.g., W. Thurston[108],p.105), such triangulations are said to be combinatorially equivalent. However, we shall avoid this terminology since, in simplicial quantum gravity, combinatorial equivalence is used as synonimous of PL- equivalence.

Fig. 2.14. Two distinct triangulations 2-dimensional PL-manifold M

T(1)

and

T(2)

of the same portion of a

Not surprisingly, in connection to the simplicial quantum gravity program natural questions to ask are among the classics of PL- Topology: (i) When can a topological manifold be triangulated? (ii) Do any two triangulations of a manifold admit isomorphic subdivisions? The answer to the first ques-

2.2 Distinct Thiangulations of the Same PL Manifold

35

tion is, for general manifolds, negative[108, 84]: Kirby, Siebenmann and Wall showed that there exist topological manifolds which do not admit combinatorial structures, and that there are non-standard structures for PL-manifolds of dimension at least 5. What Kirby and Siebenmann proved[84] is that for n ~ 5 there is a single obstruction k(M) E H 4 (M, Z2), (where Hi(M, Z2) denotes the i-th cohomology group of M with Z2 coefficients field), to the existence of a piecewise linear structure on a topological manifold M, and that the vanishing of this obstruction (k(M) == 0) allows for the existence of a worth of IH 3 (M, Z2)1 distinct PL structuress. Moreover, if n :::; 7, every PL manifold admits a compatible differentiable structure which is unique, (up to diffeomorphisms), if n :::; 6. The Hauptvermutung is the subject of question (ii), and it is false in general. For instance, as hinted before, the 5-sphere §5 admits PL-inequivalent triangulations. In the dimensions of interest to simplicial quantum gravity, the situation is more encouraging. Indeed, for dimension n :::; 3 both questions (i), and (ii), have an affermative answer. However, it is not yet known if every n- dimensional topological manifold is homeomorphic to a polyhedron for n ~ 4. Actually, (see the paper by T. Januszkiewicz[78]), there is a closed topological four- manifold M with the following properties: (i) M is not homotopy equivalent to a P L manifold; (ii) M is not homemorphic to a simplicial complex. The construction of this example is rather involved and exotic (the basic ingredient being the E s homology manifold), but shows once more, if necessary, that the interplay between the topology of 4-dimensional manifolds and quantum gravity cannot be easily tackled. For this reason when discussing the 4-dimensional case, we shall not consider explicitly topological 4-manifolds, and work directly in the PL-category. When discussing deformations of dynamical triangulations we shall need to consider smooth structures, but this is simply a technical issue since as already recalled the category of smooth 4-manifolds is equivalent to the category of PL 4-manifolds. Another natural question to ask is under what conditions a given set of integers N i is the f-vector of a triangulated polyhedron. This problem is of some relevance to simplicial quantum gravity, since it can be related to the counting of distinct triangulations, (the Entropy problem). Indeed, a possible way of stating the entropy problem is to enumerate all distinct triangulations, on a manifold of given topology, for a given choice of possible f-vectors. However, for general triangulated n-manifolds the characterization of all possible f-vectors is beyond reach. Partial answers can be obtained only under rather severe constraints. Typically, one has to assume that the polyhedron in question is realized as a convex polyhedron in Euclidean ddimensional space, jRd. In this case, the possible f-vectors are characterized by Stanley's theorem [106], according to which the Dehn-Sommerville relations are necessary but not sufficient (they suffice in dimension 2 if we add the obvious condition No ~ 4 expressing the fact that to bound a solid takes more

36

2. Triangulations

than three vertices). In general, for a sequence of integers No, N1 , ... , Nd-l, set N_ 1 = 1 and m = [~]. For 0 :::; p :::; d, define hp by

(2.35) and for, 1 :::; p :::; m, set

(2.36)

gp~hp - h p- 1 ,

then we have[106]

Theorem 2.2.1. A sequence of integers No, N 1 , ... ,Nd - 1 occurs as the number of vertices, edges, ... , (d - 1)- faces of a d-dimensional convex sim-

plicial polytope if and only if the following conditions hold: h p = h d - p , for (these are just a rewriting of Dehn-Sommerville relations); 9p ~ 0 for 1 :::; p :::; m; and, if one writes

o:::; p :::; m, 9p

=(

~ ) + ( ;~~ ) + ···+ ( ~

with n p > np-l > ... > n r 2:: r an integer in this way), then s

<(

gp+l -

np + 1 )

P+1

+(

)

(2.37)

2:: 1, (there is a unique way of decomposing

np-l

p

+1

)

+ ... + (

nr r

++11 )

(2.38)

for 1 :::; p :::; m - 1.

This result was conjectured by McMullen[88] , sufficiency and later necessity was proven respectively by Billera and Lee[27] , and Stanley and McMullen[106],[89]. Finally, it is worth noticing that for the class of all spheres the determination of all I-vectors has been conjectured by Stanley[107]. Owing to the strong limitations imposed by the convexity constraint, it is not very convenient to formulate the entropy problem in terms of the characterization of the possible I-vectors of simplicial manifolds, and in enumerating distinct triangulations we shall follow a different strategy. However, it must be stressed that progress in Stanley's theory, (and its unexpected connections with algebraic geometry) may shed light on this question.

2.2 Distinct 'friangulations of the Same PL Manifold

Fig. 2.15. Great expectations? (Himmelfahrt der Magdalena, 1507-1510)

37

3. Dynamical Triangulations

In this chapter, we will present the basic abstract formalism for dynamical triangulations. Let T -7 M = ITI be a simplicial manifold, (henceforth, if there is no danger of confusion, we shall use the shorthand notation M = ITI for denoting such a triangulated manifold). A triangulation T can be used to produce a metric on the underlying PL manifold M = ITI by declaring that all simplices in the triangulation are isometric to the standard simplex of the given dimension. This procedure provides a collection of compatible metrics on M, which can be extended to a piecewise-flat metric on M in an obvious way, here we follow the elegant construction provided by Frohlich[57]. Let us choose an orthonormal frame {e a(a n )}:=l in the interior of each simplex an E T, (this frame can be conveniently located at the barycentre of an), and introduce another frame by choosing n directed edges {EJL(an)}~=l in the boundary of an. We can write EJL(a n ) = 2:::=1 t~(an)ea(an), where the n x n matrix, the n-bein, (t~ (an)) is, for each an, a regular matrix, an being a non-degenerate simplex. Fixed edge-length metric. In terms of the chosen edges EJL{a n ), we can write the remaining edges in the boundary of an as EJLv{a n ) = EJL{a n ) - Ev(a n ), with 1 ~ J-t < v ~ n. The metric inside each simplex an is assumed to be the usual Euclidean metric, 8. It can be expressed in terms of the matrix 8(a n ) = 9JLv(a n ) = (E JL , E v )8. We assume that each edge EJL(aa) or EJLv(a n ) has a fixed length a = IEJL(an)1 = IEJLv(an)l, this characterizes fixed edgelength triangulations as compared to the variable edge-length triangulations typical of Regge calculus. In this way we get a metric in each simplex an E M which ~s extended to M by specifying that the metric is continuous when one crosses the faces of the an's, and that each face of a simplex is a linear flat subspace of the simplex in question. Thus the metric on a fixed edge-length PL manifold is entirely specified by the fixed distance elements a between nearest neighbor vertices of the corresponding simplicial manifold, viz.,

Definition 3.0.2. For any a E jR+, a Dynamical Triangulation T a of a polyhedron M = IT\, with distance cut-off a, and with Nn(Ta) simplices an, is a division of a simplicial manifold M = ITI generated by glueing, along their adjacent faces, Nn(Ta) equilateral simplices {an} with edge-length \EJL(an )\ = \EJLv(a n ) \ = a, Van E T.

40

3. Dynamical Thiangulations

Fig. 3.1. The frames {EJL(a n )} and {ea(a n )}, (here n simplex an

= 2),

associated with a

Note that in order to avoid overcounting dynamical triangulations differing only by a rescaling of the edge-lengths, one always refers to a given value of the cut-off, (typically a == 1), when characterizing distinct triangulations.

3.1 Dynamical Triangulations as Length-Spaces Contrary to what is generally believed, the metric 91-£v generated by extending over M == ITal the flat metrics 91-£ v (an ) of the simplices {an}, is not the object of· primary interest for unravelling the metric geometry of a dynamically triangulated manifold. For one thing, this metric is manifestly singular around the (n - 2)-dimensional simplices, (which have conical neighborhoods), for the other there is nothing sacred in 91-£v and one can capture more explicitly the geometry of M by looking at all curves between given points PI and P2 in M and by setting the distance d(Pl' P2) equal to the infimum of the length of these curves. Incidentally one can reconstruct 9 at any (regular) point P from the distance function. Setting f(p) ~ [d(p,p)]2, it is easily verified that 9 is provided, to leading order, by the second differential D 2 f(p)lp=p' In order to discuss some aspects of the interior geometry of a dynamical triangulation thought of as a metric space, we introduce few preliminary notions hinting to a characterization of geodesics on a dynamical triangulation without using calculus[15, 60, 37]. A curve in a dynamically triangulated manifold M is a continuous map c: I ~ M == ITal, where I is an interval. The length L(c) of the curve c: [a, b] ~

3.1 Dynamical Triangulations as Length-Spaces

41

M is then defined according to j

L(c) ~ sup

L d(C(th-l), C(th)),

(3.1)

h=1

where the supremum is taken over all subdivisions a == to < tl < ... < tj == b of [a, b], and de) denotes the distance function associated with the given dynamical triangulation Ta ~ M == ITa I. We assume that c is rectifiable, (i.e., L(c) < (0), and introduce the arc length parametrization as the nondecreasing continuous surjective map

s: [a, b]

~

[0, L(c)]

(3.2)

defined by s(t) ~ L(cl[a, t]). In terms of the arc-length, the curve c is travelled at unit speed, viz.,

c: [0, L(c)] ~ M, with c(s(t))

~

(3.3)

c(t), and

£(cl[sl' S2]) ==

lSI -

(3.4)

s21·

Similarly we can characterize a curve c: I the curve is such that

~

M travelled at speed v 2::

°

if

(3.5)

for all tl, t2 E I. If a curve c: I ~ M has constant speed v 2:: tEl the curve c has a neighborhood U in I such that

°and if for any (3.6)

for all tl, t2 E U, then such a c is called a geodesic in the dynamically triangulated manifold M == ITal. Such a geodesic is said to be minimizing if (3.6) holds for all tl, t2 E I. The interior metric dint associated to a dynamical triangulation T a is defined according to[15, 60, 37]

dint (x, y)

~

inf{L(c)},

(3.7)

where inf is taken over all curves from x to y. It is easily verified that for a dynamical triangulation such distance function coincides with the original distance function d, dint == d, and in such a sense a dynamical triangulation is a Length space [65, 52]. It is also a geodesic space [15] since for any pair of points x, y in M there is a minimizing geodesic from x to y. At first sight these facts may seem trivialities, (at least for a dynamically triangulated manifold), however they have important consequences when discussing rigorously the way a dynamically triangulated manifold approximate smooth riemannian manifolds. A first application will be encountered when discussing curvature in the PL-setting.

42

3. Dynamical Triangulations

3.1.1 PL Connections and the Incidence Matrix In general, associated with a PL metric 9J.tv(a n ) == (EJ.t, E v )8, an E T, there is also a unique connection (the Levi-Civita connection) which can be characterized by the set of matrices, r(ai, ai+1)' describing the change of bases n {EJ.t(ai)}~=l ~ {EJ.t(ai+1)}~=1' (regarded as different bases oflR ), in passing between two adjacent simplices in M, sharing a common face F(ai, ai+1)' In Regge calculus [57], the Levi- Civita connection so defined is uniquely determined by the length of the edges and by the incidence matrix of T, the Nn(T) x Nn(T) matrix I(ai,aj), whose entries are 1 if ai and aj are glued along a common face, 0 otherwise. It follows that when the metric 9 is generated by the fixed edge length prescription, the corresponding Levi-Civita connection is uniquely determined by I (ai, aj).

Definition 3.1.1. The family Q g ~ {r(ai,ai+1)}i=1, ... with ai and ai+1 adjacent in T a , is the Levi-Civita connection associated with the dynamical triangulation T a of the PL manifold M == ITI. Notice that if in place of the change of bases {EJ.t(ai)}~=l ~ {EJ.t(ai+1)}~=1 we consider the corresponding change of orthonormal bases {e a (ai)}:=l ~ {e a (ai+1)}:=1' in passing from one simplex to another, then the connection Q g can be equivalently expressed in terms of orthogonal matrices O(ai, ai+1) given by

r(af,af+1) == t- 1(af)O(af, af+1)t(af+1)'

(3.8)

A transparent definition of curvature in a simplicial manifold can be provided if together with the standard notion of curve on M, we introduce the combinatorial notion of simplicial path. A simplicial path in a simplicial manifold M is defined as a sequence of simplices {aj };=1' l == 2, 3, ... with aj E M and such that aj and aj+1 i= aj share a common face. If == then we speak of a simplicial loop in the manifold M. With any simplicial loop w, we can associate the parallel transporter

ar ar

II

R w ==

r(aj,aj+1)'

(3.9)

O'j ,O'j+l Ew

where the product is path-ordered. Let B denote the generic bone in T a , and let w(B) that unique loop winding around the bone B. If aj(B) denotes the generic simplex containing B, then[57]

w(B) == {ar(B), a~(B), . .. ,a~(B) == ar(B)},

(3.10)

with ar(B) n ... n a~(B) == B. The corresponding rotation matrix

RW(B) ==

II R(ai(B), ai+l(B))

(3.11)

is such that all vectors in the (n - 2)-dimensional hyperplane spanned by the bone B are eigenvectors with eigenvalue 1, and in the 2-dimensional plane

3.1 Dynamical Triangulations as Length-Spaces

Fig. 3.2. Parallel transport around the link of a bone in dimension n by the monkeys of Durer's Affentanz, (1523)

43

= 3, as seen

44

3. Dynamical Triangulations

Fig. 3.3. A path and a corresponding simplicial path

orthogonal to the bone B, angle ¢(B) is given by

RW(B)

reduces to a rotation, where the rotation

q(B)

¢(B) ~

L

8j (B),

(3.12)

j=l

q(B) being the number of simplices an incident on the bone B, and 8j (B) is the angle between the unique two faces of aj containing B. 3.1.2 Curvature Assignments

In Regge calculus, the quantity r(B) == 21r - ¢(B) is the deficit angle at the bone B. Since the angle between two faces sharing a common bone B, in an equilateral simplex an, is given by cos-11/n, we can write 1

¢(B) == q(B) cos- 1 -, n

(3.13)

where q(B) denotes the number of simplices an incident on the bone B. In defining curvature for dynamically triangulated manifolds one has to change the perspective on Regge calculus somewhat (as advocated by Hamber and Williams [75]) and view the geometry of the triangulation as representing an approximation to some smooth geometry and the local curvature as some

3.1 Dynamical Triangulations as Length-Spaces

45

Fig. 3.4. A simplicial path and a simplicial loop

average curvature for a small volume, (in the next section we formalize somewhat this point of view by adopting the Gromov-Hausdorff topology on the space of dynamically triangulated manifolds). Thus, to each (n - 2)dimensional bone B we assign an n-dimensional volume density given by

voln(B) == (2

n n+1

L

)

vol(aj),

(3.14)

ujEw(B)

where {aj} is the set of n-dimensional simplices in the unique simplicial loop w{B) winding around the given bone B. The volume voln{B) can be considered as the natural share of the volumes of the n-dimensional simplices to which the bone belongs. It follows that every bone B in a dynamical triangulation T a carries a curvature given by

K(B)

2[21r - q(B) cos- ~] v:~:(1~), 1

=

(3.15)

where VOl n -2{B) is the usual n - 2-dimensional (Euclidean) measure of the bone B. Recall that the Euclidean volume of a j- dimensional equilateral simplex of edge-length a is

vol(ai )

i

= a y'(J+I)

(3.16)

j!V2J

thus, we get

K(B)

= n2 (n 2 -1) In -1 a2 n +1

1

[211" - q(B) cos- ~]

q(B)

.

(3.17)

46

3. Dynamical Triangulations

Fig. 3.5. A 4-dimensional bone carrying curvature

For instance, in dimension four we have VOl n-2(B) == ~J3a2, voln(a n ) == 4 g1 V5a , and (3.15) reduces to 6

== 240 ~ [21r - q(B) cosK (B ) a2 V"5 q(B)

1

*]

(3.18)

.

Up to the choice of the cut- off edge- length, a, and of a numerical factor depending only from the dimension n, curvature in a dynamical triangulation T with N n - 2 (T) bones B(a), a == 1,2, ... ,Nn - 2 (T), is directly provided by the sequence of integers {q(a)}, where q(a) denotes the number of simplices an incident on the bone B(a). Henceforth, when speaking of curvature assignments to the bones {B (a)} of a dynamical triangulation Ta , we shall explicitly refer to the sequence of integers {q(a) }.

3.1.3 The Einstein-Hilbert Action for Dynamical Triangulations The Regge version of the Einstein-Hilbert action (with cosmological term) for a PL-manifold M, (without boundary), is given by[99],[7, 41, 114, 74]' [57]

SRegge(A, G) ,;" A

L vol(an ) un

16~G L B

r(B)vol(B).

(3.19)

3.1 Dynamical Triangulations as Length-Spaces

47

Fig. 3.6. A dynamically triangulated portion of ]R2. This dynamical triangulation correspond to a non-flat PL-manifold: there are bones (vertices) carrying negative curvature (those with 8 triangles incident on them), and bones around which the curvature is positive (the ones with incidence number 4)

Fig. 3.7. An unusual dynamical triangulation of the 2-sphere §2. The sharp vertices like VI carry no curvature. Curvature is concentrated on vertices like V2

48

3. Dynamical Triangulations

We refer the reader to [38] for a detailed mathematical analyis of the geometry underlying this expression, and in particular for assessing the delicate issue of the convergence of (3.19) to the Einstein-Hilbert action as the (Regge) triangulation becomes finer and finer. When the PL-manifold M is dynamically triangulated T a - t M, SRegge(A, G) simplifies considerably reducing to

(3.20) where we have set according to standard usage

(3.21 )

kn -

. vol(a n -

2

=

8G

2

)

(3.22)

.

Thus, for dynamical triangulations the gravitational action reduces to a combinatorial object with two (running) couplings kn - 2 and kn . The former proportional to the inverse gravitational coupling, while the latter is a linear combination of 1/161rG with the cosmological constant A. Accordingly, the formal path integration (1.1) is replaced (see (1.6)) on a dynamically triangulated PI manifold M, (of fixed topology), by the grand-canonical partition function Z[kn -

2,

L

kn ] =

e-knNn+kn-2Nn-2.

(3.23)

TET(M)

One can introduce also the canonical partition function defined by W(kn- 2 )eff

=

L

(3.24)

TET(Nn

)

where the summation is extended over all distinct dynamical triangulations with given N n , (i.e., at fixed volume), of a given PL-manifold M. Finally, we shall consider the microcanonical partition function

W[Nn -

2 , b(n, n

- 2)]

=

L

1 CT'

(3.25)

TET(Nn iN n-2)

where the summation is extended over all distinct dynamical triangulations with given N n and N n - 2 , (i.e., at fixed volume and fixed average curvature b(n, n - 2) = !n(n + 1)(Nn /Nn - 2 )), of a given PL-manifold M. Here CT denotes the symmetry factor of the triangulation. This factor is present for closed manifolds for the same reason as the special symmetry factor for Feynman vacuum diagrams. CT is equal to the order of the automorphism group

3.2 Dynamical Triangulations as Singular Metric Spaces

49

of the graph associated with the triangulation T. In general, since we shall explicitly consider rooted triangulations, we shall assume CT == 1. We will mainly interested in estimating the microcanonical partition function, this is simply the number of distinct dynamical triangulations with given volume (ex: N n ) and fixed average curvature ex: b(n,n-2)), of a given PLmanifold M. Thus, we shall often call it the entropy function. Similarly we often refer to the canonical partition function as to the effective entropy , since it is the number of distinct dynamical triangulations, with given vol.. ume, averaged over all possible average curvatures b( n, n - 2) with respect to the weight exp[kn - 2 N n - 2 ]. The action S[kn - 2 , kn ] and the associated partition function are so deceiptively simple that one may wonder under which rug we are dumping all the notorious problems affecting the Einstein-Hilbert action when coming to a path-integral quantization. Obviously most of these problems are now shifted in characterizing the summation over distinct triangulations LTET(M) which replaces the formal path integration over distinct riemannian structures, namely in characterizing the dynamical triangulation counterpart of the formal measure D[g(M)]. In general, these sort of deals hardly are true bargains. However, as we shall see, this trade off in difficulties will payoff since a rather complex problem (Path Integration over Riem(M)/Diff(M)), the mathematical formulation of which is rather ambiguous, has been transformed into a well-defined counting problem : enumerate all distinct dynamical triangulations T E T(M) admitted by a PL manifold of given topology. However, before addressing this problem, we have to formalize in more precise terms in which sense dynamical triangulations do approximate a riemannian structure on a smooth manifold.

3.2 Dynamical Triangulations as Singular Metric Spaces The fact that curvature for dynamical triangulations is a combinatorial object has a number of deep consequences, mainly of topological nature, having an important bearing on simplicial quantum gravity. These properties follow from the fact that a dynamically triangulated manifold M == ITa I, with Nn(Ta ) n-simplices and with a given average number b(n, n - 2) of nsimplices incident on the bones a n - 2 , (see (2.8)), is a piecewise-flat singular metric space of bounded geometry, a concept which goes back to A.D. Alexandrov [4]. There is a definite way in which such singular metric spaces approximate (or are approximated by suitably smooth) manifolds, and this will provide the rationale for approximating riemannian manifolds with dynamical triangulations. To start with an intuitive description, let us consider two geodesic rays starting from the some point p in a Riemannian manifold V. Thus, suppose that l1:[0,1] -+ V and l2:[0,1] -+ V are (minimal) geodesics with l1(0) == l2(0) = p. Fix t, S E (0,1) and construct the plane Euclidean triangle with

50

3. Dynamical Triangulations

(linear-parametrized) lines L 1 : [0,1] -t ~?, L 2 : [0,1] -t]R2 such that L 1 (0) == L 2(0) == (0, 0), II L 1 ( 8 ) II == d(II (0), II (S ) ), II L 2(t ) II == d(II (0), I2 (t )), and

IIL 1(8) - L2(t)II == d(II(S), I2(t)).

(3.26)

If the sectional curvatures of V are everywhere non-negative, then the geodesics II and 12 tend to come together compared with the corresponding lines L 1 , £2; i.e., (3.27)

whereas non-positive sectional curvatures force the goedesics to diverge faster than in the Euclidean situation: IIL 1 (1) - L 2 (1)11 ~ d(ll(1),l2(1)).

Fig. 3.8. Behavior of geodesic rays: (top) in a negatively curved manifold; (bottom) in a positively curved manifolds

Once one has a notion of geodesics, the above characterization of positive or negative curvature makes sense for any length-spaces, and in that case we speak of a space of positive (or negative) A1exandrov curvature . Actually, similar characterizations can be carried out in order to define spaces of Alexandrov curvature bounded above of below by a constant k E IR, by comparing the geodesics ll' I2 with the corresponding geodesics on the ap-

3.2 Dynamical Triangulations as Singular Metric Spaces

51

propriate simply connected surface of constant Gauss curvature k. By definition an Alexandrov space is a complete finite dimensional inner metric space which satisfies distance comparison in the sense of Alexandrov. These comparision techniques which are concerned with studying spaces satisfying curvature inequalities are a standard tool in global riemannian geometry, (see e.g., Chavel's book[37]).

3.2.1 Geodesic Triangles For k E lR, let Ek denote the model surface of constant Gauss curvature k. Motivated by the above remarks, we characterize dynamical triangulations as singular metric spaces with bounded Alexandrov curvature, by comparing suitable triangles in Ta with triangles in model surfaces Ek. Since in a dynamical triangulation curvature is localized around bones a n - 2 , we have to consider geodesic triangles which encircles bones. This can be done according to the following procedure. Let a ~ a n - 2 be the generic bone in a dynamical triangulation M. Consider its associated link, link(a n- 2) ~ §1, and three distinct vertices, labelled 1, 2, 3 in link(a n- 2). If q(a) simplices an are incident on the bone a, then link(a n- 2) has a length Illink(a n- 2)11 == q(a) . a, (3.28) a being the edge-length of the given dynamical triangulation. The vertices 1,2,3 are chosen (up to a rotation) in such a way that their pairwise distance, d(i,k), i =I- k == 1,2,3" measured along the link link(a n- 2), satisfies

dei, k)

~ [q~a)],

(3.29)

where the bracket function [x] stands for the greatest integer not exceeding x. Consider now (in the closure of the open star of the bone a) the three geodesic segments pairwise joining the vertices so chosen, viz., Cl ~ l(2,3), C2 ~ l(l, 3), and C3 ~ l(l, 2), then

Definition 3.2.1. The geodesic triangle L1(a) ~ (Cl' C2, C3) consisting of the three geodesic segments Cl, C2, C3 in M, (called the sides of the triangle), is the geodesic triangle encircling the bone a EM. A geodesic triangle .d ~ (ci, C2, c"3) in Ek is called an Alexandrov (or comparision) triangle for L1(a) if length(Ci) == length( i;), i == 1, 2, 3. It is not difficult to verify that such a comparision triangle exists, (and it is unique up to a congruence), and we have the following

Lemma 3.2.1. A dynamically triangulated manifold M == ITal, with Nn(Ta) n-simplices and with a given average number b(n, n - 2) of n- simplices incident on the bones a n- 2, is a piecewise-fiat singular metric space of bounded Alexandrov curvature k, with kmin ::; k ::; k max , where kmin , kmax are characterized by

52

3. Dynamical Thiangulations

"- '.

Fig. 3.9. A geodesic triangle around the bone of a 3-dimensional dynamical triangulation

53

3.2 Dynamical Thiangulations as Singular Metric Spaces

~ a ."qcos -1(1/) _ sin(a~) n .rr:-- ' 2

(3.30)

1 -1(1/) sinh(a~) -2 a· qmax cos n == r-r;:---

(3.31)

7T

V

kmax

and v-km~

7T

where, (see (2.30)) _ n(n+l) (b(n,n-2)-ii)N qmax - 2F(n) b(n, n _ 2) n·

(3.32)

Proof We start by showing that comparision triangle exists for any bone of the given dynamical triangulation. In order to prove this, let us first remark that the geodesic triangle Ll (a) has sides satisfying the triangle inequality length(ci)

+ length(ci+1)

~ length(ci+2),

(3.33)

indices taken modulo 3. However, this is not sufficient to insure the existence of a comparision triangle in Ek since, if k > 0, the perimeter of Ll(a) must satisfy the bound

length(cl)

+ length(c2) + length(c3) <

27T ..jk'

(3.34)

0c

(this bound is trivially extended to the case k ::; 0 by setting ~ 00 for k ::; 0). To prove that (3.34) holds, and in order to avoid carrying around factors, let us rescale the cut- off a by the factor (i. e., annoying a ~ JJa), let x be any point in the relative interior of the given bone a, and consider the set of all tangent vectors to the geodesic rays (lines) emanating from x, of length a, pointing into an , (the generic simplex incident on a), and orthogonal to a. This set of vectors forms a spherical (§1 )-simplex. Since this set is, up to an isometry, independent of x E a, we shall denote this set as §llink(a, an): the spherical link of a in an [78], [38]. Geometrically, the spherical link §llink(a, an) can be thought of as the arc of circumference circumscribed to link( a, an), the link of the bone a in an. If the Regge curvature around the given bone is positive, then

JJ

27T q(a) < cos-1(1/n).

JJ,

(3.35)

Since cos- 1 (1/n) ~ ~ for n ~ 2, we get that q(a) < 6. The union of the spherical links §llink(a, an) over all q(a) simplices an incident on a is then a piecewise smooth closed curve (generated by arcs of circumference) of length a·q(a) cos- 1 (1/n) < 27Ta. This curve can be thought of as the space of normal directions associated with the bone a. The fact that it is not isometric to the standard circumference (of radius a), reflects the fact that the bones are metrically singular, (equivalently, we could have expressed such singularity

54

3. Dynamical Triangulations

in terms of the tangent cone over such a spherical link). Consider now the sphere, §2(r), of radius r, such that a great circle of radius a on §2(r) is isometric to the union of the spherical links §ll in k(a, an), namely it has a circumference of length a . q(a) cos- 1 (1/n). An elementary computation characterizes the radius r of such §2(r) in terms of q(a) as 1 1 . a -2 a·q(a)cos- (l/n)=rsln(-), ~ r

or, by introducing the curvature k ~

(3.36)

/2

of §2(r)

~a. q(a) cos- 1(1/n) = Sin~). k

2~

(3.37)

Since the geodesic triangle Ll(a) is inscribed in the piecewise circumference of length a . q(a) cos- 1 (1/n), we get that (3.34) holds with k given in terms of q( a) by (3.37). By developing (3.37) to leading order in a, we get, not unexpectedly, that k is given by 6 k ~ 2" (2~ - q(a) cos- 1 (1/n)) (3.38) a namely, up to a numerical factor, by the Regge curvature associated with the bone a. This elementary analysis can be extended to the case of negative Regge curvature obtaining

1

2~ a

()

. q a cos

-1(1/)

n =

sinh(aH)

H

(3.39)

thus proving that a dynamically triangulated manifold M = ITa I, with Nn(Ta ) n-simplices and with a given average number b(n, n -- 2) of n- simplices incident on the bones a n - 2 , admits comparision triangles on model surfaces, E k , of constant curvature k, where k is provided by (3.37) and (3.39). Since on such set of dynamical triangulations the possible incidence numbers {q(a)} are bounded between

q ~ q(a)

(3.40)

~ qmax,

(see (2.30)), the above argument shows that a dynamical triangulation M = ITal, as a singular metric space, is an Alexandrov space with bounded curvature k min ~ k ~ kmax where k min and k max are respectively provided by

1 qcos " -1(1/) -a· n = 2~

sin(a~) Vkmax

(3.41)

and 1 -1( / ) _ sinh(a~) -a' qmax cos 1 n r-r;:-, 2~ y -kmin

as stated.

(3.42)

3.2 Dynamical Triangulations as Singular Metric Spaces

55

The notion of bounded geometry associated with the characterization of dynamical triangulations as Alexandrov spaces becomes especially interesting if one considers the set of all dynamical triangulations satisfying the hypotheses of lemma 3.2.1. Guided by the intuition of what happens in similar circumstances for ordinary riemannian manifolds[65, 59, 52],[68, 70, 71], we expect that the metric space structures of such dynamical triangulations is controlled by such curvature bounds. To formulate this kind of controlled behavior, it is useful to introduce a topology in the set of dynamical triangulations which arises directly from the classical idea of Hausdorff distance between compact sets in a metric space[65, 52]. The basic rationale is that given a length cut-off €, two metric spaces are to be considered near in this topology, (one is the €-Gromov- Hausdorff approximation of the other), if their metric properties are similar at length scales L ~ €, namely if there is a way of fitting them in a larger metric space so that they are close to each other. To formalize this intuition, we introduce the notion of rough isometry. Consider two compact (finite dimensional) metric spaces M I and M 2 , let dMl(·'·) and dM2 (·,·) respectively denote the corresponding distance functions, and let ¢: M I ~ M 2 be a map between M I and M 2 , (this map is not required to be continuous ). If ¢ is such that: (i), the €-neighborhood of ¢(MI ) in M 2 is equal to M 2 , and (ii), for each x, y in MI we have (3.43) then ¢ is said to be an €-Hausdorff approximation. The Gromov-Hausdorff distance between the two metric spaces M I and M 2 , da(M I , M 2 ), is then defined according to[65, 59, 52] Definition 3.2.2. da(MI , M 2 ) is the lower bound of the positive numbers € such that there exist €-Hausdorff approximations from M I to M 2 and from M 2 to MI. Alternatively[98], da(M I , M 2 ) ::; € if there is a metric, d(·, .), on the disjoint union M I 11 M 2 extendings the metrics on M I and M 2 and such that M I C {x E M I 11M2 :d(x,M2 )::; €}, M 2 C {x E M I 11M2 :d(x,MI )::; €}. Namely, two compact metric spaces M I and M 2 are Gromov-Hausdorff nearby if we can find metrics on the disjoint union of M I and M 2 such that M I and M 2 look like the same when they are close to each other. The notion of €-Gromov-Hausdorff approximation is the weakest largescale equivalence relation between metric spaces of use in geometry, and is manifestly adapted to the needs of simplicial quantum gravity, (think of a manifold and of a simplicial approximation to it). Is not obvious that da, so defined is a distance since it is hard to prove, from the notion of €-Hausdorff approximation, that the triangle inequality holds. However the alternative characterization recalled above does easily yields for such a proof and we get a metric space in which the set of isometry classes of all compact metric spaces, is Hausdorff and complete. This enlarged space does naturally contain riemannian manifolds and metric polyedra. As

56

3. Dynamical Triangulations

stressed in [98], the importance of this notion lies not so much in the fact that we have a distance function, but in that we have a way of measuring when metric spaces look alike. It is apparently surprising, due to its apparent coarseness, that the Gromov-Hausdorff topology can yield anything useful, however, this topology is actually quite strong as soon as one makes rather weak assumptions (in the comparision sense) on curvatures and few other natural geometrical parameters.

Fig. 3.10. Approximating Riemannian manifolds

3.3 Approximating Riemannian Manifolds Through Dynamical Triangulations In this section we prove that, dynamical triangulations T a , with a given number Nn(Ta) of n-simplices and a given ratio b(n, n - 2), are in the Gromov-

3.3 Approximating Riemannian Manifolds

57

Hausdorff closure of the co-dimensional compact metric space characterized by the following

Definition 3.3.1. For r a real number, D and V positive real numbers, and n a natural number, let us define the associated space of Bounded Geometries, R(n, r, D, V), as the Gromov-Hausdorff closure of the space of isometry classes of closed connected n-dimensional riemannian manifolds (M, g) with Sectional curvature bounded below by r: inf {inf{Sec(u,u):u E TxM, luxl == I}} ~ r,

xEM

(3.44)

volume bounded below by V: (3.45)

Vol(M) ~ V, and diameter bounded above by D, viz.,

diam(M) ==

sup

(p,q)EMxM

dM(p, q)

~

D.

(3.46)

It is rather immediate to recast this characterization in our current setting so as to prove the following approximation property (actually a statement of the Lebesgue covering property) which has far reaching consequences for simplicial gravity:

Theorem 3.3.1. Let DTn(a, b, N) denote the set of n-dimensional, (n == 2,3, 4), dynamically triangulated manifolds Ta ~ M == ITa!, with Nn(Ta) == N top-dimensional simplices and average incidence b(n, n-2). Let R(n, r, D, V) denote a space of bouded geometries. Then, for any given (suitably small) € > 0, there exists a pair (a(€), N(€)) such that for any riemannian manifold vn E R(n, r, D, V) we can find a dynamically triangulated manifold M E VTn(a(E), b, N(E)) such that

da(Vn,M) < E.

(3.47)

In other words, the set of dynamical triangulations DTn (a, b, N) uniformly approximates riemannian manifolds of bounded geometry.

This result can be seen as a particular case of the Burago-GromovPerel'man compactenss theorem, [32] according to which the class of Alexandrov spaces with dimension ~ n, curvature ~ k and diameter uniformly bounded above is compact in the Gromov- Hausdorff topology, and if M is the Gromov-Hausdorff limit of a sequence of riemannian manifolds with sectional curvatures ~ k, then M is an Alexandrov space with curvature ~ k. It is perhaps worth emphasizing here the fundamental difference between Riemannian manifolds and Alexandrov spaces, a difference familiar when working with dynamical triangulations: the space of directions in a Riemannian manifold is the unit sphere, whereas for an n-dimensional Alexandrov space it can be any positively curved n - I-dimensional Alexandrov space.

58

3. Dynamical Triangulations

In order to prove lemma 3.3.1, we need to recall some standard results related to the compactness properties of R(n, r, D, V). A sequence of compact metric spaces {Mi } converges to a compact metric space M provided that da(Mi , M) ~ 0 as i ~ 00. Thus we have to prove that there is an a == a(E) > 0 such that the Gromov- Hausdorff distance between the generic riemannian manifold Y E R(n, r, D, Y) and a suitably chosen dynamical triangulation Ta , is lesser than or equal to E. The proof of the above theorem is a rather immediate consequence of the following well-known properties[68, 70, 71]:

Theorem 3.3.2. Let yn E R(n, r, D, Y) be the generic n-dimensional riemannian manifold of bounded geometry, and let dv(·,p) denote the corresponding distance function from a chosen point P E y n . Then, for any given E > 0, it is always possible to find an ordered set of points {PI, ... ,Pm} in yn, so that: (i) the open metric balls, (the geodesic balls), BV(Pi' E) == {x E ynld(X,Pi) < E}, i == 1, ... , m, cover yn; (ii) the open balls BV(Pi' ~), i == 1, ... ,m, are disjoint, Le., {PI, ... ,Pm} is a minimal E-net in yn.

We recall that the filling function of a minimal E-net is the number m of points {PI, ... ,Pm}, while the first order intersection pattern of a minimal E-net in yn is the set of pairs {(i,j)li,j == 1, ... , m; B(Pi' E) n B(pj, E) f= 0}.

Fig. 3.11. A geodesic ball packing on the two-sphere

It is important to remark that on R(n, r, D, Y) neither the filling function nor the intersection pattern can be arbitrary. The filling function is always bounded above for each given E, and the best filling, with geodesic balls of radius E, of a riemannian manifold of diameter diam(yn), and

3.3 Approximating Riemannian Manifolds

59

Ricci curvature Ric(Vn ) ~ (n - l)H, is controlled by the corresponding filling of the geodesic ball of radius diam(vn) on the space form of constant curvature given by H, the bound being of the form[68, 70, 71] N~O) ~ N(n, H(diam(V n ))2, (diam(Vn))/E). More explicitly, given the minimal E-net PI,'" ,Pm in V n , let p denote any point in the simply connected, complete n-dimensional space form Vli of constant curvature H. By following a standard argument or"K.Grove and P.Petersen[68], we can apply the Bishop-Gromov volume comparision theorem (see e.g., Chavel's book[37]' pp. 123) in order to obtain the bound

N(O) €

< -

Vol(V n ) < Vol [B(p, D)] Vol[B(P io,E/2)] - Vol[B(p,E/2)] '

(3.48)

where B(Pio,E/2) is the ball of smallest volume, and Vol [B(p, D)] denotes the volume, in the space form VIi, of the geodesic ball of radius equal to D = diam(V n ). Thus (0)

N€

<

I~ sinhn -

1

n

I

/2

- J; sinh

-

( J!HTt)dt ( J!HTt)dt

(3.49)

.

The multiplicity of the first intersection pattern is similarly controlled through the geometry of the manifold to the effect that its average degree, s(vn), (i.e., the average number of mutually intersecting balls), is bounded above by a constant as the radius of the balls defining the covering tend to zero, (i.e., as E ~ In order to estimate such constant[68]' let us fix a point P E vn and let Pi!" .. ,Pik the points in the E-net such that B(p, E) n B(Pii' E) =1= 0. It is easily checked that the set of balls {B(Pii' E) }iI,... ,ik is contained in the ball of radius ~E centered at p. Thus

°).

(3.50) where the second bound is obtained by enlarging enough the radius of the generic intersecting ball B(Pik' E/2). On applying the Bishop-Gromov volume comparision theorem we get[68]

s (V

n)

<

J;€/2sinhn-I(J!HTt)dt /2

- J; sinhn -

I

(

J!HTt)dt

.

(3.51 )

Note that as E ~ 0, the right hand side of (3.51) tends to 9n . Thus, as E ~ 0, the degree of the intersection pattern is bounded above by a constant independent from E, (as expected, its leading dependence is in the dimension n, since the intersection pattern of a minimal E-net is directly related to the covering dimension of the underlying space).

60

3. Dynamical Triangulations

Definition 3.3.2. Two metric spaces (e.g., two riemannian manifolds) vIn and v2n in R(n, r, D, V) are called €-equivalent if and only if they can be equipped with minimal €-nets {PI, ... ,Pm}1 and {PI, ... ,Ps}2, respectively, such that s

== m and with the same intersection pattern.

We can introduce a natural minimal geodesic ball covering associated with a dynamical triangulation Ta according to the following

Lemma 3.3.1. Let (M == ITal, Ta) denote a dynamically triangulated manifold with fixed edge legth a. With each vertex Pi ~ a? belonging to the triangulation, we associate the largest open metric ball contained in the open star of Pi. Then the metric ball covering of M == ITa I generated by such balls {B i } is a minimal geodesic ball covering. It defines the geodesic ball covering associated with the dynamically triangulated manifold (M, T a ).

It is immediate to see that the set of balls considered defines indeed a minimal geodesic ball covering. The open balls obtained from {B i } by halfing their radius are disjoint being contained in the open stars of {Pi} in the baricentric subdivision of the triangulation. The balls with doubled radius cover (M, T a ), since they are the largest open balls contained in the stars of the vertices {Pi} of Ta . We are now in a position for proving lemma 3.3.1. Proof. Given a riemannian manifold vn E R(n, r, D, V), endowed with with a minimal €-net {qi}i=I, ... ,m we can associated with it a dynamical triangulations Ta , with a == a(€), having the same combinatorial intersection pattern {Pi == a?}i=l, ... ,m and such that Id(qi,qj) - d(pi,pj)1 < € for all i, j == 1, ... , m. In order to characterize such a dynamical triangulation, (which generally is not unique), we consider a smooth triangulation T of the riemannian manifold vn the vertices of which are provided by the minimal €-net {qi}i=I, ... ,m, and such that its I-skeleton is provided by the first order intersection pattern of the given geodesic ball covering, viz., {(i,j)li,j == 1, ... , m; B(qi, €) nB(qj, €) =I 0}. We denote by N(€) the number of n-dimensional simplices of such a triangulation, and in general by Ni (€), i == 0, ... ,n the components of its f-vector. Note that for characterizing T, we are simply using the first order intersection pattern of the geodesic ball covering and not its full intersection pattern giving rise to the skeleton of the covering, (since, generally speaking, the combinatorial structure of this skeleton is not associated with a triangulation). As a consequence, there are distinct smooth triangulations T with the same underlying I-skeleton (given by the geodesic ball covering first-order intersection pattern), but with distinct k-skeletons, (k 2: 2). Note that if

B(qi' €)

n B(qj, €) =I 0,

(3.52)

then (3.53)

3.3 Approximating Riemannian Manifolds

61

Fig. 3.12. A metric ball packing (and covering) on a portion of a dynamically triangulated manifold T a . The packing is evidentiated by the dotted disks

62

3. Dynamical Thiangulations

We can construct a dynamical triangulation T a associated with the smooth triangulation T and hence with the pair (yn, {qi}) by straightening the simplices of T, viz., by declaring that all the n-simplices of T are isometric to the standard equilateral simplex un in jRn with edge- length

a(f)==K·f

(3.54)

where, according to (3.53), K is a fixed constant such that 1 ~ K ~ 2. By denoting by {Pi == U?}i=l, ... ,m the vertex set of such T a , we get that Idv(qi, qj) - dM(Pi,Pj)1 < f for all i,j == 1, ... , m, as claimed. On the disjoint union yn Ta we can define the metric d( x, y) == min{dv(q, qi) + f + dM(Pi,P): i == 1, ... , m} for q E yn, P ETa. More explicitly, this metric is constructed by declaring d(Pi' qj) == f, and by setting, for P E yn and q ETa, the distance d(p, q) equal to the infimum of the sums d(p, Xl) + d(Xl,X2) + ... + d(Xl,q) where each consecutive pair (Xi,Xi+l), (or (P,Xl), or (Xl,q)) either has both elements in one of yn or T a or has Xi == Pm, Xi+l == qm. The metric so defined is such that yn C {x E ynIITa:d(x,Ta) ~ f}, T a C {x E MIITa:d(x,Y) ~ f}, and thus da(yn, T a) ~ 2f, which implies the stated approximation result.

II

Since a given intersection pattern may be common to distinct dynamical triangulations, there can be many distinct dynamical triangulations that are 2f-rough isometric to a given riemannian manifold. This is not surprising, the equivalence between riemannian manifolds associated with minimal f-nets is very crude, and there are distinct riemannian manifolds sharing the same fnet, (i.e., which are 2f-rough isometric). The point is that all such manifolds as well as all such dynamical triangulations have a Gromov-Hausdorff distance between each other which is smaller than or equal to 2f, and as f ~ 0, viz., as the dynamical triangulations become finer and finer such dynamical triangulations converge to a well defined metric space in R(n, T, D, V), (not necessarily a nice riemannian manifold!). Since the intersection pattern of a geodesic ball covering of a Riemannian manifold yn is rather rigidly controlled by the curvature, (actually by the dimension as f ~ 0), we can bound the (Alexandrov) curvature of the dynamical triangulation T a in terms of the curvature of yn. The largest curvature of Ta is characterized in absolute value by the largest incidence number (of n-simplices on bones) qmax. Let u~-2 the bone for which q(u~-2) == qmax is attained. The incidence number q(u~-2) is bounded above by the number of edges uJ incident on any of the vertices of the bone u~-2. Finally, this latter number is bounded above by the (maximum) order, s(Y), of the geodesic ball intersection pattern of the riemannian manifold yn, (see (3.51)). Thus qmax ~ s(yn). Actually, the proof of the bound to s(yn), (see (3.51)), shows that there exists a constant 5 < c ::; 9, (this is just a rough estimate for such a c), not depending on Y and f, such that

3.3 Approximating Riemannian Manifolds

63

Note that in this expression we are not interested in taking the limit € ~ 0 since we are dealing with dynamical triangulations Ta which are finite approximation to V n . The integrals appearing in (3.55) provide the volume of the geodesic balls (of radius c€/2 and €/2, respectively) on the space form VJ}. To leading order in €2, we get

<

qmax -

n

1- _H_(g)2

C 1

6(n+2) 2 _ _ H_(.f)2 6(n+2) 2

+ o(c2 €2) + 0 (2) € (3.56)

If we introduce the cut-off a(€) == K . €, associated with the dynamical triangulation Ta €-approximating vn, we eventually get n qmax ::; C -

H ( )2 24(n + 2)K2 a f. + ... ,

(3.57)

which provides a nice upper bound to the curvature of the approximating dynamical triangulation Ta in terms of the Riemannian curvature of the given riemannian manifold v n . It is worth noticing that it is possible to obtain a converse of the above approximation results, indeed [68, 70, 71], any polyhedral compact subset of a Euclidean space is the Gromov-Hausdorff limit of a sequence of closed Riemannian manifolds. The density results just obtained provide the mathematical rationale for using dynamical triangulations in simplicial quantum gravity. The original idea was related to the intuition that fixed-edge length triangulations, T a , (of fixed topology M == ITal) provide a grid of reference manifolds laid upon the oo-dimensional space of riemannian structures allowed on such M. The compactness results associated with the use of Gromov-Hausdorff convergence support this picture, and the uniform approximation result of lemma 3.3.1 makes precise this intuition by proving that dynamically triangulated manifolds do provide a reference grid as long as we look at riemannian manifolds and dynamical triangulations of bounded geometry. At first sight it may appear surprising that no strict topology control is required on such triangulations, but this is actually a bonus coming automatically along with the requirement of bounded geometry, (i.e., fixed N n (Volume) and fixed average incidence b(n, n - 2) (curvature bounds)). As we shall see in the next section these natural bounds severely control topology, by avoiding such patologies as infinite genus surfaces (in dimension n == 2), or wild homotopy

64

3. Dynamical Triangulations

M

Fig. 3.13. Given a riemannian manifold V E R(n, r, D, V), endowed with with a minimal €-net {qi}i=l, ... ,m we can associated with it the dynamical triangulations T a , with a = a( €), having the same combinatorial intersection pattern {Pi = a?}i=l, ... ,m and such that Id(qi' qj) - d(Pi,Pj)! < € for all i,j = 1, ... , m. On the disjoint union V IJ T a we can define the metric d( x, y) = min{d(q, qi) + € + d(Pi,p): i = 1, ... , m} for q E V, P ETa. The metric so defined is such that V C {x E VIJTa:d(x,Ta) :::; €}, T a C {x E VIJTa:d(x,V) :::; €}, and thus da(V, T a ) :::; 2€

3.4 Topological Finiteness Theorems for Dynamical Triangulations

65

types , etc. One may question the naturality of such bounds in the sense that dynamical triangulations is an approach to quantizing gravity in the sense of a statistical field theory, and in seeking the associated critical phenomena we eventually need to go to the infinite-volume limit. In this limit the bounds on N n and b(n, n - 2) are removed and one consequently loses control on everything. The point we wish to make is exactly that the above results tell us how to carry out such sort of limits. One must rigorously establish estimates for dynamical triangulations, the associated partition functions and m- points correlation functions, in a given space of bounded geometry VTn(a, b, N) c R(n, r, D, V), and then consider a nested sequence VTn(al, bl , N I ) c VTn (a2, b2, N 2) c ... C VTn(ai' bi , N i ) ... in which the convergence properties of the partition function and the associated infinitevolume limit can be safely discussed. As we shall see, the actual characterization of the infinite volume limit does not call for such a sophisticated machinery. However the discussion of a possible continuum limit may do require the introduction of a suitable projective limit procedure for DTn(a, b, N).

3.4 Topological Finiteness Theorems for Dynamical Triangulations of Bounded Geometry [Topological Finiteness Theorems for Dynamical Triangulations of Bounded Geometry] Since dynamical triangulations, (for given Nand b(n, n-2)), are in a Gromov space of bounded geometries, R( n, r, D, V), we can control to a considerable extent their topology in terms of the parameters n, r, D, and V, (namely in terms of the number of top-dimensional simplices Nand b( n, n - 2), in the range of dimension considered). This control follows from the compactness properties of R(n, r, D, V), and from the fact that topology related function can be shown to be locally constant on R( n, r, D, V), and as such they can take only finitely many distinct values. We start with result expressing finiteness of homotopy types of dynamically triangulated manifolds of bounded geometry [68, 70, 71, 69, 118]. Theorem 3.4.1. For any dimension 2 ~ n ~ 4, the number of different n(a, b, N) homotopy-types of dynamically triangulated manifolds realized in c R(n, r, D, V) is finite and is a function of n, V-I Dn, and r D 2 •

vr

(Two polyhedra M I = ITa (l)1 and M 2 = ITa (2)1 are said to have the same homotopy type if there exists a continuous map ¢ of M I into M 2 and f of M 2 into MI , such that both f . ¢ and ¢ . f are homotopic to the respective identity mappings,IM1 and 1M2 . Obviously, two homeomorphic polyhedra are of the same homotopy type, but the converse is not true). This homotopy finiteness property of dynamically triangulated manifolds is an obvious adaptation of a more general result proved in a truly remarkable

66

3. Dynamical Triangulations

series of papers by P. Petersen, K. Grove, and J. Y. Wu [68, 70, 71, 69]. Actually, for dynamical triangulations, the above theorem can be sharpened so as to provide (rather weak) conditions under which two distinct dynamically triangulated manifolds share a common homotopy type, and we can estimate the number of distinct homotopy types . We have the following

Theorem 3.4.2. For any dimension 2 :::; n :::; 4, and for a given value a of the cut-off edge-length, the number of different homotopy-types of dynamically triangulated manifolds realized in DTn(a, b, N) c R(n, r, D, V) is bounded above by ~Homotopy[DTn(a,b,N)] :::; [C(n)-l. Nvol(a n ). a- n ]4, (3.58) where C(n) is some universal constant depending only on the dimension n, and vol(a n ) is the euclidean volume of the simplex an (of edge-length a). Proof Recall[53, 54],[70, 71] that a continuous function 'l/J: [0, a) ~ jR+, a > 0, with 'l/J(O) = 0, and 'l/J(t) ~ t, for all t E [0, a), is a local geometric contractibility function for a (finite dimensional) compact metric space M if, for each x E M and t E (0, a), the open ball B(x, t), of radius t centered at x E M, is contractible in the larger ball B(x, 'l/J(t)) , (which says that in M small metric balls are contractible relative to bigger balls). On a dynamical triangulation T a E DTn (a, b, N), consider the smallest open ball containing the star of the generic vertex. This is a ball or radius a which is contractible within itself. Thus, for any such Ta we may consider the local geometric contractibility function given by

'ljJ(€)

=

€

(3.59)

on [0, a]. We can now exploit in our setting a result of K.Grove and P.Petersen[98, 68] to the effect that any two dynamical triangulations T! and T; in DTn(a, b, N) with Gromov-Hausdorff distance da(T!, T;) < t* are homotopy equivalent if €* ~ a/(32n 2 ), (this 'bound is not optimal, and similarly to what happens for riemannian manifolds with criticality radius bounded below, one should get a sharper bound of the form a/[25(n + 1)][68]). Since ITa I can be covered by the open metric balls of radius a centered on the vertices of Ta , it also admits a refinenement of this covering generated by open metric balls of radius t* /2. If N(t* /2) denotes the number of such balls, then an argument due to Yamaguchi [118] can be used to show that DTn(a, b, N) contains less than [N(t* /2)]4 distinct homotopy types. In order to estimate N(t* /2), notice that the volume of a metric ball, B(t* /2), of radius t* /2 in ITa I is bounded below by

vol[B(t* /2)] ~ q(ao) . vol(a n)€*/2 = (n + 1) . c(n)(t* /2)n,

(3.60)

where q(aO) = (n + 1) is the minimum number of n- dimensional simplices sharing a vertex, and vol(a n )€* /2 = c(n)(t* /2)n is the euclidean volume of a standard euclidean simplex an with edge-length t* /2, (this characterizes the

3.4 Topological Finiteness Theorems for Dynamical Triangulations

constant c(n) as c(n) ~ vn + l/(n!y'21i")). Since the volume of by N . vol(a n ), we get N( * /2) €

n

N . vol(a )

~ (n + 1) . c(n)(€* /2)n·

ITal

67

is given (3.61 )

Since f* ~ (a/32n 2 ), by introducing (3.61) in the Yamaguchi estimate we get . h C( n ) ==. c(n)(n+1) h d resuItWIt testate (64n 2 )n .

.A similar result, obtained by adapting to dynamical triangulations results obtained by Yamaguchi, Grove, Petersen and Wu[68, 70, 71, 69, 118], allows also to control the Betti numbers, (with generic field coeffiecients F), of any M == ITal E Drn(a, b, N). Explicitly we have Theorem 3.4.3. Let bi(M, F) denote the i-th Betti number with field coefficients F, .then

L bi(M, F) ::; (n + 1)(n+1) [C(n)-1 . Nvol(a n ) . a-n]n . n

(3.62)

i=1

Proof. The proof is an obvious rewriting of a result of P.Petersen , (see the corollary at p. 393 of P.Petersen's paper[98]).

The above results clearly show that when going to the infinite-volume limit, limN~oo,a~o < N > vol(a n ) ~ const., (where < N > denotes the statistical average of N with respect to a canonical ensemble of dynamical triangulations), more and more distinct topological types of manifolds come into play. However the resulting topological complexity is not too wild since the riumber of distinct homotopy types (and/or the Betti numbers) grows polynomially with the inverse cut-off a -1 . By considering separately the dimension n == 2, n == 3, n == 4, we can obtain a more specific topological finiteness theorem. This result follows again from the local geometric contractibility which by construction characterizes dynamically triangulated manifolds. Since sufficiently small distance balls on a dynamically triangulated manifold of bounded geometry are always contractible, we get, by specializing a theorem of Grove , Petersen , Wu [68, 70, 71, 69], Theorem 3.4.4. The set of dynamically triangulated manifolds in 1)Tn(a, b, N) C n(n, r, D, V) contains (i) finitely many simple homotopy types (all n), (ii) finitely many homeomorphism types if n == 4, (iii) finitely many diffeomorphism types ifn == 2 (orn 2:: 5, in case we remove the constraint 2 ::; n ::; 4).

Note that quantitative estimates associated with these finiteness results have the same structure of the ones associated with theorems 3.4.2 and 3.4.3, and

68

3. Dynamical Triangulations

can be easily worked out along the same lines. Note also that finiteness of the homeomorphysm types cannot be proved in dimension n = 3 as long as the Poincare conjecture is not proved. If there were a fake three-sphere then one could prove[53, 54] that a statement such as (ii) above is false for n = 3. Finally, the statement on finiteness of simple homotopy types , (which actually holds in any dimension), is particularly important for the applications in quantum gravity we discuss in the sequel. Roughly speaking the notion of simple homotopy is a refinement of the notion of homotopy equivalence, relating (in our case) triangulated manifolds (more in general CW complexes) which can be obtained one from the other by a sequence of expansion or collapses of simplices, (for details see [39], [103]). Roughly speaking, it may be thought of as an intermediate step between homotopy equivalence and homeomorphism. The fact that for a dynamical triangulations with a given number, N, of n-dimensional simplices an, and a given average bone incidence b(n, n - 2) we have a good a priori control of topology is the basic reason that allows us to enumerate distinct dynamical triangulations. An important step in this enumeration is to understand how we can reconstruct the triangulation by the knowledge of easy accessible geometrical data. At this stage it is worth recalling that the Levi-Civita connection for a dynamical triangulation is uniquely determined by the incidence matrix I (ai ,aj) of the triangulation itself, thus in order to enumerate distinct dynamical triangulations we can equivalently characterize the set of distinct Levi- Civita connections Q g = {T(ai, ai+l)}i=l,.... The problem we have to face is thus reduced to a rather standard question, familiar in gauge theories, where one needs to reconstruct the holonomy of a gauge-connection from data coming from the Wilson loop functionals. Here, we need to reconstruct all discretized LeviCivita connection from Wilson-loop data which reduce to the possible set of curvature assignments to the bones, and by taking due care of the non-trivial topological information coming from the moduli of locally homogeneous manifolds, (e.g., constant curvature metrics), which need to be described in terms of deformations of dynamical triangulations. In order to address this problem we exploit the holonomy representation associated with the Levi-Civita connection Qg.

4. Moduli Spaces for Dynamically Triangulated Manifolds

Let (Ta, M == ITal) be a dynamically triangulated manifold. If we denote by [}u(M) the family of all simplicial loops in M, starting at a given simplex 0'0, it easily follows that flu(M) , modulo ao-based homotopic equivalence, is isomorphic to the fundamental group of M, 1fl (a~ , M), based at the given simplex 0'0' Moreover, by factoring out the effect of loops homotopic to the trivial ao-based loop, the mapping

R: W

E

[}u(M)

~

(4.1)

Rw

yields a representation of the fundamental group 1fl (0'0' M). The following definition specilizes to the PL setting some of the well known properties of the holonomy of riemannian manifolds .

Definition 4.0.1. The holonomy group of the (fixed edge length) PL manifold (M,Ta ) at the simplex 0'0, Hol(a is the subgroup of the orthogonal group O(n) generated by the set of all parallel transporters R w along loops of the simplicial manifold T based at 0'0' The subgroup obtained by the loops which are homotopic to the identity, is the restricted holonomy group of (M, T) at aD, HolO(ao).

o)

If we change the base simplex aD to the simplex aD and fix some path p from aD to aD, then Hol(ao) == ApHol(a~)A;l, (and similarly for the restricted holonomy group). As a consequence the holonomy groups at the various simplices of T a are all isomorphic. And we can speak of the holonomy and restricted holonomy group of the (fixed edge length) PL manifold M == ITal: H ol (M), and H olo (M), respectively. Moreover, if Wo and WI are simplicial loops based at a~ which represent the same element, [w], of 1f1(M, 0''0), then Rwo and RW1 belong to the same arc component of Hol(M). From this observation and from the definition of restricted holonomy, it naturally follows that there exists a canonical homomorphism

1rl(M)

---+

Hol(M) HolO(M) '

(4.2)

which defines the Holonomy Representation of 1f1 (M). Holonomy representations of this sort are of relevance to dynamical triangulations. This connection is very subtle and it is related to the well-known fact that generally we cannot triangulate locally homogeneous spaces with

70

4. Moduli Spaces for Dynamically Triangulated Manifolds

equilateral flat simplices. For instance, in order to model a space of constant curvature with a dynamical triangulation we have to slightly deform the triangulation (following a suggestion anticipated many years ago by Romer and Zahringer [101]). This deformation procedure is apparently trivial in the sense that a small alteration of the chosen edge-length allows one to fit the triangulation on a constant curvature background. However subtle problems arise if there are non- trivial deformations (i.e., moduli) of the underlying constant curvature metric, (e.g, distinct flat tori). Since we work with a fine approximation to riemannian manifolds, (i. e., the edge- length a is kept fixed), we are not actually Gromov-Hausdorff approximating a riemannian manifold as close as we wish. If the riemannian manifold we are approximating possess a non-trivial moduli space (e.g., constant curvature surfaces), we need to take care of this further piece of information when stating that a given dynamical triangulation, or its deformations (in the sense of Romer-Zahringer), is an approximation of a given riemannian manifold. Stated differently, we have to say how we are deforming the triangulations since there may be a finite dimensional set of distinct locally homogeneous riemannian structures nearby the given one, and they may be not resolved at the given cut-off a. Had we addressed the counting of just distinct (in the sense of Tutte) dynamical triangulations on a PL manifold of given volume and topological type, the above problem would have not explicitly appeared. The fact is that in the definition of equivalence according to Tutte, no mention is made of the metric properties of the triangulation considered, and thus in the process of counting, we enumerate also triangulations potentially fitting on locally homogeneous spaces. The requirement of triangulating with equilateral simplices, is a restriction of no consequence as long as we consider generic riemannian manifold, but quite effective when coming to triangulating riemannian manifolds of large symmetry. We enumerate distinct triangulations basically by considering distinct curvature assignments and distinct triangulations with given curvature assignements, viz., namely using as labels for distinct triangulations the distinct Levi-Civita connections generated by distinc curvature assignments. This strategy can be considered as the discretized counterpart of the techniques underlying uniformization theory for 2-dimensional surfaces or the geometrization program for 3-dimensional manifolds. The simplest example is provided by 2-dimensional surfaces. In such a case, a given riemannian metric can be conformally deformed to a metric of constant curvature. The conformal factor is related (via the conformal Laplacian) to the gaussian curvature of the surface, and in this sense the datum of the local curvature seems sufficient to reconstruct the original surface. This fails since there are nonequivalent infinitesimal deformations of the base constant curvature metrics (moduli), and in order to reconstruct the metric, we need curvature assignments and the datum of which constant curvature metric, (e.g., which flat metric, if the surface in question is a 2-torus), is used. Actually, there is a further subtle problem related to the fact that the action of the semi-direct

4.1 Romer-Zahringer Deformations of Dynamically Triangulated Manifolds

71

product of the group of diffeomorphisms and of the conformal group makes difficult an actual reconstruction of the metric from the datum of the local curvature and of the moduli. But at this heuristic level, the parallel between the geometric setup of uniformization theory and the reconstruction of dynamical triangulations from the curvature assignments is appropriate. Roughly speaking since curvature corresponds to curvature assignments, we should expect that we have a rather large space of distinct dynamical triangulations with the same curvature assignments, and as we approach a locally homogeneous manifold, this space should reduce to a moduli space of topological origin. In a rather direct sense the situation is akin to the standard parametrization of hyperbolic surfaces through the distinct way of tesselating the hyperbolic plane with hyperbolic triangles (or polygons). In dimensions higher than 2, one may feel that such worries about moduli are not so relevant since in many cases, (in particular for hyperbolic manifolds), rigidity phenomena may occur. The most well-known result in this direction is the Mostow rigidity theorem, which, roughly speaking says that if two complete hyperbolic n-manifolds, (n ~ 3), of finite volume are homotopy equivalent, then they are isometric. Although this theorem implies that the moduli space of hyperbolic structure on a finite volume hyperbolic n- manifold, (n ~ 3) is a point, we may wish to consider the conformal analog of Teichmiiller space for the given manifold. The space of such conformal structures C(M) does not reduce to a point, and we may have non-trivial deformations [80]. In general we may have non-trivial deformations for locally homogeneous geometries, and thus the correct understanding of how dynamical triangulations approximate such spaces is a non-vacuus issue. Formally, non-trivial deformations of locally homogeneous geometries are described by the cohomology of the manifold with coefficients in the sheaf of Killing vector fields of the manifold, namely with local coefficients in the (adjoint representation of the) Lie algebra g of the group G of isometries considered. This cohomology is isomorphic to. the cohomology of the fundamental group 1fl (M) with value in the holonomy representation of 1fl (M) in g, and this is one of the basic mechanisms that calls into play topology in dynamical triangulations. Such issues can be very effectively dealt within the framework of deformation theory in algebraic geometry, (as hinted in the above formal sheaf-theoretic remarks), however, we think appropriate here a more pedagogical approa~h, also because this issue has not really been considered previously in dynamical triangulation theory.

4.1 Romer-Zahringer Deformations

of Dynamically Triangulated Manifolds As a warm up after the above introductory remarks, let us consider in particular the flat Levi-Civita connections. Since the connection is flat, the parallel

72

4. Moduli Spaces for Dynamically Triangulated Manifolds

Fig. 4.1. In dimension n 2:: 3 is not possible to tesselate jRn with flat equilateral simplices. This simply follows from the fact that cos- 1 ~, n 2:: 3, is not an integer fraction of 27r

transport along a closed simplicial loop w based at 0"0 depends only on the homotopy class of w. In this case the parallel transport, through its associated holonomy, gives rise to a representation of the fundamental group O:1rl(M,a~)

-7

Hol(M)

C

O(n),

(4.3)

the image of which is the holonomy group of the flat Levi-Civita connection considered. Such 0 is a homomorphism of 1fl(M, ao) onto Hol(M), and since 1fl (M) is countable, Hol(M) is totally disconnected. In this case one speaks of 0 as the holonomy homomorphism. It is a well-known fact that connected compact flat n-dimensional riemannian manifolds are covered by a flat n-torus p: Tn -7 M, and that the associated group of deck transformations is isomorphic to Hol(M), viz., M = ynjHol(M). Quite surprisingly, any finite group G can be realized as the holonomy group of some flat compact connected riemannian manifold, (this is the Auslander-Kuranishi theorem, see e.g.[117]). Accordingly, flat riemannian manifolds are completely determined by homomorphisms of their fundamental group 1fl (M) onto the (finite) group G, up to conjugation. The kind of difficulties encountered in dynamically triangulating flat riemannian manifolds are particularly clear when discussing the geometry of their covering space. Flat tori. It is well known that at least in dimension n ~ 3, it is not possible to build up a flat space (e.g., a flat three-torus) by glueing flat equilateral simplices {an}, this simply follows from the observation that the angle cos- l

*-'

4.1 Romer-Zahringer Deformations of Dynamically Triangulated Manifolds

73

n ~ 3, is not an integer fraction of 21f. Thus, for n ~ 3, dynamical triangulations cannot directly describe flat tori. Superficially, this does not seem to be a serious problem, since we can defor~ a few simplices in such a way to match with the 21f- flatness constraint, (see next paragraph for the details). However, this is hardly a sound prescription since, as follows from their definition, there are distinct flat tori. More formally, let us fix a basis (el' ... , en) of jRn. Then with each n- pIe of integers (a 1 , ... , an) E Zn we can associate a transformation, A, of jRn given by A(x) == x + E;=l ajej. This results in a transformation group isomorphic to Zn acting properly discontinuously on jRn. Upon quotienting jRn by this action we get a manifold diffeomorphic to the n-dimensional torus Tn. Since jRn is thought of as endowed with its standard flat metric, the resulting Tn is likewise flat. However, by changing base to jRn we change the group of transformation associated with A, and the resulting flat torus is not necessarily isometric to the original one. Thus, the naive deformation procedure suggested above is ambiguous as long as it does not specify which particular flat torus one is modelling. In particular, in dimension n == 2, let al and a2 be the vectors generating the lattice, and assume that al is the first basis vector of jR2, al == el, and that a2 == (x, y) lies in the first quadrant of jR2. Then the resulting flat tori are parametrized by the points in the plane region

(4.4) to the effect that flat tori generated by lattices corresponding to two distinct points of this region are distinct. Thus, even in dimension n == 2, where the flatness 21f-constraint can be met, it follows that the lattice generated by equilateral triangles, corresponding to a dynamically triangulated torus, can only discretize (up to dilation) the flat torus coming from exagonallattices, (x == ~, y == 1'): we cannot dynamically triangulate with equilateral triangles all remaining flat tori, (e. g., all the rectangular tori a2 == (0, y) ), (obviously, this obstruction persists only if we want the triangulation and the lattice to be consistent, otherwise we can dynamically triangulate any 2- torus; in higher dimension we completely loose this freedom). The difficulties in associating dynamical triangulations with manifolds of large symmetries is not restricted to the case of flat tori just described. Similar problems are encountered in associating dynamically triangulations to any riemannian manifold endowed with a geometric structure , namely manifolds locally modelled on homogeneous spaces, for instance manifolds of constant curvature, basically because such manifolds do not generally admit regular tilings by euclidean equilateral simplices. Deforming a triangulation. Even if there is no regular way to generate locally homogeneous PL manifolds by glueing equilateral simplices, (e.g., flat tori), we can always assume that in trying to model a locally homogeneous riemannian manifold M with a dynamical triangulation T a , this can be done in such a way that the Gromov- Hausdorff distance between M and Ta is min-

74

4. Moduli Spaces for Dynamically Triangulated Manifolds

Fig. 4.2. A flat torus y2 generated by quotienting ]R2 by a lattice

Fig. 4.3. Dynamical triangulation for a generic flat 2-torus. The defyning lattice of the torus is distinct from the lattice structure associated with the triangulation

4.2 Dynamical Triangulations and Locally Homogeneous Geometries

75

imal. As a matter of fact, we can, at least in line of principle, construct an approximating sequence {TJL }JL=1,2,... of triangulated models of M, interpolating between M and Ta , by using triangulations of M with simplices which are nearly equilateral, (the essence of this remark was very clearly suggested by Romer and Zahringer[101]). With each triangulated manifold TJL of this sequence we can associate the edge-length fluctuation functional

(4.5) where a{TJ.t) == La 1 ~:j(<71)1 is the average edge-length in Tw The functional s(TJL) has reasonable continuity properties in the Gromov-Hausdorff topology, and since the Gromov space of bounded geometries is compact[65, 59, 52], we can always choose the sequence {TJL }JL=1,2,... in such a way that it minimizes s(TJL). However, if there are non-trivial (infinitesimal) deformations of the riemannian structure M, preserving the local homogeneous geometry, (e.g., inequivalent flat tori, or moduli of Riemann surfaces), the Romer-Zahringer procedure may associate distinct moduli to the same approximating dynamical triangulation. Thus, the question remains of how to keep track of which locally homogeneous manifold is Gromov-Hausdorff approximated by the dynamical triangulation considered. This can be done by exploiting the GromovHausdorff continuity of the homotopy type of metric spaces of bounded geometry and by generalizing the previous remark relating the holonomy of flat connections to the representations of the fundamental group, (see eqn. (4.3)).

4.2 Dynamical Triangulations

and Locally Homogeneous Geometries In order to formalize the above remarks, we need some preparatory material on riemannian metrics on homogeneous spaces and the way a dynamical triangulation can approximate such particular riemannian structures. From the mathematical point of view, most of this material is standard and as a guide, the reader may found profitable to consult the papers by W.Thurston [108] and by W.M.Goldman [62] where the classical geometry of deformations of locally homogeneous riemannian manifolds is beautifully discussed. As far as applications·to dynamical triangulation are concerned, the point of view developed here and in the following paragraphs is quite new and hopefully helps to unravel the subtle connection between topology and simplicial quantum gravity.

76

4. Moduli Spaces for Dynamically Triangulated Manifolds

4.2.1 Locally Homogeneous Geometries

Let G be a Lie group, H eGa closed (compact) subgroup, we denote by X == G/ H ~ {gH: 9 E G} the set of cosets. To each 9 E G we can associate the left translation Tg:G/H ~ G/H given by Tg(g'H) == (gg')H which makes G act as a transitive Lie transformation group on the homogeneous space X. Such a space X is riemannian homogeneous if G / H is endowed with a riemannian metric relative to which T g is an isometry for every 9 E G. A locally homogeneous riemannian manifold M is a riemannian manifold locally modelled on the geometry of a homogeneous space (X, G). Constant curvature metrics are a typical example. They are locally isometric either to euclidean space, the sphere, or hyperbolic space, depending on wether the curvature is zero, positive, or negative respectively. More generally, locally homogeneous riemannian manifolds include all geometric structures used to uniformize surfaces or to discuss Thurston's geometrization program for 3dimensional manifolds[104]. To be more precise, let X be a real analytic

M

Fig. 4.4. Charts for a locally homogeneous manifold

manifold and let G be a finite dimensional Lie group acting analytically and faithfully (i. e. gx == x for all x E X implies 9 == id) on X. We say that a manifold M has a (G, X) structure if we have an open covering {Ui } of M, and coordinate charts cPi: Ui ~ X with transition functions

(4.6)

4.2 Dynamical Triangulations and Locally Homogeneous Geometries

77

where each lij agrees locally with an element of the group G. Let M be a (X, G) manifold and M its universal covering. If we fix a base point Xo in M, M can be identified with the quotient of the set of pairs (x, (3) , where x E M and (3 is a path from Xo to x, with respect to the equivalence relation (x, (3)

~

(x*, (3*)

(4.7)

if and only if x == x* and (3*(3-1 ~ 1 in 1rl(M,xo). The projection p:M ~ M being defined by p[ (x, (3)] == x. We wish to recall the characterization of the Developing map D: M ~ X associated with a locally homogeneous manifold M. The basic idea is that given a (X, G) structure on a manifold M it is natural to try to make the charts {(Ui ,
'l/Jk ~ 'Yo 1 (Xl)'Y12(X2) ... 'Yk-l,k(Xk)
(4.8)

The adjusted charts 'l/Jl, 'l/J2,' .. ,'l/Jn form the analytic continuation

a
(4.9)

of the chosen
Definition 4.2.1. For a given choice of the initial chart (Uo, cPo) and of the basepoint Xo E Uo, the developing map of a locally homogeneous manifold M is the map D: M ~ X defined by D ~
78

4. Moduli Spaces for Dynamically Triangulated Manifolds

~_~.... o

.--_

~

..

M

x Fig. 4.5. The analytic continuation of a given locally homogeneous chart along a path Q

4.2 Dynamical Triangulations and Locally Homogeneous Geometries

79

case of such a construction). Let a be an element of the fundamental group of M, 11"1 (M, xo). To such a loop there is associated a covering transformation Ta:M ~ M in the universal cover defined by Ta[(x, T)] ~ [(x, aT)]. Analytic continuation along the loop a provides <prJ == 'l/Jn which is well-defined at the basepoint xo, (since Xn == xo), and depends only on the homotopy class'of the loop a. We call Holonomy of M the map H: 1r1(M) ~ G defined by a ~ h(a) where h a is that element of G such that
For our purposes we need to consider not just a particular locally homogeneous structure (M, g*) on a compact manifold M, but rather a space of such structures. To this end[62] , mark a basepoint Xo E M and let 1r1(M,xo) and M ~ M be the corresponding fundamental group and universal covering space. We consider the set of homomorphisms of 1r1 (M, xo) into the structure group of the geometry G, H om( 11"1, G) endowed with the pointwise convergence topology, i. e., a sequence On E H om( 1r1 (M), G) is said to converge to a homomorphism 0 if limn -+ oo On(w) == O(w) for each w E 1r1 (M, xo). Let us consider, as before, the coordinate map
80

4. Moduli Spaces for Dynamically Thiangulated Manifolds

Fig. 4.6. The development map of an affine torus, i.e., a torus generated by identifying, through the action of the affine group, the opposite sides of a quadrilateral. Such a torus can be correspondingly triangulated. In general similar identifications through the action of a group G can generate lattices (or generalized dynamical triangulations) implementing the symmetry group G. Examples realizing such a construction can be easily seen in nature (the pattern of the florets of a Sunflower) or in Art (Michelangelo:the Campidoglio square)

4.2 Dynamical Triangulations and Locally Homogeneous Geometries

I(x,G)(M) == {(D, 0)10 E Hom(1f1 (M), G)},

81

(4.11)

and where D:!VI ~ X is a O-equivariant nonsingular smooth map, viz., a developing map. If we topologize I(x,G)(M) using the Coo topology on the developing map D, then the map

h~l:I(x,G)(M) ~ Hom(1f1 (M), G),

(4.12)

which associates to the developing pair (D,O) the corresponding holonomy representation 0 is continuous. Notice that if Dif fo(M, xo) is the identity component in the group of all diffeomorphisms M ~ M fixing xo, then[62] Dif fo(M, xo) acts properly and freely on I(x,G)(M) and the map h~l is invariant under this action. It is also easily verified that h~l is G- equivariant also respect to the natural G-actions of G on the developing map and on the representation space Hom(1f1(M), G). In the quoted Goldman's paper[62], the intersted reader can find the statement and the proof of an equivariant slice theorem which allows to parametrize the space I(x,G)(M) of based geometric structures on M in terms of a representation 0 E H om( 1f1 (M), M) and of a diffeomorphism h E Diffo(M,xo)), (see the deformation theorem at p. 178 in[62]). We state this result in a somehow less technical fashion

Theorem 4.2.1. The map h~l:I(x,G)(M) ~ Hom(1f1 (M), G) is an open map, and for any given locally homogeneous structure u E I(x,G) (M) there is a corresponding neighborhood W C I(x,G)(M) such that for any two (based) locally inhomogeneous structures U1, U2 E W such that h~l( U1) == h~l( U2) == o E Hom(1f1(M),G) there is a (based) diffeomorphism h E Diffo(M,xo) and, E G with (h, ,) . U1 == U2. In other words, in a sufficiently small neighborhood of a given locally homogeneous structure, any two locally homogeneous structures with the same holonomy representations are equivalent, and we can use the holonomy homomorphism as a sort of coordinate map for the space of inequivalent locally homogeneous structures that we can obtain from the given one by local deformation. It also follows from the above theorem that the map

" . I(x,G)(M) hol:I(x,G)(M) = Diffo(M,xo)

--t

Hom(nt(M), G)

(4.13)

is a local homeomorphism. If we define[62] the Deformation Space of locally homogeneous structures on M as the quotient

D f e

(X,G)

(M) - i(x,G)(M) -

G

'

(4.14)

then one would expect that the map hol extends to a local homeomorphism from Def(x,G)(M) and the orbit space Hom(1f1 (M), G)jG (endowed with the quotient topology). In general this is the case only if we have more information

82

4. Moduli Spaces for Dynamically Triangulated Manifolds

on the structure of the orbit space under the G-action. Typically hol is wellbehaved if we restrict our attention to the subspace in H om(1rl (M), G) / G generated by the conjugacy classes of stable representations. In any case we shall work mainly with hol: i(x,G)(M) ~ Hom(1rl(M), G), and simply assume, at least at this stage, that the representations considered are such that the local properties of the representation variety Hom( 1rl (M), G) / G describe, through the holonomy map hoi, the local properties of the deformation space De!(x,G)(M) of locally homogeneous structures on M. This is an important point for it will imply that the representations in Hom( 1rb(M),G) provide the missing piece of information needed for describing the inequivalent deformations of dynamical triangulations approximating a given locally homogeneous riemannian manifold. By exploiting the preparatory material so introduced, we now describe in details the procedure for parametrizing the deformation space of dynamical triangulations.

4.3 Moduli of Dynamical Triangulations Consider on a dynamical triangulation (M, T a ) the set of metric balls of radius a, {Ui(M)}, centered at the baricenters, Pi, of the Nn - 2 (M) bones {a n - 2 } of M. Each such Ui(M) is homeomorphic to B n - 2 x C(link(a n - 2 )), where C(link(a n - 2 )) is the cone on the link of a n - 2 , and Bk denotes the topological k-dimensional ball. Since C(link(a n - 2 )) ~ B 2 , we get that Ui(M) ~ B2 X Bn-2 ~ Bn, viz. each metric ball Ui(M) is indeed a topological ball. Since their radius is a, the collection of such Ui(M), i = 1, ... , N n - 2 (M), generates an open covering of M. Assume that the dynamically triangulated manifold (M, Ta ), endowed with the covering {Ui(M)}, is Gromov-Hausdorff near to a locally homogeneous (X, G) riemannian manifold (of bounded geometry) (5, 9). This assumption can be translated into the requirement that the riemannian manifold (5,9) admits a geodesic ball covering {qi}i=l, ... ,m E 5 generated by metric balls Ui (5), of radius E = a, such that (4.15)

Since, by a standard argument already exploited various times, this bound implies that the Gromov-Hausdorff distance between M and 5 is such that dG(M,5) :::; 2a. By itself, the dynamically triangulated manifold M cannot be endowed, for any fixed value of the cut-off a, with a locally homogeneous metric geometry, (with the exception of some particular cases in dimension n = 2). As already stressed this may be a source of serious ambiguities. In particular, let us assume that the given locally homogeneous riemannian manifold (5, 9) admits non-trivial infinitesimal deformations (moduli) De!(x,G) (5), then, up to the cut-off length scale a, the dynamically triangulated manifold (M, T a ) is roughly-isometric to any of the inequivalent locally

4.3 Moduli of Dynamical Thiangulations

83

Fig. 4.7. The holonomy map associated with an xo-based loop (J E 7rl(M, xo). Analytic continuation of the local chart (Uo,
84

4. Moduli Spaces for Dynamically Triangulated Manifolds

homogeneous manifolds associated with the moduli of (5, g), i. e., according to (4.15),

dG(M, 5*)

~

2a,

(4.16)

for any 5* E Def(X,G) (M). Thus, (M, Ta ) can be roughly-isometric to any locally homogeneous structure in Def(X,G) (M). In order to avoid this ambiguity, we need to declare which locally homogeneous geomet~y (M, T a ) is actually approximating. To enforce such control, let us consider (M, Ta ) endowed with the covering {Ui(M)}. Choose a basepoint Po among the baricenters {Pi} of the bones of M, and let Uo(M) denote the corresponding metric ball in the cover {Ui(M)}. Finally, mark an n-dimensional simplex aD among those incident on the bone O"~-2 associated with the given choice of Po. Notice that with these markings we can associate to Uo(M) a well- defined element, Ro(O"~-2), of the orthogonal group O(n): Ro(O"~-2) is the rotation around the bone a~-2 of the vectors defining the marked simplex 0"0. Together with these elements, let us also choose a representation (}:1fl(M,po) ~ G of 1fl(M,po) into the structure group G of the locally homogeneous geometry approximated by

(M,Ta ).

Since (M, Ta ) is Gromov-Hausdorff near to a locally homogeneous manifold (5, g), there is by hypothesis an open set Uo(5) on 5, corresponding to Uo(M), centered around qo E 5, and a corresponding map cPo: Uo(5) ~ (X, G) associated with the element, Ro(a~-2), of the orthogonal group describing the rotation around the bone a~-2. If we think of Ro(a~-2) as an element of G associated with the chosen marked simplex 0'0, we can generate ¢o by analytical continuation of ¢(qo) ~ Ro(a~-2) in Uo(5) along the unique simplicial loop winding around a~-2. Thus, associated with the markings of (Uo(M), 0"0) on M there is a local chart (Uo(E), cPo) for the locally homogeneous geometry of E. Since Gromov-Hausdorff nearby metric spaces share a common homotopy type, it also follows that 1fl (M, Po) c::: 1fl (5, qo), (provided that a is small enough so that M and 5 are sufficiently near in the Gromov-Hausdorff distance), and the representation of 1fl(M,po) induces a corresponding representation that we still denote by (), (): 1fl (5, qo) ~ G. The relevance of such interplay between the markings of (Uo(M), 0"0) on M and the corresponding local chart (Uo(E), cPo) on 5, is twofold and follows from the properties of locally homegeneous geometries recalled in the previous paragraph. First, the local chart (Uo(5), cPo) can be used as a basepoint for the development pair (D, (): 1fl (5, qo) ~ G) defining the locally homogeneous structure (5, g) considered. Moreover, according to the Goldman-Thurston deformation theorem 4.2.1, if the geometry (5, g) admits non-trivial deformations Def(x,G) (5), such deformations are locally parametrized by the representation variety Hom(1fl(5),G)/G. Thus, the data needed in order to specify unambiguosly which locally homogeneous geometry we are approximating with (M, T a ) is basically contained in the association of a holonomy homomorphism (): 1rl(M,po) ~ G to the dynamical triangulation (M, T a ).

4.3 Moduli of Dynamical Thiangulations

85

Summing up, we have the following

Theorem 4.3.1. Let (D,O:1rl(5,qo) ~ G) be the developing pair of a locally homogeneous geometry (5,9) with a non-trivial deformation space Def(x,C) (5), and let (M,Ta) E DTn(a,b,N) denote a dynamical triangulation approximating (5, 9). Then the distinct locally homogeneous riemannian manifolds in Def(x,C) (E) are Gromov-Hausdorff approximated by the pairs [(M, Ta)root, 0] where: (i) (M, Ta)root is the rooting of the dynamical triangulation (M, Ta) generated by the marking of a bone a~-2 of (M, Ta), of a simplicial loop Wo winding around a~-2, and of a base simplex 0'0 E wo, while (ii) 0 is a holonomy homomorphism O:1rl(M,po)

~

G,

(4.17)

varying in a suitable neighborhood of 0 in the representation variety (4.18)

Proof. The markings in (i) are simply the markings characterizing the data (Uo(M), 0'0) on M inducing a local chart (Uo(5), cPo) for the locally homogeneous geometry of E, through which 5 can be unrolled by means of the developing map D. These data are necessary for providing a well-defined holonomy representation. Henceforth we shall refer to dynamical triangulations with such marked simplices as rooted triangulations for short. As we have recalled[62], the map

h~l: De!(x,c) (5) ~ Hom(1rl(E), G),

(4.19)

which associates to the developing pair (D, 0), (i. e., to the locally homogeneous manifold (E, 9) ), the holonomy representation 0 is a continuous and open map. Two locally homogeneous manifolds 51 and 52 in Def(x,c)(E) are near in the Coo topology (on the respective developing maps D 1 and D 2 ) and the corresponding geodesic ball coverings {Ui (E 1 )} and {Ui (E 2 )} are such that dc (5 1 , 52) ~ € where € depends on the Coo-norm on the respective developing pairs II (D, ( 1 ) - (D, ( 2 ) 1100. It follows that (M, Ta ) is GromovHausdorff near any locally homogeneous manifolds in Def(x,C) (5). By choosing, if necessary, a sufficiently small neighborhood of (5, 9) in De!(x,c) (5), we can assume that (M, Ta) shares a common homotopy type with all (5*,9) in De!(x,c)(5). Thus an immediate application of the Goldman-Thurston deformation theorem 4.2.1 implies that we can establish a one-to-one correspondence between each pair [(M, Ta)root' 0] and a corresponding locally homogeneous manifold in Def(x,C) (5), simply by declaring that a given [(M, Ta)root' 0] is associated with the locally homogeneous manifold in Def(x,c)(5) defined by the holonomy homomorphism 0, and this correspondence is well-defined for 0 varying in a suitable neighborhood of 0 in Hom(1f~(M),C.

86

4. Moduli Spaces for Dynamically Triangulated Manifolds

Thus, with a slight abuse of language, it seems appropriate to consider the representation variety Hom( 1rJ(M),G as the Moduli Space for a dynamically triangulated manifold M E DTn(a, b, N). Since we are interested in closed manifolds M, the study of this moduli space is basically equivalent to the study of discrete cocompact subgroups r of the maximal isometry group G that can be ammitted by M, (recall that a discrete subgroup reG is called cocompact if the quotient space G/ r is compact). The underlying geometric rationale is that the Lie group in question is the isometry group of the simply connected homogeneous riemannian manifold X which can be approximated by the (universal cover of the) dynamical triangulation M. The discrete subgroup r appears (if it has no torsion) as the deck transformation group of the covering X ---+ E where E = X / r is the smooth riemannian manifold approximated by M, thus r ~ 1rl (M), (if r has torsion we will be dealing with orbifolds). It is intersting to remark that even if not every semisimple Lie group G may be realized as the isometry group of some symmetric space X, every such G is locally isomorphic, (viz. the isomorphism is at the level of the universal coverings of the groups in question), to'the isometry group of some symmetric space X. The existence of a non-trivial moduli space for dynamical triangulations (of closed manifolds) is thus reduced to the study of the rigidity of lattices in such groups G[66]. For the convenience of the reader, we have collected in an Appendix some basic material on the geometry of the representation variety Hom( 1rd(M),a. Here we stress that since 1rl (M) is a finitely generated group, (say with m generators), Hom(1rl(M), G) is an analytic subvariety [62, 63, 76, 93, 110] of Gm , (as a variety is defined by the m-tuples (8(1), ... ,8(m)) E G m such that pi(8(1), ... ,8(m)) = 1 for each relation Pi associated with the given presentation of the group 1rl (M)). Note that on applying a theorem due to X. Rong[102], one can actually provide a upper bound to the minimal number of generators for 1rl(M) for M E VTn(a, b, N), (the bound depends on the parameters a,b, and N).

4.4 Gauge-Fixing of the Moduli of a Dynamical Triangulation Far most of our efforts will be bent toward an enumeration of the distinct dynamical triangulations a PL manifold can carry. Since distinct dynamical triangulations (of bounded geometry) are meant to approximate distinct riemannian structures, we are actually addressing the characterization of a measure on the space of riemannian structures. As we have seen in the previous sections, the correspondence between a dynamical triangulation and a riemannian structure must be handled with care when the dynamical triangulation in question is meant to approximate, (for a given, sufficiently small,

4.4 Gauge-Fixing of the Moduli of a Dynamical Triangulation

87

(=,9)

\ }

. ~y

"'fa .

)

(X,GJ Fig. 4.8. Rooting of a dynamical triangulation M homogeneous riemannian manifold (:5, g)

= Ta

approximating a locally

88

4. Moduli Spaces for Dynamically Triangulated Manifolds

cut-off a), a locally homogeneous manifold. The point is that distinct moduli of locally homogeneous riemannian manifolds must correspond, in some regular way, to distinct Romer-Zahringer deformation, of the given dynamical triangulation. There are plenty of such deformations. In order to carryover an enumeration of distinct dynamical triangulations while preserving a oneto-one correspondence between moduli of locally homogeneous riemannian structures and Romer-Zahringer deformations of dynamical triangulations approximating such structures, we have have to impose a suitable gauge fixing on such deformations. The situation is akin to the one familiar in gauge theories where, in order to introduce a suitable path integration on the configuration space of the theory, we have to remove gauge freedom, (by using a slice theorem for the action of the gauge group), to obtain a local (formal) measure on the orbit space labelling the gauge equivalent configurations. Eventually, as in any gauge fixing procedure, in order to obtain the correct count of the nunlber of distinct dynamical triangulations, we have to divide the number of distinct holonomies by the local volume of the moduli space parametrizing the conjugacy-classes of representation 8:1rl(M;a~) ~ G in a neighborhood of the chosen representation. From a combinatorial point of view, the necessity of this factorization comes from the fact that the number of distinct holonomies provides the number of distinct dynamical triangulation in a given holonomy representation of the fundamental group, namely it is a labelled or rooted enumeration. The unrooted enumeration can be obtained from the rooted one by dividing by all possible labellings of the inequivalent representations.

4.5 A Measure on the Moduli Space According to the above remarks, in order to complete our gauge-fixing procedure, we have to introduce a measure on the moduli space M(Ta ) of a given dynamical triangulation. This procedure is related to the characterization of measures on the representation variety Hom(1rh(M) ,G) . In dimension two, the study of such measures has been considerably developed by exploiting its connection with the semiclassical limit of Yang-Mills theory on Riemann surfaces[115, 116, 79]. For dynamical triangulations in dimension greater than two, the connection with measure theory on Hom( 1rb(M),G) has been stressed in[16, 35] as the origin of the possible critical behavior in higher dimensional simplicial quantum gravity. Let us consider the representation, fJ, of 1rl (M) on the Lie algebra g generated by composing the given representation 8: 1rl (M) ~ G with the adjoint action of G on g, viz., ():1rl(M) ~e G ~Ad End(g), (henceforth we will always refer to this representation ). The tangent space to Hom( 1rb(M),G) corresponding to the conjugacy class of representations [fJ] is provided by H 1 ( 1rl(M), g), the first cohomology group of 1rl(M) with values in the Lie

4.5 A Measure on the Moduli Space

89

algebra g. It is a known fact that this cohomology group is isomorphic to the first cohomology group of the dynamically triangulated manifold M == ITa I, with coefficient in the adjoint bundle ad(B) defined by the representation B, i.e., Hl(M, ad(B)), (for more details see the Appendix). Notice that Hl(M, ad(B)) is only a formal tangent space to Hom (1rb(M) ,G) , since the representation variety has singularities in correspondence to reducible representations and/or linearization unstable representations. The choice of a well-defined holonomy representation amounts to the choice of a map (basically a section) 8: Hom( 1rh(M),G) ~ Hom(1fl(M), G) which associates to a conjugacy class, [B], of representations, the holonomy representation B: 1fl (M, Po) ~ g, based at the chosen basepoint. Let 8G denote the generic (infinitesimal) deformations, at B, of the chosen slice S. Note that 8G E Zl(1fl(M);g), the space of l-cocycle on 1fl(M) with coefficients in the Lie algebra g. This is the (formal) tangent space to H om(1fl (M), G) at the chosen representation B. A component of the deformation 8G may be trivial in the sense that it may lie in the tangent space to the Ad-orbit containing B. This part of the deformation maps, by Ad-conjugation, S(B) to a deformed representation 0 in a nearby slice 8(0) with [0] == [B]. The non-trivial part of the deformation lies in the tangent space to the chosen slice 1r 8, viz., T8[Hom( b(M),G)] ~ Hl(1fl(M), g). This latter part of the deformation maps B in a neighboring representation which cannot be obtained by Ad-conjugation from B. As recalled, deformations in T8[Hom( 1rh(M),G)] can be interpreted as inequivalent Romer-Zahringer deformations of the given dynamical triangulation. A natural[115, 116, 79],[58] volume element on the space of such deformations is the product of a volume form on the Lie algebra 9 and of a volume on TS[Hom( 1rh(M),G)]. In order to define the former, consider an Haar measure on the group G, with total volume Vol(G), and let Yl, Y2, . .. ,Yt be Euclidean coordinates on the Lie algebra g. We choose the Yh in such a way that the measure TI~ dYh on 9 is the chosen Haar measure at the identity element of G. Analogously, we can introduce a measure, dJ.-t, on the tangent space to the chosen representative of the (smooth component of the) variety 8[Hom( 1rb(M),G)] C Hom(1fl (M), G). Recall that Hom( 1rb(M),G) is an analytic subvariety of Gm, m being the number of generators of 1fl(M). The fixed Ad-invariant inner product on 9 induces a Riemannian structure on Gm, and hence on TS[Hom( 1rh(M),G)], (at least on its smooth component, to which we are restricting our considerations). The corresponding inner product between tangent vectors in Hl(M, ad(B)), will be denoted by « ',' », and dJ.-t can be thought of as the associated volume measure. We shall denote by (4.20)

the volume of the moduli space Hom( 1rh(M),G) in this measure. We cannot use dJ.-t as such since we want to localize, as the representation variety becomes

90

4. Moduli Spaces for Dynamically Triangulated Manifolds

smaller and smaller, the choice of the representations () around the trivial representation. This is dictated by the requirement that as V ---+ 0, (e.g., for simply connected manifolds), the measures should converge to the atom concentrated on the representation [w,Po] I----t Ida, where [w,Po] denotes the trivial loops based at the chosen basepoint Po.To this end, we normalize the product measure

TI~ dYh dJ.L x Vol(G)' on g

X

(4.21)

TS[Hom( 7rb(M),a)], with the heat kernel on the analytic manifold

Hom(1rl(M), G) c Gm ,

(4.22)

and set

d~ ~ K (Nnvol(an)V, 0, Id) dJ.L x V;l~~)'

(4.23)

where () is the given representation in Hom( 1rl (M), G), I d is the trivial representation, vol (an) is the volume of the elementary simplex an, and

K(t,x,y):lR+ x Hom(1rl(M), G) x Hom(1rl(M), G)

---+

lR

(4.24)

is the fundamental solution of the heat equation on Hom(1rl(M), G), (since Hom(1rl(M), G) is an analytic subvariety of G x G x ... x G, K(t, x, y) can be constructed with the heat kernel on G). Notice that Nnvol(an)V can be interpreted as the volume of the dynamically triangulated manifold M as expressed in terms of the volume of the representation variety H om(1rl (M), G) / G. Thus, the natural normalizing volume element we consider in the procedure of gauge fixing the Romer-Zahringer deformations of a dynamical triangulation is

(4.25) 4.5.1 Moduli Asymptotics

If we exploit the standard properties of the heat kernel, we can get an explicit asymptotic expression for Vol[M(Ta )] in the limit of small volume for the representation variety. Theorem 4.5.1. For a given value of N n and for Vvol(a n ) near zero, the leading contribution to Vol[M(Ta )] is given by

4.5 A Measure on the Moduli Space

A(V) . (21rNnVvol(a n ))-D/2(1

+ O(NnVvol(a n))),

91

(4.26)

where D = ~dim[Hom(1rl(M),G)], and A(V) is a function of V

(The explicit expression of A(V) is provided in the proof). Proof. We shall consider H om( 1rl (M), G) c Gm as a compact manifold with a Riemannian structure induced by the Cartan- Killing metric on g. According to standard estimates[44] on the heat kernel on a compact Riemannian manifold, there is a neighborhood U of the diagonal in H om(1rl (M), G) x Hom(1rl(M), G) such that for any pair of representations (f)1,f)2) E U and Vol[Hom(1rl(M),G)/G] near 0, we can write K (NnVvol(a n ), f)l, f)2) =

(271"NnVvol(an))-D/2exp ( -

2~lv~:Z(~n)) (1 + O(NnVvol(a n)))(4.27)

where If)l - f)21 denotes the riemannian distance, (in the induced CartanKilling norm), on Hom(1rl(M), G) C Gm. Set t~Nn Vvol(a n ), and let us consider[80],[110] the representations f), for t near 0, as one-parameter family of representations f)t f)t = exp[tu(a) + O(t 2)]f)(a), (4.28) where a E 1rl(M), and where u: 1rl(M) ~ g. Given a and b in 1rl(M), if we differentiate the homomorphism condition f)t (ab) = f)t (a)f)t (b), we get that the u are one-cocycles of 1rl (M) with coefficients in the 1rl (M)-module go, u(ab) = u(a)

+ [Ad(f)(a))]u(b),

(4.29)

namely, u E Zl(1rl(M),g), the tangent space to Hom(1rl (M), G). Among the vectors u E Zl(1rl(M), g), those tangent to the Ad-orbit are of the form u(a) = [Ad(f)(a))]h - h, for some h E g, these are 1- coboundaries on 1rl(M) with coefficients in the representation, i.e., elements of B1(1rl(M), g). We can decompose u in a component PHu E Hl(1rl(M), g) = Zl(1rl(M),g)/B1(1rl(M), g) and in a component Pgu E g, PH and Pg denoting the respective projection operators. Thus f)t - Id = t(PHU + Pgu) + O(t 2). (4.30) Introducing this in (4.27) we get K (NnVvol(a n ), f), I d) = (21rN nVvol(a n)) -DH /2-Dg / 2 •

(4.31) 1r H is the dimension of the smooth component of Hom( b(M),G) and 1r D g = dimG. Integrating (4.31) over g X S[Hom( b(M),G)] and evaluating the gaussian integral over g we get to leading order:

where

D

92

4. Moduli Spaces for Dynamically Triangulated Manifolds

1r[detPg . detPgl* s

-1/2

exp

(

2

2N IPHUl Vvol(a n

n)

)

'

(4.32)

which, upon setting

A(V)

~ l[detPg . detPgtl/2 exp ( - 2N~~:;1;an)) ,

(4.33)

yields the stated result. Note that the computation of the integral A(V), extended over the moduli space Hom( 1rh(M),G) , can be related to the theory of Reidemeister torsion[115, 116, 79],[58].

4.5.2 Moduli Asymptotics for DT-Surfaces Both the presence of the volume of the representation variety and of its dimension, are an important aspect of (4.26). In dimension n == 2, we can be more explicit, and if we gather information on the dimension of Hom(1r~}M),G) [63, 76, 93, 110],[20], then in the case of 2-dimensional surfaces theorem 4.5.1 specializes to the following

Lemma 4.5.1. For n == 2 and for Vvol(a n ) near zero, the leading contribution to Vol[M(Ta )] is given as a function of surface topology by: (i) The sphere §2:

Vol[M(Ta )] ~Vvol«(7n)«l ~ 1.

(4.34)

(ii) The torus 'lr2 : Vol[M(Ta )] ~Vvol«(7n)«l~ A(V)· (21rNn Vvol(a n ))-S/2.

(4.35)

(iii) Orientable surfaces Eh of genus h > 1, and for representations of ~ G whose center has dimension dimZ:

1r1(Eh)

Vol[M(Ta )] ~Vvol«(7n)«l ~ A(V) . (21rN n Vvol(a n ) )-[(2h-1)dimG+dimZ]/2.

(4.36)

(If we have a non-trivial centralizer for the representation, then we have to slightly alter the argument in theorem 4.5.1).

4.5 A Measure on the Moduli Space

93

Proof. For a sphere dim[H1(1Tl(M), g)] == 0, and trivially V == o. As V ~ 0, the measure dN weakly converges to 8(Id), where 8(Id) is the Dirac distribution concentrated on the trivial representation. This immediately yields the stated result. The torus '[2, can be actually considered as a limiting case of (iii). From [62, 63, 76, 93, 110] it follows that if z(O) denotes the centralizer of O(1Tl (L'h)) in G, then the dimension of Hom(1Tl (L'h) , G) is given by

(2h - l)dim(G)

+ dim(z(O)).

(4.37)

By taking G == 8L(2, lR), and setting h == 1 we immediately get the stated result for '[2. The general result for h > 1 is a trivial consequence of the dimensionality formula (4.37). We wish to stress that these results (in particular the asymptotics of the volume of moduli space) refer only to the smooth component of the representation space Hom(1Tl (M), G)jG. Singularities are present in correspondence with reducible representations and around representations which are linearization unstable [63, 76, 93, 110]. Also, Hom(1Tl (M), G)jG may have many distinct connected components. For example, in the relevant case of G == P8L(2,lR), Hom(1Tl(M),P8L(2,JR)) has 22h +1 + 2h - 3 connected components, [62, 63, 76, 93, 110]. In 2d-Yang- Mills, it is a known fact that, at least in the semiclassical limit, the singular component of the representation variety does not contribute to the volume of the moduli space[58] , [115, 116, 79], [83]. Since there are many points of contact between our approach in evaluating the asymptotics of the volume of the moduli space of Romer-Zahringer deformations of a dynamical triangulation, and 2d-YangMills, we may reasonably assume that the singular part of the representation variety is not playing any relevant role in our case, at least in dimension n == 2. For dimension n > 2 this is an open issue under current investigation. As we shall see in due course, the asymptotic of the volume of the moduli space of Romer-Zahringer deformations plays a basic role in simplicial quantum gravity being related to the entropy exponent of the theory, thus these matters are quite relevant in addressing on analytical grounds the existence of the continuum limit of the theory. 4.5.3 A Compact Formula for Surfaces

We conclude this section by providing a compact expression for the the asymptotics of the volume of the moduli space of Romer-Zahringer deformations of a dynamical triangulated surface. In the 2-dimensional case, according to lemma 4.5.1 the small volume behavior of Vol[M(Ta )] is

Ah (V) . N~(Eh) , (4.38) where the constant Ah (V) ~A(V) (211"Vvol ((1n ))- [(2h-l )dirnG+dirnZ]/2 , (for genus h 2:: 1, whereas Ah(V)~A(V) for the sphere), and e(L'h) is an exponent depending from the genus, and the group G. This exponent can be given a

94

4. Moduli Spaces for Dynamically Thiangulated Manifolds

more compact expression if we introduce an effective dimension, De!!' of Hom(1fl (M), G) according to Deff~(2h - l)adim(G)

+ adim(z(O)) + {3,

(4.39)

where the parameters a and {3 are determined by fitting (4.39) with the dimensions of the singular case provided by the sphere §2 and the limiting case of the torus 'lr2 . Explicitly, according to lemma 4.5.1, we have

~(L\) = -(2h -

~ + 1,

1)idim(G) - idim(z(B)) -

(4.40)

with

~dim(G)

-

~dim(z(B)) - f!. + 1

~(§2)

2 2 2

-~dim(G) - ~dim(z(B)) - f!. + 1

~ (1r

2 2 2

2

== 1,

) == 1 -

5 2.

(4.41)

In this way we get

{3

5 2dimG' ~(1- dim(z((}))) 2 dimG'

(4.42)

and (4.43) where X(Eh)~2 - 2h is the Euler-Poincare characteristic of the surface Eh. These observations establish the following Lemma 4.5.2. The small volume limit asymptotics of the volume of the moduli space of Romer- Ziihringer deformations of a dynamical triangulated surface is given by

Vol[M(Ta )]

~Vvol((1n)«l~

--

2x(E h )_2

Ah(V) . N,i

2.

(4.44)

5. Curvature Assignments for Dynamical Triangulations

According to the analysis of the previous sections, the topological information associated with a dynamical triangulation is encoded in the moduli space of its Romer-Zahringer deformations. Here, our purpose is to characterize the metric properties of the triangulations in terms of the (restricted) holonomy of its natural Levi-Civita connections, and reduce the enumeration of distinct dynamical triangulations to two manageable counting problem: the enumeration of distinct curvature assignments, and the estimation of the number of distinct dynamical triangulations with a given set of curvature assignments. In order to answer such a question we exploit the connection between curvature assignments an the theory of partitions of integers.

5.1 Partitions of Integers and Curvature Assignments Recall that in order to interpret a dynamical triangulation Ta as a GromovHausdorff approximation of a Riemannian manifold, we need to consider T a as an Alexandrov space of bounded geometry, namely as an element of Drn(a, b, N), (see theorem 3.3.1), with curvature (in the comparision sense) bounded below by some real number r. This description requires that the curvature assignments {q( a)} of a dynamical triangulation T a E Drn (a, b, N) are bounded above by an integer q(O) ~ q(a)o>o that we can assume as representing a distinguished curvature assignment at the marked bone a == o. From a combinatorial point of view, q(O) can be interpreted as a sort of root blob of curvature. However, we stress that this rooting is here not just as a convenient enumerative means as often happens in combinatorics, but its presence is dictated by the geometry underlying the dynamical triangulations approach. According to (2.7) the curvature assignments {q(a)} of a dynamical triangulation Ta are further constrained by the total number Nn(T(i)) of nsimplices in T a . We have

q(o) +

L q(a) = A

0=1

1

2n(n + l)Nn (T(i)),

(5.1)

96

5. Curvature Assignments for Dynamical Thiangulations

where the summation extends over the bones of T~i), and where, for notational simplicity, we have denoted by A+l == N n - 2 (T) the number of bones in the dynamical triangulation T a , (note that we have included in the summation the marked bone, by considering also the incidence number q(O)). Another natural constraint comes about from the fact that in order to generate an n-dimensional polyhedron there is a minimum number ij(n) of simplices an that must join together at a bone, (according to the introductory remarks, in dimension n == 2, n == 3, n == 4 we have ij(n) == 3, otherwise we have no polyhedral manifold. Rather than use this explicit value, we keep in using the general notation ij(n) ). If we define

Q(O) Q(a)

~ ~

q(O) - ij(n), q(a) - ij(n),

(5.2)

then we can put these two constraints together by writing

(5.3) (recall that N n _ 2 (T(i)) == A + 1, also recall that b(n, n - 2) == ~n(n + 1)(Nn (T(i))/Nn _ 2 (T(i))) is the average number of simplices an, in T(i), incident on a bone). Thus, we have A

Q(O)

+L

0=1

Q(a) == [b(n, n - 2) - ij(n)] (A + 1).

(5.4)

",'

We shall now analize the possible solutions of (5.4) as an equation in integers. To this end, recall that a partition of a positive integer n is a representation of n as the sum n == 1· 'fJl + 2· 'fJ2 +... + n· 'fJn, where 'fJl' 'fJ2,' .. ,'fJn are non-negative integers, and 'fJl + 'fJ2 + ... + 'fJn determines the number of summands of the partition. This remark suggests that for each given value of Q(O), b(n, n - 2), and A, the set of curvature assignments q(a) == Q(a) + ij satisfying the constraint (5.4) may be related to (some of) the possible partitions of the integer [b(n, n - 2) - ij(n)] (A + 1). Among the attractive features of such an interpretation is the possibility of exploiting the large body of knowledge we have on partition theory, a well-established chapter of combinatorics and analytic number theory. In order to exploit these techniques, we need to modify a little (5.4) so that it resembles more directly a partition. As we have seen in the introductory chapter, (see (2.30)), the largest curvature assignments can well be of the order of A, and the largest curvature assignments which simultaneously minimize the remaining curvature assignments on the other bones are realized on particular triangulations, (see the discussion following (2.30)). In such triangulations, most of the

5.1 Partitions of Integers and Curvature Assignments

97

top-dimensional simplices are incident on n distinguished bones, (bounding an embedded §n-2). Notice that, when generated with equilateral simplices, these particular triangulations have a discrete simmetry associated with the interchange of the n distinguished bones carrying most of the curvature. Since we have to consider (5.4) as q(O) varies from q to qmax, we have to factor out these residual symmetries, and we do so by marking not just one bone, (the one providing the basepoint Po), but rather n bones, a n - 2 (0), a n - 2 (1), . .. ,an - 2 (n - 1), which support, in the extreme case described above, the largest curvatures (with q(O) == q(l) = ... == q(n - 1)) while minimising the incidence on the remaining bones {a n- 2 (0:)}a=n, ... ,,X. The original marked bone a~-2 is one of those n distinguished bones, and can be selected by the required marking of an n-simplex 0'0 associated with the gauge fixing procedure for the moduli space of dynamical triangulations. Thus, we start by replacing (5.4) with ,X

nQ(O)

+ L Q(o:) == [b(n, n - 2) - q(n)] (A + 1).

(5.5)

a=n

Since nQ(O) ~ Q(o:), we can refer the curvature assignments Q(o:) to the chosen value for the curvature assignment nQ(O) around the marked bones. To this end, let us add and subtract to (5.5) the expression nQ(O)('\ + 1), which allows us to eventually rewrite (5.4) as ,X

L Q(o:) == [nQ(O) - b(n, n - 2) + q] (,\ + 1) - n(n -

l)Q(O),

(5.6)

a=n where

Q(o:)~nQ(O) - Q(o:).

(5.7)

This form (5.6) of (5.4) can be interpreted as providing a partition of the integer [nQ(O) - b + q](A + 1) - n(n - l)Q(O) in at most A + 1 - n integers Q(o:), (some of the Q(o:) can be zero), each bounded above by (b - q)(A + 1). Note that is not a priori true that each partition of the integer [nQ(O) -b+ q](A + 1) - n(n -l)Q(O) can be generated by curvature assignments through the Q(o:). Thus, we shall denote by p~urv

([nQ(O) - b + q](,\ + 1) - n(n - l)Q(O),)

(5.8)

the number of partitions of [nQ(O) - b(n, n - 2) + q](A + 1) - n(n - l)Q(O) into at most A + 1 - n parts, (each ~ (b(n, n - 2) - q)(A + 1)), arising from actual curvature assignments. Remark 5.1.1 (A point of notation). Note that in comparing with standard formulae for partition theory[5], a more standard notation would have been p~U;f-n ([nQ(O)

- b + q](A + 1) - n(n - l)Q(O)) ,

(5.9)

98

5. Curvature Assignments for Dynamical Triangulations

however, in order not to burden the notation further, we shall adopt the above simplified version; in some cases, if there is no danger of confusion, we shall simply write p~urv for the above expression and for other similar expressions involving partitions. Trivially p~urv < P.x, and a first estimate of the asymptotic behaviour of can be obtained by the asymptotics of p.x:

p~urv

Lemma 5.1.1. Let DTn(a, b, A) denote the set of distinct dynamical triangulations, of an n-dimensional PL manifold M, with N n- 2 (Ta ) = A + 1 bones, and average incidence b(n, n - 2) = b. Then, for large A the number of distinct partitions of the integer [nQ(O) - b + q](A + 1) - n(n - l)Q(O) (into A + 1 - n parts) coming from curvature assignments of a dynamical triangulation Ta E DTn(a, b, A) is exponentially bounded above according to p~urv

::;

e[2+ln(b-q)]A A-2.

(5.10)

Proof. The asymptotic behavior of p.x([nQ(O) - b + q](A + 1) - n(n -l)Q(O)) follows by noticing that (see e.g[73]' pp. 32) P.x ([nQ(O) - b+ q](A + 1) - n(nl)Q(O)) depends only on the value of [nQ(O) - b + q](A + 1) - n(n - l)Q(O) modulo (A + 1 - n)!, and that it is a polynomial of degree A- n whose leading term is

{[nQ(O) - b + q](A + 1) - n(n - l)Q(o)}.x-n (A + 1 - n)!(A - n)!

(5.11)

On applying Stirling's formula, we get for A >> n

p.x([nQ(O) - b + q](A + 1) - n(n - l)Q(O)) e 2.x

{[nQ(O) - b + q]A}.x

21f A2 .x[nQ(0) - b(n, n - 2)

+ q]A·

~

(5.12) (5.13)

As Q(O) varies, p.x([nQ(O) - b+q](A+ 1) -n(n-1)Q(0)) is bounded above by the right side of the above expression evaluated for max[nQ(O)] = (b-q)A, which yields P.x ~ e[2+ln(b-q)].x A-2. Since p~urv < P.x, the result follows. The bound (5.10) may be consistent as well with a subexponential growth for p~urv, a subexponential growth that, if present, may make the actual control of p~urv quite unassailable. However, more can be said on the actual asymptotics of the curvature assignments in DTn(a, b, A). The following result shows that, up to a rather simple exponential rescaling, the asymptotics of the number of curvature assignments is actually provided by the number of distinct partitions of the integer [nQ(O) - b + q](A + 1) - n(n - l)Q(O) (into A + 1 - n parts).

Theorem 5.1.1. Let DTn(a, b, A) denote the set of distinct dynamical triangulations, of an n-dimensional PL manifold M, with N n- 2 (Ta ) = A + 1

5.1 Partitions of Integers and Curvature Assignments

99

bones, and average incidence b(n, n - 2) == b. Then there is a constant s ~ 0, possibly depending on inf b(n, n - 2), sup b(n, n - 2), q and the dimension n, such that, for A large enough, we have ~

f'.J

p~urv - e

-s(b-q) ..

(5.14)

.

Proof. For a given value of b, Q(O), and A » 1, let C(b, A, Ta ), (C for short), be the set of partitions associated with curvature assignments of the rooted triangulations T a of bounded geometry in DTn(a, b, A). Let us index the elements of C, so that C == {hI, h2 , h3 , ... } with hk denoting a partition {Q(a)} of the integer [nQ(O) - b + q](A + 1) - n(n - l)Q(O) into A + 1 - n parts coming from curvature assignments. We observe that ICI ~ Card[C] == p~urv. Let P == {k l , k 2 , k 3 , ... } denote the set of generic partitions of the integer [nQ(O) - b + q](A + 1) - n(n - l)Q(O) into A + 1 - n parts not necessarily coming from curvature assignments. Obviously C c P. On the pair (P, C) we can introduce the Hamming distance (5.15) defined to be the number of positions in which the two partitions k == {k l , k2 , ... } E P and h == {hI, h2 , ... } E C differ, (it is easily verified that this is indeed a distance; it is often used in the theory of codes and de'sign where provides a natural distance between addresses). By exploiting dHwe can partition P relatively to C into disjoint classes. First, let us assume that, for any A, there exists an integer 0 < m< +00 such that the sets Si ~ {k E P:dH(hi,k) ~ m} as hi varies in C, cover P. Notice that this assumption states that, for large A, there are sufficiently many partitions into curvature assignments with respect to the set of generic partitions. This is a natural hypothesis, which however we have been not able to prove starting from more elementary assumptions. Define

1r(h l )

~

{k E P: dH(k, hI)

~

m}.

(5.16)

Extend this inductively as

1r(hj ) ~ {k E P / U~~{-I1r(hi): dH(k, hj ) ~ m}. Then,

P

=

Il 7r(h

(5.17)

(5.18)

i ).

hiES

Since Card[P] == p).. ==

and Card[C] == p~urv we get

L Card[1r(h

hiEC

and

p)..,

i )],

(5.19)

5. Curvature Assignments for Dynamical Triangulations

100

p).. 1 p=rv = pcurv )..

)..

~

(5.20)

LJ Card[7r(h i )]. hiEC

Thus, not unexpectedly, the ratio p)../p~urv appears as the average value over C of the number of partitions which are not curvature assignments. Consider two independent sets C1 , and C2 , associated with the curvature assignments of dynamical triangulations with distinct rootings Q(O)[(], ( = 1,2. And let p~urv[(], denote their respective cardinalities, (i.e., the number of distinct partitions of the integer [nQ(O)[(] - b + q](A + 1) - n(n -l)Q(O)[(] into A + 1 - n parts). Starting with such C1 and C2 we may construct a third curvature assignments set C3 associated with dynamical triangulations with rooting given by Q(0)[3] ~ Q(O)[I] + Q(0)[2] - 2. In order to define this new set of curvature assignments, consider triangulations T(l) and T(2), respectively associated with C1 and C2. The rootings Q(O)1 and Q(0)2 define unique simplicial loops winding around a marked bone a n - 2 (1) and a n - 2 (2), respectively. Let an (l) be a simplex chosen among the nQ(O)[I] n-simplices incident on the marked bone a n - 2 (1), and let a n (2) be a simplex chosen among the nQ(0)[2] n-simplices incident on the marked bone a n- 2(2). Set

T(3)

~

(T(l) - 0-(1)) Uf (T(2) - o-n(2)) ,

(5.21 )

where 0-(.) denotes the interior of the given a's, and f is an homeomorphism from the boundary of a n (l) to the boundary of a n (2). In this way, with any pair of triangulations T(l) and T(2) associated with curvature assignments in C1 and C2 , we associate a triangulation T(3) with curvature rooting given by Q(0)[3] = Q(O)[I] + Q(0)[2] - 2. The average number of partitions of

[nQ(0)[3] - b + q](>'l

+ >'2 + 1 - ~n(n + 1)) -

n(n - 1)Q(0)[3]

(5.22)

which are not curvature assignments of triangulations like T(3), is then provided by

p~>;'v IQ(O)[3] = (pfu>;'v IQ(O)[l]) . (p~>;'v IQ(O)[2)) .

(5.23)

On the other hand, this number cannot be greater than the average number of partitions of [nQ(0)[3] - b+ q](AI + A2 + 1- (1/2)n(n + 1)) - n(n -1)Q(0)[3] which are not curvature assignments of generic triangulations, (with Al + A2 + 1 - (1/2)n(n + 1) bones), with curvature rooting Q(0)[3]. Namely, we have

p~>;'v IQ(O)[l)+Q(O)[2] ~ (p~v IQ(O)[l)) . (p~Arv IQ(O)[2)) .

(5.24)

This results shows that _In(p)../p~UrV) is subadditive with respect to the assignment of the root curvature nQ(O). As A ~ 00, nQ(O) may go to 00 as max nQ(O) = (b - q)A, then it makes sense to consider the limit

5.1 Partitions of Integers and Curvature Assignments

101

Fig. 5.1. Joining two triangulations T(l) and T(2) with rootings Q(O)[l] and Q(O)[2] we can generate a triangulation T(3) with rooting Q(O)[l] + Q(O)[2] - 2

102

5. Curvature Assignments for Dynamical Triangulations

(1 · - n P>.) -- . 11m

Q(O)-+oo

(5.25)

p~urv

By subadditivity, this limit is characterized by

lim

(-ln~) == p~urv

_1_

Q(O)

Q(O)-+oo

. fn 1 -(- n 1 P>.) I -

Q(O)

Q(O)

(5.26)

p~urv'

where the inf is taken over all possible values of Q(O). Since max nQ(O) goes at 00 as (b-q)A, where b varies in inf b(n, n - 2) ~ b(n, n - 2) ~ sup b(n, n - 2), there is a constant s ~ 0, possibly depending on inf b(n, n - 2), sup b(n, n - 2), fj and the dimension n, s=.

· f In

1

(>.,b)E(O,oo)X[bTnin,b Tnax ] Q(O)(A,b)

(1

P>.) -n-

< 0,

p~urv-

(5.27)

such that, for A large enough, we may write ~ rv curv P>.

e

-s(b-q)>.

(5.28)

,

as stated. According to (5.28) the average p~urv is, in the large A limit, either a constant P>.. (if s == 0), or an exponential function of A, inf b(n, n - 2), and sup b(n, n - 2), (if s > 0). Since we are eventually interested not so much in the asymptotics of p~urv but rather in the large A behavior of the sum curv P unroot

==

~ pcurv

L.-J

'

(5.29)

Q(O)

representing the number of unrooted curvature assignments, the above theorem implies that in place of P~~~ot == 2: Q (O) pcurv we may directly consider Punroot == 2: Q (O) P>.· The advantage of such a trade is that we can evaluate exactly 2: Q (O) p(z) as we shall see in the next chapter. Then, we can evaluate P~~~ot by multiplying 2: Q (O) p(z) by a rescaling function which is a monotonically decreasing function of A. This remark is the backbone of our enumeration strategy.

5.2 Distinct Dynamical Triangulations with Given Curvature Assignments For small A's the actual distribution of partitions corresponding to curvature assignments is rather accidental, however according to theorem 5.1.1,

5.2 Distinct Dynamical Triangulations with Given Curvature Assignments

103

for A >> 1 a large fraction of partitions corresponds to actual curvature assignments. A basic question we need to address at this point is to what extent the datum of curvature assignments characterize a dynamical triangulation. Let {T~i)} the set of distinct dynamical triangulations of a given PLmanifold, and let {T~i)}curv={q(a)} C {T~i)} the subset of distinct triangulations sharing a common set of curvature assignments {q(Q) }~=o, (often we shall write {T~i)}curv for short if there is no danger ~f confusion). Since for distinct curvature assignments {q( Q) }~=o =1= {q({3) }~=o, the sets . • d··· t an d thelr . coI { fT1(i)} .La curv={q(a)} an d {fT1(i)} .La curv={q(,B)} are paIrWIse ISJOln, lection is finite, we can apply the rule of sums in enumeration theory and write (i) Card{Ta(i) } = ~ ~ Card{Ta }curv={q(a)} , q(a)

(5.30)

where the sum is over all distinct curvature assignments (5.31 ) whose average incidence is the given b( n, n - 2). If we introduce the average value of Card{T~i)}curv={q(a)} over all such curvature assignments:

< Card{TJi}}curv >~ c:rv P)...

L Card{TJi}}curv={q(a)} ,

q(a)

(5.32)

then we can eventually write Card{T~i)}

= p~urv < Card{T~i)}curv > .

(5.33)

This relation provides the connection between the enumeration of distinct dynamical triangulations and the enumeration of distinct curvature assignments. Since p~urv grows exponentially with A, (5.33) suggests that, at least asymptotically, distinct curvature assignments correspond to distinct dynamical triangulations, in the sense that for large A, Card{T~i)} dominates over < Card{T~i)}curv >. Formally, it is instructive to prove this by the elementary probabilistic methods which are typical in discrete mathematics whenever attention is on asymptotic properties. We implement such methods by exploiting, (in a rather trivial way), the ergodicity of the Pachner moves (see the Appendix), mapping a given triangulation into a distinct one. However, most of this section is devoted to a detailed analysis providing an asymptotic estimation of the size of the set of distict dynamical triangulations with given curvature assignments, < Card{T~i)}curv >. As a matter of fact, the subdominat tails in the asymptotics of Card{T~i)} associated to < Card{T~i)}curv > provide important corrections to the naive counting through unrestricted partitions of integers. Such corrections characterize the transition from the strong to the weak coupling limit in simplicial quantum gravity.

104

5. Curvature Assignments for Dynamical Triangulations

Let us start by noticing that it is obviously false that every possible set of distinct curvature assignment corresponds to triangulations having distinct incidence matrices. Indeed, one may easily construct particular examples showing that {T~i)}curv is not trivial. Consider for instance two dynamically triangulated (exagonal) flat tori. By flipping links, we may generate two curvature bumps on each torus, (on each torus, the bumps may correspond to distinct curvature assignments) . By inserting in each torus a copy of these curvature bumps in such a way that their distance is different in the two tori, we get distinct triangulations with the same curvature assignments. As another more general example, (again in dimension 2), consider a 2-dimensional triangulation with a large number of vertices, let abc, acd, and efg, egh four triangles pairwise sharing a common edge (ac for the former pair, and eg for the latter), but otherwise largely separated, (so that their incidence numbers on the respective vertices are uncorrelated). Let us assume that the corresponding incidence number are given by: q(a) == a, q(b) == ,,/, q( c) == (3, q( d) == ~, for the first pair of triangles, while for the second we set: q(e) == "/ + 1, q(f) == a-I, q(g) == ~ + 1, q(h) == {3 - 1. It is immediate to check that a flip move will interchange the curvature assignments of the first pair of adjacent triangles with those of the second pair, thus changing the incidence relations in the triangulation. Nonetheless, the sequence of curvature assignments remains unchanged, and again we have two distinct triangulations with the same sequence of curvature assignments. Clearly, these counterexamples work either because we chose a particularly symmetric triangulation or because we adjusted the curvature assignments to the particular action of the flip move. In any case, there are plenty of them even in less particular situation. However, when moving from a dynamical triangulation to another, with a set of ergodic moves (Pachner ([96, 97, 67])), the curvature assignments are not fixed, and in this sense the above counterexamples are not generic, we can prove that as ,\ » 1, {T~i)}curv is in a suitable sense a small subset in {T~i)}. We can formalize this remark according to the

Lemma 5.2.1. For a PL-manifold M, let 8: 7rl(M;po) --t G, be a given holonomy representation, and let {T~i)}A denote the set of distinct dynamical triangulations of M == IT~i) I with N n- 2(T~i)) == A + 1 bones, and let {T~i)} AEZ be the associated inductive limit space generated as ,\ grows. Let

PT ~ ({T~i)}AEZ,B,mT) be the probability space endowed with the normalized counting measure mT on every non-empty open set B

C

{TJi)}AEZ. For

any Ta E {T~ i)} A' let {q(a)} describe the sequence of integers providing the incidence 01 the simplices {a~}J.Vn(Ta) over the ,\ + 1 bones {an-2}Nn-2(Ta). ~ ~=1 k k=l Then, as ,\ --t 00, the curvature assignments {q(a)} characterize with probability one the dynamical triangulation Ta[q(a)].

5.2 Distinct Dynamical Triangulations with Given Curvature Assignments

105

Proof. We start by creating a probability space PT whose elements are the distinct triangulations in {T~i)} AEZ, and whose Borel probability measure, mT, is the normalized counting measure on every non-empty open set B C {T~i)hEZ' Note that we have to consider triangulation with variable >., since the known set of ergodic moves ([96, 97, 67]) mapping a triangulation T~i)(l)A into a distinct triangulation T~i)(2)A' (with the same A), go through intermediate steps which do not preserve volume (hence the number of bones A at a given b(n, n - 2)). If, for a given value of A, {T~i)}curv is the subset of distinct triangulations with a same set of curvature assignments, we can write (A fixed) (i)

(i)} ] _ Card{Ta }curv mT [{Ta curv (') . Card{Ta~ }

(5.34)

Recall ([96, 97, 67]) that on (PT, B, mT), the Pachner moves characterize a continuous and ergodic dynamical system with respect to mT. If (")

· Card{Ta~ }curv 11m (')

A~OO

Card{Ta~}

0

>,

(5.35)

then by Birkhoff ergodic theorem[112] we would get (")

mT[{Ta~

}curv] > 0,

(5.36)

and the Pachner moves would fix mT-almost everywhere the set {T~i)}curv thus contradicting their ergodicity. Hence we must have (") t

lim Card{Ta ~curv A~OO Card{T~~)}

= 0,

(5.37)

which reflects the fact that Pachner moves can fix the curvature assignments only under special circumstances. Despite this very general, formal result, we still have to face the fact that distinct dynamical triangulations associated to a same set of curvature assignments may give rise to significant subleading corrections to the counting of distinct triangulations of a given PL-manifold. In the previous chapter we have partially addressed this issue by considering the characterization the space of deformations, M(Ta ) of a dynamical triangulation approximating a locally homogeneous riemannian manifold. In that case, the set of possible deformations may be thought of as parametrizing a set of distinct dynamical triangulations (with the same curvature assignments of the given, undeformed, dynamical triangulations) with suitably deformed distances among vertices. Such deformed triangulations are meant to approximate, (in the limit a ~ 0), the distinct locally homogeneous manifolds (moduli) infinitesimally near the given manifold. According to theorem 4.5.1 the number of distinct deformations grows polynomially with N n with

106

5. Curvature Assignments for Dynamical Triangulations

Fig. 5.2. By increasing the diameter and by shrinking the neck of the dumbell we can generate distinct riemannian structures with the same volume and the same curvature (up to exponentially small curvature correction terms in the regions joining the neck with the spheres). Similarly, there are distinct dynamical triangulations with the same volume and the same curvature assignments.

an exponent of topological origin, showing that the counting of {TJi}}curv is a non-trivial issue. While the asymptotics of M(Ta) C {T~i}}curv takes care of the topological aspects associated with the enumeration of {TJi}}curv, we have still to estimate the contribution to {T~i}}curv coming from dynamical triangulations not necessarily approximating locally homogeneous manifolds. As we have seen in chapter 4, the subset M(Ta) C {T~i}}curv is characterized by the representations of the fundamental group 1fl (Ta ) into the structure group of the geometry, thus the topological control in counting {T~i}}curv is realized if we consider, as usual, rooted triangulations in {T~ i) } curv according to theorem 4.3.1. We have the following result providing a non-trivial, bystable asymptotics for the average number < {T~i)}curv >, (see (5.32)), in function of the average incidence b(n, n - 2). Theorem 5.2.1. If n

~ 3, there is a critical value bo(n), of the average incidence b(n, n - 2), sufficiently near to the lower kinematical bound b(n, n2)min, such that if

b(n, n - 2)min :s; b(n, n - 2; {q(k)}) :s; bo(n),

(5.38)

then, as,\ ~ 00, the rate of growth of the average number < Card{T~i}}curv > of rooted triangulations in {T~i}}curv is at most polynomial, viz., there are constants, j..t(b(n,n - 2)) > 0, and T(b(n,n - 2)) ~ O,(possibly depending on b(n, n - 2)), such that

< Card{T~i}}curv

>~ j..t(b(n,n -

2))· N nT (b(n,n-2}.

(5.39)

Note that this polynomial rate of growth also holds in the two- dimensional case for < Card{T~i)}curvln=2 >. Conversely, if bo(n) < b(n, n - 2) :s; b(n, n - 2)max,

(5.40)

5.2 Distinct Dynamical Triangulations with Given Curvature Assignments

107

then the asymptotics of < Card{T~i)}curv > is exponential. Namely there is a constant m(b(n, n - 2)) > 0, possibly depending on the average incidence b(n, n - 2), and an nH ~ n such that

< Card{T~i)}curv

>~

j-t(b(n, n - 2)) . exp[ -m(b(n, n - 2) )N~/nH]N~(b(n,n-2» ,

(5.41)

as N n goes to infinity. We have been rather allusive concerning the statement of this theorem, in particular for what concerns the actual value of the parameters j-t(b(n, n- 2)), r(b(n,n - 2)), m(b(n,n - 2)), nH ~ n, and a more explicit characterization of the critical average incidence bo(n) around which the transition between polynomial and subexponential asymptotics occurs. We wish to stress that at the moment of writing these notes we do not yet have a quantitative control on the explicit structure of such parameters. Such a characterization is an important open issue since it is rather plausible that the transition from a polynomial to a subexponential asymptotics in Card{T~i)}curv can shed light on the nature of the critical point of simplicial quantum gravity.

Proof. Let {T~i)}curv={q(k)} denote the set of rooted distinct dynamical triangulations (with A + 1 bones one of which marked), having a given set of curvature assignments {q(k)}O~k~A. According to the rooting prescription of theorem 4.3.1, we have to mark a simplex ao E T~i), among the q(O) incident around the marked bone. We can realize such a choice by selecting !n(n + 1) curvature assignments, (the same for all T~i»), {q(a)}0~a~n(n+l)/2-1 E {q(k) }O~k~A' and declare that these numbers are the incidence numbers associated with the ~n(n + 1) bones of such a marked simplex. Denote by T~i)(ao), the triangulation obtained by removing from T~i) E {T~i)}curv the interior ao of the marked simplex ao, (this removal is not strictly necessary, but technically it simplifies the proof). On the generic T~i) (ao) so defined, we can consider simplicial geodesic balls, of simplicial radius R, B(i) (ao; R) centered at the boundary 8ao of the marked simplex ao, namely B(i) (an. R) ~ {{ aT!'}· d(8a n 8 a T!') < R} (5.42) 0'

0'

J.

J

-

,

where d(Bao,8aj) is the length (in T~i») of the simplicial path with endpoints Bao, and Baj, Ban denoting the boundary of the given simplex. Given {q(k)}O~k~A' let {q(k)}(B(R)) C {q(k)}O~k~A a set of possible curvature assignments in the gedesic ball B(i) (aD; R). We can correspondingly define, (up to an a dependent factor), the volume of the ball according to

Vol[B(i)(ao;R)]=nn

2

1

L

( + ) {q(k)}(B(R))

q(k).

(5.43)

108

5. Curvature Assignments for Dynamical Thiangulations

Note that if we mark 0'0 by providing the curvature assignments at its bones, then Vol[B(i)(ao;R = 1)] is fixed and, up to an a dependent constant, is given by Vo ~ Vol [B(i) (an; R o

= 1)] ~

2 n(n+1)

~

LJ

q(a).

(5.44)

O~a~n(n+I)/2-I

With these preliminary remarks along the way, let us start by noticing that if there are distinct triangulations, say Ta (l) and Ta (2) in {T~i)}curv, sharing the common set of curvature assignments {q( k) }O~k~..\, then the relative distances between bones with the same curvature ass.ignments in Ta (l) and Ta (2) will be in general different. This implies in particular that the distributions of curvature assignments in the balls B(I)(ao; R) C Ta (l) and B(2) (0'0; R) C Ta (2) will be different, and correspondingly Vol[B(I)(a~;R)]

f: Vol[B(2)(a~;R)].

(5.45)

Thus, an indication of how large is the set {T~i)}curv can be obtained by discussing the possible distribution over {T~i)}curv of the volumes of the geodesic ball B(ao; R). To this end, let

{T~) [V; R]}curv

C

{T~i)}curv

(5.46)

denote the set of distinct (dynamical triangulations of the) simplicial balls B(i)(ao; R), such that Vol[B(i)(a~;R)] = V,

(5.47)

and with given curvature assignments in

{q(k) }O
(5.48)

In order to avoid any misunderstanding, note that the distinct simplicial balls {T~) [V; R]}curv, are the distinct triangulations of the polyhedron IB(a'[); R)I induced by the distinct triangulations in {T~i)}curv. Moreover, the parameter R is fixed, since we wish to discuss the distribution of the number of distinct simplicial balls, Card{T~) [V; R]}curv, as a function of their volume V. Since the curvature assignments are given, the set {T~i) [V; R]} curv, for an arbitrary choice of V, may well be empty, and we need to characterize, in terms of the curvature assignments, the values of V corresponding to which the set {T~i) [V; R]}curv gets its leading contribution. This characterization can be obtained by a simple subbaditive argument, combined with volume estimates related ·to the boundedness of the metric geometry of the balls (i) {TB [V; R]}curv. Let us start by noticing the following simple but important property. Let VI and V2 be given, and consider any two triangulated balls of radius R, say B(l) and B(2), both centered at a marked simplex 80'0, with Vol[B(l)] = VI and Vol[B(2)] = V2, and with curvature assignments in {q(k)}O~k~A' (recall

5.2 Distinct Dynamical Triangulations with Given Curvature Assignments

109

that the curvature assignments {q(a)}0~Q~n(n+I)/2-I at 80'0 are given). At the expenses of increasing the amount of negative curvature around the bones of 80'0' we can glue B(l) and B(2) together so as to generate a new ball, of the same radius R, and with volume VI + V2 . Explicitly, set

B(3)

~

(B(l)) Uf (B(2)),

(5.49)

where f is an homeomorphism from 80'0(1) to 80'0(2). This construction allows us to associate with any pair of such triangulated balls B(l) and B(2), a triangulated ball B(3), again centered at 80'0 and such that

Vol[B(3)]

= VI + V2 .

(5.50)

Note that the number of distinct triangulated balls that we may construct according to the above glueing procedure is given by

IAut(J) I [Card{B(i)[V1;R]}curv] x [Card{B(i)[V2 ;R]}curv],

(5.51)

where the factor \Aut(f)\ is the number of distinct way of glueing 80'0(1) to

80'0(2). It is important to stress that even if this glueing does not alter the curvature assignments, {q(k)}n(n+I)/2~k~A' it changes the assignments at 80'0: it decreases the Regge curvature along the bones of 80'0 since the curvature assignments at 80'0 in B(3) are now given by (5.52) The glueing just described is a familiar surgery operation on triangulated manifolds, and it is of importance, for instance, in characterizing the so called minimal-bottleneck-universes, minbus. The feasibility of this construction around a particular marked simplex 0'0 does not force any particular constraint on the triangulation. However, this construction becomes quite more selective if we require that it can be carried out along the boundary of any simplex an, viz., that it holds for any possible marking 0'0 of a base simplex an. First of all, in order to have the set of distinct balls {B(i)[V;R]}curv, with given curvature assignments in {q(k)}O
(5.54)

110

5. Curvature Assignments for Dynamical Thiangulations

Roughly speaking, if the set of distinct balls {B(i) [V; R]}curv, with given curvature assignments, is closed under the glueing procedure (5.49) it means that we are considering the generic triangulation as a space of non-positive Alexandrov curvature, (i. e., endowed with a sort of hyperbolic structure). As a matter of fact, the condition (5.53) is a simple-minded version of the large link condition (here applied to the links of the bones) which is required in the Gromov's hyperbolization of a piecewise-flat manifold. Recall that [78] hyperbolization is a procedure devised by Gromov which, (possibly by modifying the topology of the underlying PL-manifold), changes the geometry so that the geodesics in the resulting metric space behave as in a negatively curved Riemannian manifold. The standard 2-dimensional example is afforded by the hyperbolization of a triangulation realized by taking a 2-fold ramified covering with branching points at the vertices of the triangulation. In this way, we get a triangulation such that each vertex is adjacent to at least six triangles, (however, for 2- dimensional dynamical triangulations if we require strictly negative curvature, some care is required-see below). The glueing precedure described above besides being a test for the large links condition (5.53), reflects another basic property of Alexandrov spaces of negative curvature: given any two such spaces, we can glue them together along a common isometric subspace so as to obtain another Alexandrov space of negative curvature. It is also interesting to note that in the smooth case, the removal of the bones {a~-2(Q)}o~a~n(n+l)/2-1 amounts, in dimension n == 2 to have a surface with three puntures, e.g., a sphere with three punctures); in dimension n == 3, a 3-manifold minus a knot, (e.g., arrange the figure-eight knot along the I-skeleton of the tetrahedron a~, and consider the complement of such a knot in §3); and in dimension n == 4, a 4-manifold minus a surface. It is well-known that in all such cases, the resulting manifold can indeed be endowed with a hyperbolic metric of finite volume [108]. Even if the above remarks are very suggestive of the properties of hyperbolic geometry in the appropriate dimensions, there is an important difference as for what concerns dimension n == 2. In dimension n == 2, and for N 2 » 1, the generic triangle a; has at least one vertex a? with incidence q(a?) == 5. Since (5.53) must hold for every choice of the bones a 2 (k), we get that for n == 2, Pn=2(k) ~ 5, and condition (5.53) imply that q(k) on the average should be < q(k) >~ 8. But this cannot hold for a 2-dimensional dynamical triangulation since the average incidence is constrained by b(n, n-2)ln=2 == 6. Thus, in the 2-dimensional case, regardless of the values of the given curvature assignments {q(k)}O~k~A' the above glueing mechanism cannot be realized for every choice of the marked bones in 8(15. This result has a simple geometrical intrepretation, that is worth discussing. Consider a complete connected and orientable surface M with Gaussian curvature == -1. According to the Gauss-Bonnet theorem [37], the finiteness of Vol(M) implies that

Vol(M) == -27rX(M),

(5.55)

5.2 Distinct Dynamical Triangulations with Given Curvature Assignments

111

where X(M) denotes the Euler characteristic. This quite rigidly fixes the volume, also to the effect that there are only finitely many topologically different surfaces M with given volume. The situation is quite different when M is approximated by a dynamical triangulation. In such a case, the DehnSommerville relations imply that

N 2 == 2No - 2X(M),

(5.56)

and the volume Vol(M) ex: N 2 loses the above topological meaning. Obviously, this is simply due to the fact that for a dynamically triangulated surface, when IX(M)I 2:: 1, the (Regge) curvature cannot be normalized to -1. On the average, curvature is zero, and on a dynamically triangulated surface there is always a rather large number of bones supporting positive curvature. The situation is rather similar to what happens for a smooth 2sphere, §2, where there is no way of eliminating positive curvature, since if sup K(§2) is the upper curvature bound, and diam(§2) the diameter, then we have (5.57) for every riemannian metric on §2, [33]. It is quite interesting to remark that an obstruction like (5.57) does not hold for the 3-sphere, since almost negatively curved metrics can exist on the 3-sphere. As a matter of fact it is possible to prove (Gromov, see [33] for a simpler argument) that for all E > 0 there exists a Riemannian metric on §3 with diameter diam(§3) and upper sectional curvature bound sup K(§3) satisfying sup K(§3) . [diam(§3)]2 ::;

E.

(5.58)

If even the 3-sphere, (but the same is true for n == 4), which cannot admit metrics of strictly negative sectional curvature, (by Hadamard's theorem), fails to do so only for the presence of a very small amount of positive cur-

vature, it does not came as a surprise that there are no particular simplicial obstructions to the onset of almost-negative curvature for dynamical triangulations for n 2:: 3. Explicitly, for n 2:: 3, the generic simplex ai has bones af-2 whose incidence number can all attain the lower bound q(af-2) == 3. Thus, in such a case, Pn>3 2:: 3, and condition (5.53) implies that q(k) on the average should be < q(k) >2:: 4, which is well in the allowed range for the corresponding b(n, n - 2). The above analysis shows that if enough negative curvature is present, we can, at lest for n 2:: 3, shorten the radius R of the typical ball of given volume. This property is reminescent of the well known fact that the typical radius of a ball of volume V behaves as (V/w n )l/n if curvature is positive, (w n being a constant related to the volume of the unit sphere), whereas it behaves like n~l In(V/w n ) when curvature is negative. By exploiting the glueing mechanism associated with the curvature condition (5.53), we can activate a rather effective subadditive argument.

112

5. Curvature Assignments for Dynamical Triangulations

Let {T~i) }curv denote the set of dynamically triangulated manifolds with given curvature assignments {q(k)}O~k~A satisfying the condition (5.53). For the generic T~i)\{lTg-2(a)}o~o:~n(n+I)/2_I in {T~i)}curv, consider the set of

triangulated balls {T~)[V == VI + V2 ; R]}curv. Then, for any choice of the curvature assignments in the given set {q(k)}O~k~A' we have (i)

Card{TB [VI

+ V2 ; R]}curv ~

] IAut(f)1 [ Card{TB(i) [VI; R]}curv ] x [ Card{TB(i) [V2 ; R]}curv,

(5.59)

i

since Card{T1 )[VI + V2 ; R]}curv certainly contains balls, like B(3), obtained by glueing along alTo smaller balls of the same radius R. Note that the relation (5.59) simply express (at the level of geodesic balls) the basic property of Alexandrov spaces of negative curvature of being closed under connected sum along isometric subspaces. As is easily verified, this property does not hold for Alexandrov spaces with positive curvature. Relation (5.59) shows that, for fixed R,

-In[Card{TB(i) [V,. R]}curv]

(5.60)

is subadditive with respect to the variable V under the curvature condition (5.53). In such a case, by considering formally the limit V ~ 00 we get, by subadditivity,

J~oo ~ (-In[Card{T~)[V;R]}curvl) = i{}f

~ ( -In[Card{T~) [Vj Rllcurvl) ~ m(R, N n ; {q(k)}),

(5.61)

where the function m( R, N n; {q( k) }) depends explicitly from the radius of the ball R, the total simlicial volume of the underlying manifold N n , and the given set of curvature assignments {q( k) }O
== V*.

(5.62)

Note that the triangulations in {T~i) }curv which give rise to simplicial balls {T1i )[V*; R]}curv, constrain a smaller number of simplices to be in B(lTo; R), with respect to the triangulations in {T~i)}curv which give rise to the simplicial balls {T~) [V; R]}curv. Thus,

Card{T~)[V;R]}curv ~ Card{T~)[V*;R]}curv,

(5.63)

and it follows that the number of simplicial balls Card{T~) [V; R]}curv is a decreasing function of V. This implies that

m(R,Nn;{q(k)}) 2: o.

(5.64)

5.2 Distinct Dynamical Triangulations with Given Curvature Assignments

113

Geometrically, m(R, Nn ;{q(k)}) -1 is the typical volume of the geodesic ball B(i)(a~; R) in {TJi)}curv, viz., most of the triangulations in {TJi)}curv have Vol[B(i)(a~; R)] ~ m(R, N n ;{q(k)} )-1.

(5.65)

In order to discuss the dependence of m(R, N n ; {q(k)}) from the radius R of the ball, consider the set of triangulations {TJi)}curv for which

Vol[B(i)(ag;R+ 1)] == ~

(5.66)

In the large V limit, this set contains also all simplicial balls {T~) [V*; R]}curv for every V* :::; V. It follows that, for 1 :::; R < 00, we have (i)

(i)

Card{TB [V + €; R + l]}curv 2:: Card{TB [V; R]}curv, for any 0

~

€

~

1. Since

E

m(R + l,-Nn ;{q(k)})

(5.67)

can be arbitrarily small, we get that ~

m(R, Nn ;{q(k)}),

(5.68)

namely, m(R, N n ;{q(k)}) is a non-increasing function of R. According to the geometrical meaning of m(R,Nn;{q(k)}), this growth property simply expresses the obvious fact, that for any triangulation in {TJi)}curv, the typical volume m(R, N n ;{q(k)} )-1 of the ball B(i) (ao; R) is smaller than the typical volume m(R + 1, N n ;{q(k)} )-1 of the ball B(i) (a o ;R + 1). It is also possible to relate m(R,Nn;{q(k)}) to the distribution of the diameters of the triangulations in {TJi)}curv. To this end, recall that for a geodesic ball B(i)(ao ;R), whose radius equals the diameter of the underlying manifold M, we have that Vol[B(i)(O"o;R)] == Vol(M). Thus, for a given R, and V ---7 N n , the number of simplicial balls Card{T~) [V == Nn ; R]}curv enumerates the distinct triangulations {TJi)}diam(T)~2R in {TJi)}curv whose diameter diam[T~i)] is not-greater then 2R, i.e., diam[T~i)] ~ 2R, (the factor 2 comes from the existence of balls of radius R that can be obtained by glueing two smaller balls of radius R, like B(3); in that case the diameter of the resulting ball is 2R). (Actually, in place of the diameter, it would be more convenient to use the radius, rad(M), of the manifold M. This is defined by the smallest R > 0 so that M == B(a n , R) for some an E T~i). In general one has rad(M) ~ diam(M) ~ 2rad(M), [69]). Since

-- Nn,. R]} curv - e-m(R,Nn;{q(k)})Nn {T(i)[V B f'.J

(5.69)

and m( R, N n; {q( k) }) is a not-increasing function of R, it follows that {T~i) [V == N n ;R]}curv is a not-decreasing function of the diameter, and to leading order Card{T~i)}curv ~

lim e-m(R,Nn;{q(k)})Nn . R-HXJ

(5.70)

114

5. Curvature Assignments for Dynamical Triangulations

Note that under the curvature condition (5.53), m(R,Nn;{q(k)}) is not identically zero. As follows from the geometrical construction described above, (see (5.49)), there are triangulated balls of the form

TB(3) ~ (TB(l) - o-~(l)) Uf (TB(2) - o-~(2)),

(5.71)

for which we have the functional relation

In[Card{T~)[Vl + V2 ; R]}curv]

=

In IAut(f)l+

In[Card{TB [V1 ; R]}curv] + In[Card{TB [V2 ; R]}curv]. (5.72) Up to the constant, (n-dependent) factor IAut(f)1 which disappears in the large Nn-limit, this equation admits continuous solutions of the form In[Card{T~)[V; R]}curv] = pV, for some constant p, (as is well kown, (5.72) admits non-measurable solutions as well, but these are typically excluded from consideration). Thus m(R, Nn ; {q(k)}) is not trivially O. In order to estimate the N n dependence of m(R, N n ; {q(k)}) recall that m(R, N n ; {q(k)}) provides, for large V, the typical volume of the geodesic ball B(i)(ao; R) in {T~i)}curv. Since the (simplicial) volume of the underlying manifold (M = IT~i) I, T~i») is given by N n , (up to the usual a-dependent factor), we can interpret (i)

(i)

Vol(M) m(R,Nn;{q(k)})Nn = Vol(B(i)(a~;R))'

(5.73)

as providing the typical number of balls B(i) (ao; R) that can be packed into M. Since we work in a space of bounded geometries, according to BishopGromov theorem [37], the ratio (5.73) is bounded above by the ratio

Vol[BH(r

= diam(T~i»))]/Vol[BH(r = R)],

(5.74)

where H is the lower bound to the curvature ofT~i), and BH(r) is the geodesic ball of radius r in the complete simply connected Riemannian manifold with constant curvature H. According to the standard estimate for the volume of the hyperbolic ball of given radius, we Can write

Vol(M) m(R,Nn;{q(k)})Nn = Vol(B(i)(a~;R)) :::; exp[(n - l)(diam(T~i») - R)v1Hf],

(5.75)

(we have assumed that H < 0). For [(n-1)(diam(T~i»)-R)v1Hf] sufficiently small, we can write, to leading order,

m(R, N n ; {q(k)} )Nn ~ (n - l)(diam(T~i») - R)v1Hf.

(5.76)

When H < 0, (and even not very large with respect to N n ), we have triangulations in {T~i)}curv where most of the simplices are inside a simplicial ball of small radius, (e.g., it can be shown that every triangulable 3-manifold can be

5.2 Distinct Dynamical Triangulations with Given Curvature Assignments

115

triangulated so that the closed star of a bone contains all the vertices, [111]). In such a case, we can parametrize the (simplicial) diameter of the resulting PL-manifold according to 1

diam(T(i)) NnH, aex:n

(5.77)

for some nH 2:: n, (note that a word of caution is in order here since the simplicial diameter is distinct from the diameter measured along edge-paths, the former being in general larger than the latter). Inserting this parametrization in (5.76) we can write --1--1

m(R,Nn;{q(k)})~An(R,H)N;:H

,

(5.78)

for a suitable constant An(R, H) depending on the dimension n, the cut-off a, the curvature bound H, and R. The expression (5.78) provides the required N n dependence of m(R, N n ; {q(k)}). Geometrically, m(R,Nn;{q(k)}) > 0 implies that for fixed V == Nn, the configurations with large diameter are damped (exponentially) with respect to configurations in which the diameter is as small as possible. As already recalled, the reason for such a behavior is that, for dimension n ~ 3, even relatively low negative curvature for the bones associated with the marked simplex 0'0, implies that the number of simplices incident on the vertices of 0'0 may be very large. In this way, the typical volume of the ball B(i)(a R), for R sufficiently small, is a significant fraction of N n . Typically, most of this volume is provided by the marked Yo, (compare with [36]; we thank S. Catterall for drawing our attention on this reference which, to some extent, inspired the present proof of theorem 5.2.1). Roughly speaking, in this situation, a relatively small fraction of simplices is left for triangulating the complement of the marked ball, and the number of distinct triangulations

o;

correspondingly drops down very fast. Thus, if we allow for enough negative

curvature (a property which depends from the given curvature assignments {q(k)}, according to (5.53)), Card{T~i)}curv is exponentially damped. One wonders what there is behind this behavior of {T~i)}curv: in other words, what are the riemannian manifolds Gromov-Hausdorff approximated by the above type of triangulations? In order to provide a tentative answer to such a question, we recall, (see e.g.,[64]), the following characterization of almost fiat manifolds. A compact manifold M is almost flat if there is a sequence of Coo Riemannian metrics {gj} on M such that {(M.g j )} converges, in the Gromov-Hausdorff sense to a point in such a way that the metrics {gj} have a uniform bound on the absolute value of the sectional curvature. By suitably scaling the metric, this characterization is basically equivalent to (5.58). Namely, there is, for each E > 0, a metric gf on M such that ISec(M,gf)ldiam(M,gf)2 ~

E,

(5.79)

where Sec(M,gf) denotes (all) sectional curvatures of (M,gf), (note that there are almost flat manifolds that do not admit any flat metric). At this

116

5. Curvature Assignments for Dynamical Triangulations

point, we might conjecture that the positivity of m(R, N n ; {q(k)}) yields for a dominance of such almost-flat manifolds in {T~i)}curv. As a matter of fact, as long as m(R, N n ; {q(k)}) > 0, in the infinite-volume limit of our dynamically triangulated model the dominating configuration will exactly be PL- manifolds of very small spatial extension. In such a phase of the theory, (which corresponds to strong coupling), there is no naive way of taking a scaling limit because every vertex of the generic triangulation is connected to another vertex by an edge-path of few units. Perhaps, a more sophisticated scaling, as the one suggested by (5.79), may help in understanding this rather pathological phase of simplicial quantum gravity, (for a more detailed description see Sect. 6.7). There is also another related explanation of the m(R,Nn;{q(k)}) > 0 induced damping, related to the fact that, contrary to what happens in dimension n == 2, (where a surface generally supports a continuum of complete metrics with curvature == -1), higher dimensional hyperbolic manifolds may have no non-trivial deformations. Mostow's rigidity theorem [117] is typical in this situation. For example, there are no deformations of a compact hyperbolic manifold M when n ~ 3, (if n ~ 4 it is sufficient to have V ol(M) < (0). This rigidity strongly supports the damping in the number of distinct triangulations when the simplicial curvature is negative and nearly constant. The condition m(R,Nn;{q(k)}) > 0 does not hold for every possible choice of the curvature assignments {q( k) }. It is not difficult to prove that if the curvature condition (5.53) does not hold, and positive curvature dominates, i. e., if 1

inf {21r - q(k) cos-I -} > 0, n

q(k)

(5.80)

then we cannot have configurations satisfying (5.59) while mantaining the constraint (5.80). Assume to the contrary that also in this case (5.59) holds. Choose VI == V2 ex wnRn, where W n is the volume of the unit ball in jRn. Then the simplicial balls in {T~)[VI;R]}curv and {T~)[V2;R]}curv saturate Card{T~i)[VI;R]}curv and Card{T~)[V2;R]}curv, since under the positive curvature constraint the volume of geodesic balls of radius R is bounded above by the corresponding volume of the ball on §n, (by Bishop- Gromov's theorem [37], [59]). On the other hand, if VI == V2 ex wnRn, positive curvature is already saturated by the balls of volume VI and V2, and for given R we cannot have any geodesic ball of radius R with V == VI + V2, thus correspondingly Card{T~i)[VI + V2;R]}curv ---7 0, and (5.59) cannot hold. This is simply a manifestation of the geometric fact that, contrary to what happens for Alexandrov spaces of negative curvature, Alexandrov spaces of positive curvature are not closed under connected sum along isometric subspaces. Rather than (5.59), under the positive curvature constraint (5.80), we have (i)

Card{TB [VI

+ V2 ; R]}curv ~

5.2 Distinct Dynamical Thiangulations with Given Curvature Assignments (i) [ Card{TB

[VI; R]}curv ]

X [ Card{TB(i)

[V2;R]}curv ] .

117

(5.81 )

This implies that, for fixed R, (i)

In[Card{TB [V; R]}curv]

(5.82)

is subadditive with respect to the variable V under the curvature condition (5.80), (whereas -In[Card{T~)[V; R]}curv] was the subadditive quantity under the curvature condition (5.53)). Since in any case Card{T~)[V; R]}curv is a decreasing function of V, it follows that under the positive curvature constraint we must necessarily have

m(R, N n ; {q(k)})

= O.

(5.83)

Thus, the large volume effects (at fixed, small diameter), are entropically suppressed under positive curvature constraint (5.80). Card{T~i)}curv, is no longer damped by the clustering of most simplices within the marked B(i)(ao; R = 1). The number of triangulations {T~i)}curv correspondingly increases, and is controlled by a polynomial (in V) asymptotics of the form Card{T~i) }curv ~ J-t( {q(k)}) . N~( {q(k)}),

(5.84)

asNn goes to infinity, and where J-t({q(k)}) and T({q(k)}) are constants depending on the given set of curvature assignments {q( k)}. Note that when {q(k)} is such that b(n, n - 2; {q(k)}) = b(n, n - 2)min, (5.80) trivially holds. Similarly, when {q(k)} is such that b(n, n - 2; {q(k)}) = b(n, n - 2)max, curvature can be sufficiently negative so that (5.53) holds. Thus, as we take the average of Card{TJi)}curv={q(k)} over the set of possible curvature assignments {q( k) }O
b(n, n - 2)min ::; b(n, n - 2; {q(k)}) < bo(n),

(5.85)

then, as N n ~ 00, the asymptotics of the average < Card{T~i)}curv > is of polynomial type. Whereas, if

bo(n)

~

b(n, n - 2; {q(k)})

~

b(n, n - 2)max,

(5.86)

< Card{T~i)}curv > is sub-exponentially damped.

Explicitly, by taking the {q(k)}O
< Card{TJi)}curv >~< e-m(R,Nnj{q(k)})Nn >~ e-m(b(n,n-2))N~/nH '5.87) (viz" m(b(n,n - 2)) is the logarithmic average of An(R,H) in (5.78)). Similarly, by taking the average of (5.84), we can define the constants J-t(b(n,n - 2)) and T(b(n,n - 2)), to the effect that we have the following generic behavior of < {T~i)}curv >. There are constants, J-t(b(n, n-2)) > 0, T(b(n, n-2)) ~ 0, and m(b(n, n2), with m(b(n, n - 2))ln=2 = 0, m(b(n, n - 2))ln~3 = for b(n, n - 2)min ~

°

118

5. Curvature Assignments for Dynamical Triangulations

b(n, n - 2) ~ bo(n) and m(b(n, n - 2))ln~3 > 0 for b(n)o < b(n, n - 2) < b(n, n - 2)max, such that

< Card{T~i)}curv >~ J-L(b(n, n - 2)) . e-m(b(n,n-2))N~/nH N~(b(n,n-2){5.88) as N n goes to infinity, as stated. It is not difficult to prove that the critical value bo(n) is located sufficiently near the lower kinematical bound b(n, n-2)min. Let b(n, n-2)min ~ b(n, n2) ~ bo(n), and let {q(k)} be a sequence of incidence numbers such that L~=o q(k) = bo(n)('x + 1). For any {3 such that 0 < (3 < bo(n), at least

(bo(n) - (3)('x + 1) sup q(k) - {3

(5.89)

of the q(k) are greater than (3. If 21r bo(n) > , - cos- 1 (1/n)

we could choose 21r .B = [ cos- 1(1/) n lint,

(5.90)

(5.91 )

where [·]int denotes the integer part of the expression in parenthesis. In such a case, most, (i.e., O(,x)), of the incidence numbers q(k) are larger than [COs-~(l/n)lint, a configuration this latter corresponding to the dominance of negative curvature, and which should not occur under the stated assumption b(n, n - 2)min ~ b(n, n - 2) ~ bo(n). Such a dominance of negative curvature occurs unless bo(n) is sufficiently near to the lower kinematical bound b( n, n2)min. It turns out that the bounds on < Card{T~i)}curv > provided by theorem 5.2.1, even if quite rough, are sufficient for our purposes. In particular, when the distribution of p~urv enhances those configurations for which b(n, n - 2) is nearby to its lower bound b(n,n - 2)min, (e.g., b(n,n - 2)min = 4 for n = 4), it follows that < Card{T~i)}curv > can, at worst, give rise to a subleading polynomial correction to the counting associated with the curvature assignments. Whereas, for b(n, n - 2) is nearby to its kinematical upper bound b(n,n - 2)max, (e.g., b(n,n - 2)max = 5 for n = 4), it follows that < Card{T~i)}curv > can, at worst, give rise to a subleading exponential correction of the form exp( -m(b(n, n - 2)N~/nH) to the counting associated with the curvature assignments. This implies that through the simple counting of the curvature assignments we can get very accurate results as far as a characterization of the infinite volume limit is concerned, (since this limit is characterized by the leading exponential asymptotics of the triangulation counting). Less accurate result should be expected as far as the characterization of the nature of the possible critical points of the theory, since these latter require a sharp location of the value bo(n) corresponding to which

5.2 Distinct Dynamical Triangulations with Given Curvature Assignments

119

< Card{T~i)}curv > changes its asymptotic behavior. As we shall see, this interplay between the asymptotics of p~urv and < Card{T~i)}curv > charac-

terizes quite effectively the phases of simplicial quantum gravity.

Fig. 5.3. The geometrical set-up for the characterization of the asymptotics of {TatC) }curv

Fig. 5.4. If enough negative curvature is present, by glueing two distinct balls in {Tli)}curv, both of radius R and of respective volumes VI and V2, we get a well defined ball of the same radius and of volume VI + V2 in {T~i) }curv

120

5. Curvature Assignments for Dynamical Triangulations

5.3 The Counting Principle The exponential bounds obtained in the previous section establish the existence of the model of dynamical triangulations in dimensions higher than two. Moreover, the structure of such bounds suggests that we can actually shapen such bounds in asymptotic estimates which can be profitably used to deduce a number of properties of our geometrical model. The possibility of actually providing an asymptotic enumeration of the distinct triangulations {T~i)} rather than just a bound, rests on theorem 5.1.1. In particular, the subadditivity (5.28) implies that the ratio

L

curv Q(O) P>.. . LQ(o) P>..

(5.92)

= en,

where 0 ~ Cn ~ 1, either is a constant or an exponential monotonically decreasing function of ,.\, (depending on b(n, n - 2) through inf b(n, n2) and supb(n,n - 2)). This monotonicity property suggests that the functional dependence of Cn on ,.\ and on the range of variation of b(n, n - 2) can be conveniently recast in the form

L

curv Q(O) P>..

~ en

= CoeanNn(Ta)+an-2Nn-2(Ta) ,

(5.93)

LQ(o)P)..

where now the parameters 0 < Co ~ 1, Q n ~ 0, and Qn-2 ~ 0 may depend only on the dimension n, and the range of variation of b(n, n - 2), (i.e., on inf b and sup b). Obviously we do not know the exact expression for such parameters, how-

ever since Cn has exactly the structure of the exponential of the dynamical triangulation action, and since LQ(o) p~urv basically enumerates distinct dynamical triangulations, we can use for enumerating distinct dynamical triangulations the much more manageable counting function LQ(o) p)... This procedure requires that in the final results we allow for a renormalization of the couplings kn - 2 , and kn , according to

kn kn -

2

~

~

+ c5n , kn - 2 + c5n - 2 , kn

(5.94)

with c5n ~ 0, and c5n - 2 ~ 0 constants depending only on the dimension n. From a physical point of view it may appear that the above assumptions underlie the construction of a mean field theory such as the one suggested in[25], (we are grateful to A. Krzywicki for a careful criticism in this connection). In other words assuming that the field governing the dynamics of our ensemble of triangulations is associated with the fluctuations of p).. rather than of p~urv may seem a way of averaging out some degrees of freedom. As a consequence one may cast doubts as to the reliability of the formalism, in particular when assessing the nature of the critical points. Indeed, a priori, urv p1 may fluctuate, as a function of b(n, n - 2), more wildly than p).., and in

5.4 A Remark on SU(2) Holonomy

121

that case the critical points associated with the distribution p~urv would be of a different nature as compared to those associated to the distribution p)... But in our case, as follows from subadditivity, (5.28), there is not such a different spectrum of fluctuations, since the ratio of the distributions p~urv and p).. varies monotonically with A in the large A limit. (As already remarked, what affects the nature of the critical point is not the use of p).. in place of p>..urv but rather the asymptotics of Card{T~i)}curv). Actually, by using p).. as an enum.erator, we are using a discrete measure on a space (the space of distinct partitions of the integer [nQ(O) - b + q](A + 1) - n(n - l)Q(O)) which contains the relevant space of curvature assignments, but which is definitely larger. The discrete measure associated with p).. induces a corresponding measure, p>..urv on the space of distinct dynamical triangulations, and the two measure scales monotonically according to (5.93), (which appears as a the discrete Radon-Nykodim derivative of the measure p~urv with respect p)..). The nature of this scaling allows us to use directly the overcounting measure p).. since in seeking the infinite volume limit and a possible scaling limit for the partition function (3.23) we have to vary both couplings kn , k n - 2 . Roughly speaking, in using directly p).. in place of p>..urv it is more or less like including tadpoles and self-energies in the triangulation counting. The shift (5.93) and (5.94) allows us to remove, in the final results, the effect of such inclusions. We formalize the results of the lemma 5.1.1, and theorems 5.2.1, 5.1.1, in the following lemma providing the rationale of our counting strategy, Lemma 5.3.1. There exists constants an :::; 0, a n-2 :::; 0, depending only on the dimension n, inf b(n, n - 2), and sup b(n, n - 2), such that the number of distinct dynamical triangulations T a E VTn(a, b, A) is enumerated, up to the scaling Card{T~i)} (i)

< Card{Ta }curv > ·l:Q(O) P)..

~

en = eanNn(Ta)+an-2Nn-2(Ta) ,

(5.95)

by l:Q(O) p).., where p).. is the set of distinct partitions of the integer [nQ(O) b+q](A+1)-n(n-1)Q(0) into at most A+l-n parts, (each:::; (b-q)(A+l)).

5.4 A Remark on 8U(2) Holonomy According to the above results, curvature assignments generically characterize the leading asymptotics in enumerating dynamical triangulations. In the 2-dimensional case this can be related to the fact that curvature assignments provide the rotation matrices defining the holonomy, and thus the triangulation. Also for dimension n == 3, and n == 4 curvature assignments provide most of the information characterizing the holonomy for dynamical triangulations. This remark may suggest a bridge between dynamical triangulations

122

5. Curvature Assignments for Dynamical Triangulations

and the connection formalism based on the Ashtekar variables, a formalism which eventually hints to the existence of a discrete underlying structure of quantum gravity. Here, as a first elementary step in this direction, we work out in some detail the connection between 8U(2) holonomy and curvature assignments for 3-dimensional dynamical triangulations. The case for dimension n = 4 can be easily discussed along the same lines by exploiting the isomorphism 80(4) ~ SU(2~2SU(2). This analysis is mainly due to J. Lewandowski

[61], to whom we wish to express special gratitude. We start by introducing the following

Definition 5.4.1. A simplicial loop v, based at a simplex 0'0, is a (small) lasso with nose at the bone B, if it can be decomposed into three simplicial curves v(B) = 7- 1 . w(B) ·7, where 7 is a simplicial path from 0'0 to one of the simplices an(B) containing the bone B, 7- 1 is the same curve going backward, and w(B) is the unique loop winding around the bone B. Since the open stars of the bones provide an open covering of M = ITa I, any simplicial loop in M, if it is homotopic to zero, is a product of lassos whose nose is always contained in the open star of some bone B, (this is the Lasso lemma [21]). Thus, in order to evaluate the holonomy around a generic contractible simplicial loop based at a marked simplex 0'0 we can proceed as follows. Let us fix a holonomy representation of the fundamental group of M, (): 1f1 (M) ~ G ~ End(g). Recall that the choice of such () calls for the marking of a bone ag- 2 , (whose barycenter provides the basepoint Po), of a simplex

0'8'

:J a~-2, and of the holonomy around the marked bone. As

the notation suggests, we naturally identify the marked simplex 0'0 with the simplex on which the simplicial lassos are based. Let us consider the set of simplicial lassos, {v(a),ao} based at the marked 0'0 and with their noses corresponding to the ,\ + 1 bones Bo,B(l), . .. ,B('\). The effect of parallel transporting, according to the Levi- Civita connection associated with T a , along any of these lassos is a rotation in a two- dimensional plane, (orthogonal to the bone considered). If w is the generic (contractible) simplicial loop based at 0'0, then the holonomy along the generic loop w based at 0'0, can be written as A

Rw(ao) =

II {A(a)-1 R(m(a)¢(a))A(a)},

(5.96)

0=0

where m(a) E Z are the integers providing the winding numbers of w around the various bones, R[m(a)¢(a)] denote the rotations in the planes orthogonal to the bones B(a), and finally, A(O), A(a) denote the orthogonal matrices describing the parallel transport along the ropes, 7 0 , of the lassos, (A(O) is associated with the trivial path 0'0)' On rather general grounds, it is known[14] that through holonomy we can reconstruct the underlying connection, (up to conjugation). In our case, as

5.4 A Remark on 8U(2) Holonomy

123

already stressed, this reconstruction procedure is particularly relevant since distinct connections are in correspondence with distinct triangulations. It follows that the conjugacy class of the (restricted) holonomy Rw(O"o) is Ginvariant and fully describes the underlying dynamical triangulation in the given representation 8:1fl(M,po) --? G. The natural class functions of the holonomy (and hence of the triangulation) are linear combination of the group characters, namely of the traces of Rw(ao), (these latters being thought of as elements of the fundamental representation of the restricted holonomy group). These traces depend from the given set of incidence numbers {q(a)} of the triangulation, and a question of relevance is to what extent such traces, and hence the curvature assignments, characterize the triangulation itself. In order to work out the 3-dimensional case in the SU(2) connection, let us consider the marked simplex 0"0, (in this case a tetrahedron), and fix an orthonormal basis {ex,y,z} in 0"0' Among the six bones (edges) in the boundary of 0"0, we can choose !n(n - 1) = 3 bones, Ti' i = 0, 1, 2, one of which is the marked one, TO, and all sharing a common vertex, (recall the need of marking n bones in interpreting (5.4) in terms of partition of integers!). These bones are defined by the vectors E( Ti) E ]R3, (with components referred to the fixed orthonormal basis {ex,y,z}). Set

Ex (Ti)

Ti =

ax IE(ri)1

Ey(Ti)

(

Ez(Ti)

+ a y IE(ri)1 + a z IE(ri)I '

5.97

)

with o"x, O"y, a z denoting the Pauli matrices. The holonomy matrix describing the 2- dimensional plane rotations generated by winding around the bones Ti can be written as the SU(2) matrix

U(Ti) = I cos(¢(i)j2) + Ti sin(¢(i)j2),

(5.98) cos- 1

where I is the identity operator, and ¢(i) = q(i) ~ denotes the rotation angle around the bone Ti. In general, we can reconstruct the holonomy matrix U(Ti) for every distinguished bone Ti in the marked base simplex ao, by giving the traces of U(Ti), and the traces of U(Ti)U(Tj), i =1= j. The traces tr[U(Ti)] are proportional to the incidence numbers q(i), i = 0,1,2, while tr[U(Ti)U(Tj)] are proportional to the scalar product in SU(2) between the spinors T i and T j, respectively associated to the bones Ti' Tj. These latter data are actually already known, since the base simplex is equilateral. Thus the SU(2)-holonomies around the base simplex 0"0 are trivially determined once we give the corresponding curvature assignments. The reconstruction of the 8U(2)- holonomies around the remaining A + 1 - !n(n + 1) = A - 5 bones T a = {0"~-2,0 < a < A,0"~-2 ¢ ao}, is similar. Formally, for such a reconstruction, we would need the assignment of the traces x(a) = tr[U(Ta )], and of x(a,j) = tr[U(Ta)U(Tj)], with j = 0,1,2. Indeed, with the characters x(a) = tr[U(Ta )] = 2 cos 4>~) we can reconstruct the rotation angles ¢(a), while with the characters x(a,j) = tr[U(Ta)U(Tj)], viz.,

x(a,j)

= 2 (cos ¢~a) cos ¢~) - < To.,Tj > sin ¢~) sin ¢~»),

(5.99)

124

5. Curvature Assignments for Dynamical Triangulations

we can reconstruct the SU(2) inner products

< To:,Tj

>~ -~Tr[To: ·Tj],

(5.100)

and hence obtain the spinors T a through which the SU(2)-holonomies U(Ta ) = I cos(¢(a)/2) + T a sin(¢(a)/2) are defined. The characters x(a) = tr[U(Ta )] with a = O,l, ... ,A are given (up to a constant factor) by the incidence numbers {q(a)}~=o, and thus are known. Since the simplices in the triangulations are all equilateral, the bones T j are not arbitrarily distributed with respect to the reference bones T a, but are in correspondance with the 12 points on the unit sphere §2 marked by the vertices of the inscribed Icosahedron, (arrange the icosahedron in space so that the uppermost vertex coincides with (0,0,1)). Hence, the SU(2) inner products < Ta,Tj > are known, and the characters x(a,j) = tr[U(Ta)U(Tj)] are given in terms of the {q(a) }~=o. It follows that the curvature assignments {q(a)}~=o provide grosso modo the SU(2) holonomy matrices, (up to conjugation), what is missing at this detailed level, (which is independent from any asymptotics), is the combinatorial data of how to put together the local holonomies so as to reconstruct the underlying dynamical triangulation Ta . As proved in the previous section, such a piece of information is far from being trivial being related to the enumeration of {T~i)}curv. This latter set of triangulations may then come into play through a set of suitable Mandelstam identities on the q(a)-induced SU(2) holonomies.

6. Entropy Estimates

According to the results of the previous sections, dynamical triangulations are to a large extent characterized by the corresponding curvature assignments {q( a) }, and the simple counting of the possible curvature assignments provides an estimate, to leading order, of the number of distinct dynamical triangulations a PL manifolds (with given fundamental group) can support. Our strategy is to further exploit the close connection between the theory of partitions of integers and the curvature assignments in order to estimate the leading asymptotics of the number of distinct dynamical triangulations. Since also the subleading asymptotics can be parametrized in a precise way according to theorems 5.2.1, 5.1.1 and lemma 5.3.1, we can get in this; way a rather reliable description of the entropy estimates of relevance in simplicial quantum gravity.

6.1 The Asymptotic Generating Functions for the Enumeration of Dynamical Triangulation Since the partitions of [nQ(O) - b(n, n - 2) + q](A + 1) - n(n - l)Q(O) into at most A + 1 - n parts, (each ~ (b(n, n - 2) - q)(A + 1)), can be thought of as enumerating, in the large A limit, distinct curvature assignments, theorem 5.2.1 implies that they also asymptotically enumerate distinct rooted dynamical triangulations. This observation establishes the following Theorem 6.1.1. Let 8: 1f1 (M; po) ~ G, be a given holonomy representation of an n-dimensional PL-manifold M, n ~ 2. The number W(8, A, b, Q(O)), of distinct rooted dynamical triangulations on M, with A + 1 bones and with a given average number, b~b(n,n - 2), ofn-simplices incident on a bone, is given, to leading order in the large A limit, by W(8, A, b, Q(O)) ==

Cn·

< Card{T~i)}curv > x

XPA([nQ(O) - b + q](A + 1) - n(n -l)Q(O)), where

Cn

(6.1)

is the rescaling factor of lemma 5.3.1.

Note that selecting the value Q(O) of the curvature at the n marked bones is equivalent to considering dynamical triangulations with a boundary, (the

126

6. Entropy Estimates

links of the marked bone), of variable volume, (the length being proportional to the chosen Q(O)). Thus we have the Lemma 6.1.1. The function W(8, A, b, Q(O)) provides, to leading order, the

number of distinct dynamical triangulations with boundary 8Ta consisting of the disjoint union of n = spheres of dimension n - 1, Sn-I, each one of volume Vol[8Ta ] = q(O)vol[a n- I ]. Proof. This trivially follows from the above theorem by noticing that by removing from Ta the open star of the marked bones, (on each of which q( 0) simplices {an} are incident), we get dynamical triangulations with a boundary 8Ta ~ II sn-l of volume Vol[8Ta ] = nq(0)vol[a n- 1 ].

6.2 Gauss Polynomials and Dynamical Triangulations By exploiting the properties of the partitions PA ([nQ(O) - b+q](A +1) - n(nl)Q(O)) we can actually characterize the asymptotic generating function for the number of distinct rooted dynamical triangulations. Theorem 6.2.1. Let 0

~ t ~ 1 be a generic indeterminate, and let PA(h) denote the number of partitions of the generic integer h into (at most) A+1-n parts, each ~ (b(n, n - 2) - q)(A + 1). In a given holonomy represention 8: 1Tl(M;a ~ G, and for a given value of the parameterb = b(n,n-2), the generating function for the number of distinct rooted dynamical triangulations with N n _ 2 (TJi)) = A + 1 bones and given number

o)

0) _ h + (b - q)(A + 1) n( A + 2 - n)

q(

"

(6.2)

+ q,

of n-dimensional simplices incident on the n marked bones is given, to leading order in A, by Q[W(8, A, b; t)]

= Cn· < Card{T~i)}curv > . LPA(h)t h == h~O

. C d{T(i)} . [(b- q)(A+l)+(A+l) -n] Cn < ar a curv > (A + 1) - n '

where 0 < Cn

~

(6.3)

1 is an exponential rescaling factor, (see lemma 5.3.1), and

n] (1 - tn)(l - tn-I) ... (1 - t rn + I ) . (t)n [ m - (1 - t n- m)(l - t n - m - 1 ) ... (1 - t) - (t)n-m(t)m' is the Gauss polynomial in the variable t.

(6.4)

6.3 Dynamical Thiangulations and oo-Dimensional Operators

127

Note that the Gauss polynomial (t)n~:(t)m is a polynomial in t of degree

(n - m)m, and that

!~ [: ] = m!(nn~m)! = ( :

),

(6.5)

(see[5] for an extensive review of the properties of Gaussian polynomials).

Proof. The proof is a straightforward consequence of the fact [5] that the Gauss polynomials are the generating function of the partitions p.\(h). From these observations it follows that the asymptotic generating function for dynamical triangulations with (spherical) boundaries is provided, up to an exponential rescaling, (see lemma 5.3.1), by a Gauss polynomial of degree (b - q) (A + 1) (A + 1 - n), and we expect that their properties bear relevance to simplicial quantum gravity. It is interesting to remark that these polynomials already playa basic role in the celebrated solution of the Hard Hexagon Model by R.J. Baxter[6].

6.3 Dynamical Triangulations and co-Dimensional Operators The fact that the partitions p.\(h) basically enumerate distinct (rooted) dynamical triangulations has an intriguing consequence related to the theory of Duistermaat-Heckman measure for Hamiltonian actions of the n-Torus on compact symplectic manifolds [72]. Let d(.\+l-n)(.\+l)(b-q) be the simplex in JR A+1- n with vertices at the origin and at the points

(,\ + 1 - n)(~ + l)(b - q) ei,

(6.6)

't

with i = 1, ... ,A + 1 - n, and where the ei's are the standard basis vectors of ~.\+l-n. It is not difficult to show that the sum (.\+l-n)(.\+l)(b-q)

L

p.\(h),

(6.7)

h~O

providing the leading asymptotics of the number of distinct dynamical triangulations (see next paragraph), is given by the number of lattice points in the simplex d(A+l-n)(.\+l)(b-q). This latter simplex is a Delzant polytope [72] , namely is a convex polytope in (JR.\+l-n)*, (* denoting the dual vector space), such that: I There are (A + 1 - n) edges meeting in each vertex p. II The edges meeting in the vertex p are rational: each edge can be written as p + tv- 0 < < 00 - t , with v· E (z(A+l-n»)*. ~,

~

128

6. Entropy Estimates

III VI, ... , V(A+I-n) can be chosen to be a basis of (z(A+I-n))*. It is possible to associate to Ll(A+I-n)(A+I)(b-q) , (henceforth Ll for short), a symplectic manifold XL!, this construction is rather delicate, and would force us to a long detour in symplectic geometry which is not related to the main purpose of these notes, thus we will not explicitly discuss this construction here. However, the basic strategy is rather easy to describe. Let L1 denote a Delzant polytope in (]Rn)* which we may consider defined through equations of the form Ui(X) ~ "1i, i = 1, ... , d, where Ui are fundamentals lattice vectors in ]Rn. Let us consider the map 1r: (]Rd, Zd) ~ (]Rn, zn), defined by associating with the standard basis vectors, el, ... , ed of ]Rd the corresponding primitive lattice vector Ui, i.e., ei ~ Ui. The map 1r induces a map between the associated tori 1r: T

d

~ Tn

(6.8)

with kernel ker(1r) ~ K. By restricting to K the Bamiltonian action of T d on Cd, one gets a Hamiltonian action of K on Cd with moment map

J(ZI, ... ,Zd) = (IZI12, ... , IZdI2) /2,

(6.9)

(actually one is here referring to the pullback i* J, where i: K ~ ]Rd is the inclusion). One sets XL! ~ (i* J)-I(O)/K,

(6.10)

which is a compact symplectic manifold of dimension 2n, with symplectic form vL!, (note that for any Delzant polytope, the symplectic volume equals the Euclidean volume). Moreover (i* J)-I(-i*("1I, ... ,"1d)) ~ XL!

(6.11)

is a principal K-bundle. For details the reader can consult V.Guillemin's book[72]. An example of such a construction is afforded by the symplectic manifold XL! associated with the standard n-dimensional simplex L1n with vertices at the origin and at the points (1,0, ... ,0), (0,1,0, ... ,0), etc.. In such a case, it turns out that XLI =
IT

and the Riemann-Roch number of XL!: R.R.(XL!)

= jeXP[VLl]

IT 1 i=1 - exp Ci Ci (

)'

(6.13)

6.4 Asymptotics and Entropy Estimates

129

Now, let us introduce the differential operator of infinite order

8

r(oh)

=

IT

8i i=l 1 - exp(oi)'

(6.14)

where 8i = 8/ 8h i , and hi are generic indeterminates. In terms of this differential operator, one can rewrite[72]

R.R.(XLl)

= r(:h)

[J

exp ([VLl]

+ ~hiCi)]

Ih=O.

(6.15)

But, Khovanskii[72],[82] has proved that the right member in the above expression is the number of lattice points in L1. According to our analysis, this volume (up to a rescaling factor en < Card{T~i)}curv », provides the number, W(A, b), of distinct dynamical triangulations in a given representation of the fundamental group, and by combining these two results we get that

8 R.R.(XLl ) = W(A, b) = r(8h)Vol(L1(h))lh=o,

(6.16)

establishing the claimed connection among the topology of the symplectic manifold XLl, the enumeration of distinct dynamical triangulation (with given A, and average curvature b(n,n - 2)), and the infinite dimensional operator r(th). The expression (6.16) suggests that the m-point functions, (i.e., the counting of distinct triangulations with m marked vertices at preassigned distances), associated with dynamical triangulations are related to the solutions of formal partial differential equations of the form (6.16). We will not push these remarks any further here, but certainly it is highly plausible that an approach along these lines may provide a deep connection between ndimensional dynamical triangulation theory (n > 2) and symplectic geometry along the lines of the 2-dimensional case[115, 86].

6.4 Asymptotics and Entropy Estimates If in the generating function Q[W(8, A, b; t)] we let t ---t 1 we get the asymptotic enumerator of the distinct dynamical triangulations, for a given representation of the fundamental group. Roughly speaking, for t ---t 1, Q[W(e, A, b; t)] reduces to the sum over all possible curvature assignments q(O) in the set of distinct dynamical triangulations in the given holonomy representation. We have the following

Theorem 6.4.1. In a given represention 8: 7fl(M; aD) ---t G, and for a given value of the parameter b = b(n, n - 2), the number W(e, A, b) of distinct dynamical triangulations with Nn_2(T~i)) = A + 1 bones, n of which are marked, is given, in the large A limit, by W(8,A,b) =

130

6. Entropy Estimates

(')t ( (b - q)(A + 1) + (A + 1) - n ) en" < Card{Ta }curv > " (A + 1) - n "

(6.17)

Proof. The sum limt-+l Q[W(8, A, b; t)] involving the partions p,\(h) can be evaluated by exploiting the property (6.5 ) of the Gaussian polynomials, viz., limQ[W(8,A,b;t)] == t-+l (,\+ I-n)(,\+ l)(b-q)

L

en· < Card{T~i)}curv > .

p,\(h)

h~O

en

. < C d{T(i)} ar

a

curv

> . ( (b - q)(A + 1) + (A + 1) (A + 1) - n

n)

'

(6 18) .

establishing the desired result. On applying Stirling's formula, we easily get that the above result yields, for large A, the asymptotics W(8;A;b)

~

en < Card{Ta(')'" }curv > (b-q+l)1-2n [(b_ q +1)b- q+l]A+1 _! ~

b-

q

(b _

q)b- q

A

(6.19) In order to obtain from (6.19)the asymptotics of the number of distinct dynamical triangulations Ta , with A + 1 bones, we have to factor out the particular rooting we exploited for our enumerative purposes. We have first of all to factor out the marking of the n bones a n- 2(O), a n- 2(1), ... ,an- 2(n-l), and the marking of the base simplex aD. To factor out the marking of the n bones, we have to divide (6.19) by An. There can be at most [(b(n,n~2)-q)](,\+ 1) simplices an sharing one of the marked bones {a n- 2 ( i) }i=O,... ,n-l, (actually the actual count is [(b(n,n~2)-q)](A + 1) where ij can be larger than q; we use q for simplicity). Since the marked simplex a'O can be incident on any of the n marked bones, we still have to divide by n. For A » 1, we get in this way a normalization factor [b(n, n - 2) - q]An+1 . We have also marked an orthogonal representation eo of 7rl(M;po) corresponding to the marked base simplex aD' and as discussed in section IV, besides the above purely combinatorial factors, we have to divide (6.19) also by the volume of the moduli space, Vol[M(Ta )], associated with the possible non-trivial deformations of the class of dynamical triangulations considered. According to theorem 4.5.1 this volume is provided by

6.4 Asymptotics and Entropy Estimates

)-D/2 b Vol[M(Ta )] ~ A(V) ( 4nVvol((1n) n(n + 1) >. ,

131

(6.20)

where V~Vol[Hom(1r~(M),G)] and D~dim[Hom(7r1(M),G)]. Hence, in order to get an asymptotic estimate for the number of distinct dynamical triangulations, we have to divide (6.19) by

)-D/2 b A(V)[b(n, n - 2) - q]>.n+l ( 4nVvol((1n) n(n + 1) >. ·

(6.21)

These observations establish the following

Theorem 6.4.2 (The Entropy function). The number of distinct dynamical triangulations, with ,\ + 1 bones, and with an average number, b == b(n, n - 2), of n-simplices incident on a bone, on an n-dimensional, (n ::; 4), PL- manifold M of given fundamental group 7rl(M), can be asymptotically estimated according to

W('\, b)

~

W . en

< Card{Ta~C) }curv > (b - q+ 1)1-2n

1r

V2i

(b - q)3

(6.22) where 0 < Cn ::; 1 is the exponential rescaling factor of lemma 5.3.1, and where we have introduced the topology dependent parameter W1r~A-1 (V) [47rVvol(a n )]D/2.

(6.23)

As we shall see momentarily, the estimate (6.22) fits extremely well with the data coming from numerical simulations in dimension n 2 3, and in dimension n == 2, (where b(n,n - 2) == 6), it is in remarkable agreement with the known analytical estimates[109]. Notice in particular that for a 2dimensional sphere the integration over the representation variety drops, and (6.22) yields 2

C2

1 [4 4 ]V_ v

W(>.; S ) ~ 24v'61r 33

2,

(6.24)

where we have denoted by v == ,\ + 1 the number of vertices of the triangulation. This estimate should be compared with the known result provided long ago by Tutte [109], viz.,

132

6. Entropy Estimates 4 [4 ]V 7 64V61T 33 v-~ .

1

(6.25)

Thus, if the exponential rescaling 0

<

c2 :::;

1, characterized by lemma

5.3.1, just reduces to a constant resc~ling (C2 ~ 24/64), the agreement be-

tween the two asymptotic analysis would be excellent.

6.5 The 2-Dimensional Case If we exploit the compact formula (4.44) of lemma 4.5.2, providing the asymptoties of the volume of Romer-Zahringer deformations for surfaces, then the agreement between our analysis and the known results in dimension n = 2 is complete. For a surface E h of genus h, and for representations of 1fl (E h ) in a general semi-simple Lie group G, the properties of the representation variety [62] exploited in lemma 4.5.2 imply that (6.22) takes the form

W(,X)ln=2 ~ W1r

44 ] A+l

•

c~ [ 3"

24y 61f 3

,X-l-h(Eh)(l

+ ...),

(6.26)

where X(E h ) is the Euler characteristic of the surface E h , and ... stands for terms of higher order. If we rewrite (6.26) as ( W1r. ( 44)A\"Ys-3 33 1\ 1+

. .. ) ,

(6.27)

where 1S is the critical exponent, then we get

1s(M)

= 2-

5

"4 X(M).

(6.28)

The exponent 1s(M) defines the universal critical exponent of pure 2Dgravity. The expression (6.28) coincides with the well-known expression obtained by the use of Liouville theory.

6.6 The n

> 3-Dimensional Case

In order to check our analytical results against the Monte Carlo simulations which up to now are the only available results in dimension n ~ 3, let us rewrite the partition function for dynamical triangulations (1.6) explicitly in terms of the entropy function W (;\, b), as

(6.29)

6.6 The n

~

3-Dimensional Case

133

where a sum over all possible ,X appears. This sum is needed since triangulations with distinct values of ,X contribute to the number of distinct triangulations with a given volume N. Notice that in dimension n == 2 this further summation is absent since, in that case, the average value of 2-simplices incident on a bone (vertex) is always b == 6. The presence of this summation over ,X shows that in dimension greater than 2, what entropically characterizes the infinite volume limit of a dynamically triangulated model as a statistical system is not just the entropy function W('x, b) , (which can be identified with the entropy in the microcanonical ensemble) but rather the effective entropy provided by the canonical partition function ,ATnax

W(N, kn - 2 )eff ==

L

W('x, N)e kn - 2 ,A.

(6.30)

,ATnin

In order to evaluate the canonical partition function and compare the results with the know numerical simulations we can limit ourselves·to considering dynamical triangulations of simply connected manifolds, since numerical data are available just for the 3-sphere §3 and the 4-sphere §4, (recently, simulations for the four dimensional torus y4 and the product manifold §3 x §l have appeared[26]). Already in this case computations are far from being trivial, and much care is needed in handling the asymptotic estimation of the sum (6.30). According to proposition 6.4.1, the entropy for simply-connected PLmanifolds is W('\ b)

,

(

C)

= en < Card{Ta' lcurv > [(b - q)(,X + l)n+l]

(b(n,n-2)-Q)(,\+1)+(,\+1)-n) ('x+1)-n '

(6.31)

where the factor [(b - q)(,X + l)n+l] comes from removing the labellings associated with the rootings, (see the discussion following theorem 6.4.1). For discussing the asymptotics of (6.31), we need Stirling's formula in the form k! ~ ..;2i(k + l)k+! e- k - 1 , which is accurate also for small k. With these caveats along the way, and setting explicitly q == 3, a long but elementary computation provides the asymptotics

W(A,N)

~

en < Card{TJi)}curv > ..;2i where

IV ~ !n(n + l)N,

,\-1/2(N - 2'\ + 1)-n

NJ(N -

3'\ + 1)(IV - 2'\ + 1)

and where we have set

eh(N,A)

(6.32) ,

134

6. Entropy Estimates A .

h(N,A) == -AlogA+log

[N - 2A + 1] (1V-2A+l) [N - 3A + 1](N-3A+l) A

(6.33)

A .

Thus, (6.30) reduces to _ en W(N , k n-2 ) ell -

X

L

rn=

y21r

/\ -1/2(N - 2/\\ + l)-n

A7nax A"'in

< Card{Ta'tC) }curv >

X

A

\

N

V(N -

eh (N,A)+k n -

2 A,

3'\ + 1) (N - 2,\ + 1)

(6.34)

where for n == 3

N 2 6 == Amin ::; A ::; Amax == gN, A

and for n == 4

N

N

5 == Amin ::; A ::; Amax == 4'

(6.35)

(6.36)

We can estimate the sum (6.34) by noticing that the function appearing in the exponent, viz.,

f(N, A)~h(N, A) + kn - 2 A == - Alog A + (N - 2A + 1) log( N - 2A + 1) (N - 3A + 1) log(N - 3A + 1) + kn - 2 A, has a sharp maximum in correspondence of the solution, A == A*(kn the equation 1

[N - 3'\ + 1]3 =-k

og '\[N _ 2'\ + 1]2

n-2-

(6.37) 2 ),

of

(6.38)

A straightforward computation provides

.

* N+1 1 A == -3-(1- A(kn - 2 ))'

(6.39)

where for notational convenience we have set

[2;

e kn - 2

+ 1-

J(2;

ekn-2

+ 1)2 -1] 1/3 -1.

(6.40)

The structure of (6.39) suggests the change of variable

A == (N

+ 1)11,

(6.41)

6.6 The n 2:: 3-Dimensional Case

135

(we wish to thank G. Gionti for this remark), and by replacing the sum (6.34) with an integration, we get W(N , kn-2 )eff

-

-

en

(i)

< Card{Ta ~

}curv

> f.!-n-l/2

X

(6.42) where

-'r/ log 'r/

+ (1 -

2'r/) log(l - 2'r/) - (1 - 3'r/) log(l - 3'r/)

+ kn- 2 'r/,

(6.43)

and 'r/min ~ 'r/(Amin) , 'r/max ~ 'r/(A max ). For n = 4, we get 'r/min ~ 1/5, 'r/max ~ 1/4, whereas for n = 3, 'r/min ~ 1/6 and 'r/max ~ 2/9.

f(fJl

Fig. 6.1. The relative position of the maximum 1J*(kn - 2 ), with respect to the two kinematical boundaries 1Jmin and 1Jmax, provides the rationale for characterizing the leading asymptotic behavior of the canonical partition function W(N, kn - 2 )ejj.

136

6. Entropy Estimates

The obvious strategy is to estimate (6.42) with Laplace method, however attention must be paid to the possibility that, as kn - 2 varies, the maximum (6.44) crosses the integration limits 'TJmin and 'TJmax. It will become clear in a moment that the quantities telling us when we are nearby these particular regions are (6.45) and (6.46) This suggests that in order to control the large N behavior of the effective entropy as k n - 2 varies, we have to use uniform Laplace estimation in terms of 'ljJ. If we note that the maximum of f('TJ) at 'TJ* localizes < Card{T~i)}curv(b) > at 'T/* , then the following general theorem provides such uniform asymptotics. We state it first for the case involving the crossing of the lower integration limit 'TJmin. A completely analogous result holds, for the upper crossing 'TJmax, and we state it as an obvious corollary . Theorem 6.6.1. Let us consider the set of all simply-connected, dynamically triangulated n-manifolds (n = 3,4). Let k~~2 denote the unique solution of the equation

(6.47)

Let 0 < € < 1 small enough, then, under the assumptions of lemma 5.3.1, and for all values of the inverse gravitational coupling kn- 2 such that in! - € < kn-2 < kin! (6.48) kn-2 n-2 + €, the large N -behavior of the canonical partition function W(N, kn- 2)e!! is given by the uniform asymptotics ~NT-n-l/2e(N+l)!(l1Tnin)-m(11*)Nl/nH X

~

x [

JNwo(1/Jmin(kn- 2)vN) + ~W-l(1/Jmin(kn-2)vN)] ,

(6.49)

where wr(z) ~ r(l- r)e z2 / 4 D r _l(Z), (r < 1), Dr-1(z) and r(l- r) respectively denote the parabolic cylinder functions and the Gamma function, and where the constants ao and al are given by

6.6 The n

~

3-Dimensional Case

137

(6.51) Note that the above expression, (in which we have explicited taken into account < Card{TJi)}curv(b) > for "1 == "1*, and where f'T/~df /d"1, fl1l1~d2 f /d"12) , provides the leading asymptotics. The full asymptotics is discussed during the proof of the theorem. Notee also that in some circumstances, (notably in the 3-dimensional case when employing as variables No, and N 3 ), there can be an equally important constribution to W(N, kn- 2)e!! due to the upper crossing "1max. In that case W(N, kn- 2)e!! is, at leading order, characterized by the sum of the uniform asymptotics around both f("1min) and f("1max).

Proof. We have to provide a uniform asymptotic estimation for large the integral appearing in (6.42), viz., I~

l

n-l/2(1 2n )-n d"1 " -" e(N+l)!('T/). l1m.in J(l - 3"1)(1 - 2"1) 11 m.ax

A

N of

(6.52)

The uniformity requirement stated here refers to the existence of a 8 E lR.+ such that for all 1'l/J(kn - 2 ) I ~ 8 the error after a finite number of terms in the asymptotic expansion should be smaller in N than the last term which is kept [28, 105]. As discussed above, the integral I has various distinct asymptotic regimes, according to the relative location of the maximum "1* with respect to the lower integration limit "1min or to the upper limit "1max. We explicitly discuss for simplicity the case where "1* may approach "1min. The case in which "1* ~ "1max, can be dealt with similarly, and we shall indicate the necessary modifications in the proof and state the final result. In order to carry out the required asymptotic estimation, we first transform the exponential in the integral I by introducing the variable P == p("1) according to (6.53) where for "1 == "1min we assume P == 0 and (d"1/dp)\p=o > 0, so that the orientation of the path of integration remains unchanged. Differentiating this expression we get

138

6. Entropy Estimates

df dp d", = -(p + 'l/Jmin) d",'

(6.54)

In order to have d'fJ(p) j dp finite and non-zero everywhere we require that p ~ -'l/Jmin as 'fJ ~ 'fJ*. Thus, by L'Hopital rule

~.

lim d",= p~-t/Jdp V-~

(6.55)

To determine the expression of the parameter 'l/Jmin(kn- 2), (see (6.45)), controlling the uniform expansion, we evaluate f('fJ) - f('fJmin) at 'fJ = 'fJ*, (viz., for p = -'l/Jmin):

/(",*) - /("'min)

= ~'l/J~in,

(6.56)

whence

'l/Jmin = ±V2[f('fJ*) - f('fJmin)].

(6.57)

The branch of the square root is selected by examining d~ [f ('fJ) - f ('fJmin)] at 'fJ = 'fJmin, (i.e., for p = 0). We get

df dplp=o

df

d p2

= d",lp=o = - dp{'2 +'l/Jminp)lp=o = -'l/Jmin.

Since (d'fJ/dp)lp=o > 0 by hypothesis, and (df /d'fJ)ITJrnin ((df jd'fJ)ITJrnin < 0, for 'fJ* < 'fJmin), we get that

(6.58)

> 0, for 'fJ* > 'fJmin,

sgn'l/Jmin = -sgn('fJ* - 'fJmin).

(6.59)

Thus

'l/Jmin(kn- 2 ) = -sgn('fJ* - 'fJmin)V2[f('fJ*) - f('fJmin)],

(6.60)

(note that 'l/Jmin(kn- 2 ) is a monotonically decreasing function of kn- 2). With these remarks out the way, we can write

I =

1

00

e(N+l)!(71m in)

G{p)e-(N+l)(t p2 +t/JminP)dp,

(6.61)

where we have extended the upper limit of integration to 00, (on introducing a Heaviside function), and where we have set

(6.62) If p = -'l/Jmin E [0, +00), then we can expand the function G(p) in a neighborhood of p = -'l/Jmin, (corresponding to 'fJ = 'fJ*). Otherwise, if p = -'l/Jmin ¢ [0, +00) we need to expand around p = 0, (corresponding to 'fJ =

6.6 The n 2:: 3-Dimensional Case

139

TJmin). According to a standard procedure[28, 105], we can take care of both cases by setting

(6.63) The integral I now becomes

+eCN+l)!C'1min)

1

00

wr(z)

~

1

p(p +

~min)Gl (p)e-CN+l)( tp

2

+1/JminP)dp.

(6.65)

<1

Recall that for r 00

(6.64)

p-re-ctp2+ZP)dp

= F(l- r)ez2/4Dr_l(Z),

(6.66)

where r(l - r) is the Euler Gamma function and Dr - 1 (z) is the parabolic cylinder function [1]. Thus we get

(6.67) where the constants ao and a1 are determined by

(6.68)

ao = G(O),

and a1 =

G(O) - G( -VJmin) III,

.

•

(6.69)

o/m~n

Namely:

TJ~~~2(1 - 2TJmin)-n

VJmin(k n- 2) J(l - 3TJmin)(1 - 2TJmin) f l1 (TJmin) ,

(6.70)

. ao a1=----

VJmin (kn- 2)

(6.71)

140

6. Entropy Estimates

The remaining integral II can be reduced to the same structure of I and an inductive argument [28, 105] shows that the asymptotics of I is provided by

{W o('l/Jmin(k n- 2)JN+!)

y'&+1

[~ Aa2i

~(N+1)i

+0

1_)]

(-A

+ W-l('l/Jmin(~n-2)JN+!) [~ ~2i+l. + 0 . N +1 ~ (N + l)t

Nm+l

(_A_1 _)]}, Nm+1

(6.72)

where the constants a2i and a2i+l are determined recursively from ao and al according to (6.73)

_ Gi(O) - Gi (-'l/Jmin) dG i I a2i+l Ill. . + -d p=-1/J, o/m'tn P and where the functions Gi(p) satisfy the equation

Gi(p)

dGi(p)

+ p~ = a2i + pa 2i+1 + p(p + 'l/Jmin)Gi+l(P)·

(6.74)

(6.75)

Thus, the leading uniform asymptotics is given by

which provides the stated result. In A similar analysis can be carried out when 'fJ* approaches 'fJmax == this case, it is convenient to make the preliminary change of variable y == 1- 3'fJ thus reducing the integral I to

-!.

Joo

== 3n

J

I - 311Trtin

1-311~ax

dy

(1

)-1/2(1 + 2 )-n '" - Y Y e(N+l)![l1(Y)]. y'y(l +2y)

(6.77)

If we set

'l/Jmax(kn- 2) == -sgn(Y('fJ*))y'2[f('fJ*) - f('fJmax)], and introduce the variable p == p(y) according to

(6.78)

6.6 The n 2 3-Dimensional Case

1[7](Y)]- 1[7](0)] = - (P;

+ 1/Jmax(kn- 2 )P) ,

141

(6.79)

we get again an integral whose asymptotic estimation can be reduced to the integral representation of the parabolic cylinder function and toa set of parameters {32i and (32i+l recursively determined from the function H(p) defined by

H(p)p-l/2

~

[ (1- y(p))-3/2 ] dy(p). Vy(p)(l + 2y(p)) dp

(6.80)

Explicitly we obtain

100 =

e(N+l)!(l1max) x

where ~o

= H(O),

(6.82)

and

6 = H(O) - H( -1/Jmax) .

(6.83)

'l/Jmax

The constants e2i and e2i+l are determined recursively from eo and el according to

(6.84) t.

_

~2'1,+1 -

Hi(O) - Hi( -VJmax) III.

o/max

+

dHi I d

P

p=-1/J,

(6.85)

and the functions Hi(p) satisfy the equation

Hi(P)

dHi(p)

+ P~ = 6i + P6i+l + p(p + 1/Jmax)Hi+l(P).

(6.86)

From these expressions it is straightforward to obtain the leading asymptotics for 1 as kn - 2 is such that "1* crosses "1max. Summing up we have the following Lemma 6.6.1. Let us consider the set of all simply-connected n-dimensional, (n = 3,4), dynamically triangulated manifolds. Let k~~~ denote the unique solution of the equation

142

6. Entropy Estimates

(6.87) Let 0 < E < 1 small enough, then for all values of the inverse gravitational coupling k n - 2 such that k:-~~ -

E < kn -

2

< +00,

(6.88)

the large N -behavior of the canonical partition function W(N, kn - 2 )eff is given by the uniform asymptotics

(6.89)

6.6.1 The Infinite Volume Limit

From the above asymptotics we can also read the critical value of the coupling k~rit corresponding to which the infinite-volume limit may be taken, namely for which we may have, as (k n - k~rit) -+ 0+, physically extended configurations dominating our statistical sum, (see next section for more details). In general, i.e., for all values of the inverse gravitational coupling kn -2, the critical line k~rit (k n - 2) is provided by

kcrit(k n

n-2

)=

r

N~oo

logW(N,kn - 2 )eff N '

(6.90)

where W(N, kn - 2 )eff is given by (6.89). By considering separately the various ranges of kn - 2 we explicitly get

kncrit(kn-2 )

n ] = ~2 n (n + 1) [1n A(k23) + 2 + N~oo 1· lnc N

'

(6.91)

which, for 0 < E(N) < 1 characterized by the condition 2'l/J2(k~2 ± E)N ~ 1, holds for all kn - 2 < k~~2 - E. Whereas, for k~~2 + E::; kn - 2 < +00 we get "t

k~r~

1 2

(k n- 2 ) = -n(n + 1)·

(6.92)

6.6 The n

I I I

~

3-Dimensional Case

POLYMER

I

I

I I I I

I

STRONG 'YJmax

Fig. 6.2. The infinite volume line k~rit(kn_2).

143

144

6. Entropy Estimates

6.7 Distinct Asymptotic Regimes As kn- 2 varies around k~~2 and k~~2' the two parameters 't/Jmin(kn- 2) and 't/Jmax(kn- 2) vary monotonically from positive values to negative values. This monotonicity allows us to describe in details the asymptotic behavior of I in terms of kn - 2 . Strictly speaking we should consider separately both a neighborhood of k~n!2 as well as a neighborhood of k::-~2. However, corresponding to the former, available numerical simulations are so polluted by finite size effects that the final confrontation between the analytical results and the numerical results would be quite unreliable. Thus we confine ourself to a detailed analysis of what happens nearby k::-~2' this corresponds to the value of the inverse gravitational couplings probed, reliably, by the Monte Carlo simulations. In any case, it is trivial to extend the conclusions we reach to in! kn-2·

inf k n-2

max k n-2

Fig. 6.3. An indealized view of the distinct asymptotic regimes as r( (k n -2) varies. Note in particular that for 'f/ = 'f/max, the resulting triangulation is not that of a round sphere, as suggested by the picture, but rather that of a stacked sphere.

6.7 Distinct Asymptotic Regimes

Thus, limiting ourself to a neighborhood of following cases:

k~~2

145

we can distinguish the

6.7.1 Strong Coupling According to theorem 5.2.1, there is a critical value bo(n) of b(n, n- 2) corresponding to which the average number of distinct triangulations with given curvature assignments < Card{T~i)}curv > jumps from a subexponential asymptotics of the form

< Card{T~i)}curv with nH

~ n,

>~ exp[-m{b{n, n - 2))N~/nH],

(6.93)

to a polynomial asymptotics of the form

< Card{T~i)}curv

>~ p(b(n,n - 2))· N nT (b(n,n-2)).

(6.94)

Associated with such a critical value bo{n) of b(n, n - 2) there is a unique solution k~r.!.~ of the equation

~(13

1 ) A(kn- 2)

= _1_, bo(n)

(6.95)

corresponding to which such transition occurs. This k~r.!.~ defines the critical value of the (inverse gravitational) coupling kn- 2. Since bo{n), is sufficiently near to the kinematical bound bmax{n, n - 2), it follows by continuity that such a critical value k~r.!.~ is sufficiently near to k~2' viz., there is a J small enough such that max _ J < k crit < k max kn-2 (6.96) - n-2 - n-2· For kn -

2

< k~r.!.~, namely for kn - 2 in a suitably small left-neighborhood of

k~~~, Card{T~i)}curv has a subexponential asymptotics. In dimension n = 4, this behavior corresponds to the range k2

< 1.387 - J,

(6.97)

for 8 sufficiently small. Computer simulations indicate for k2rit the value 1.24, this corresponds to a value of the critical

bo(n)ln=4

~

(6.98)

4.07,

indeed quite near to the kinematical bound b(n, n - 2)minln=4 = 4. Since kn - 2 is proportional to the inverse gravitational coupling, this asymptotic behavior describes a strong coupling regime for simplicial quantum gravity. For such a range of values of kn- 2, TJ* < TJmax and we get

'lfJmax(kn- 2)

= =

sgn(TJ* - TJmax)J2[f(TJ*) - f(TJmax)] J2[f(TJ*) - f(TJmax)] < o.

(6.99)

In such a case the appropriate asymptotic expansion for the parabolic cylinder function is

146

6. Entropy Estimates

rv -z2/4 r (1- r(r -1) Dr (Z ) -e z 2

2z

J2(ff r1ri z2/4 - r - l - r( -r) e e z

- 2)(r + r(r -l)(r 4 2· 4z

3) _

...

)_

(1 + (r + l)(r + 2) + ... ) , 2z

(6.100)

2

Izi »

(which holds for all complex z such that

1,

Izi »

r, and

i <

arg(z) < ~7I"). Setting z = ttPmaxViV + 1 and r = -1, -2 in (6.76) we get that the leading asymptotics for I is

(6.101) The corresponding expression for the effective entropy in this range of kn - 2 is given by n - 2) + W(N , kn-2 ) eff -_ en ((A(k 3A(k - ) n

2))

-n N-n-1

2

[

()

exp -m T/* N

l/n H ]

.

(6.102) where en is the scaling factor of lemma 5.3.1. Thus, in this phase we have a exp[-m(1J*)N 1 / n H] subleading behavior which changes as kn - 2 ~ k~r.!.~. This exponential subleading asymptotics is of relevance to the Monte Carlo simulations since it appears consistent with the numerical data for kn - 2 < crit k n-2· What are the leading configurations in such a strong coupling phase? If we go to the grand-canonical ensemble and consider the associated partition function Z[kn -

2

,

kn ] =

L

e-knNn+kn-2Nn-2,

(6.103)

TET(M)

(see (3.23)), we easily see that the presence of the exp[-m(b(n, n-2))N 1 / n H] subleading behavior implies that the (grand-canonical) average value of the simplicial volume N, goes to 0 as (k n - k~rit) ~ 0+. Thus, we have a collapsed (crumpled) phase. The dominant configurations in the grand-canonical partition function are realized by rather degenerate PL metrics, whereby one concentrates most of the topology and curvature in a set of very large volume and very small diameter. As a matter of fact, Monte Carlo simulations show that in such a phase the vertices of the triangulations have a very large coordination number, of the same order of magnitude than the total number of vertices of the triangulation. Note that the curvature corresponding to such

6.7 Distinct Asymptotic Regimes

147

PL metrics is negative but not necessarily very large. In other words, whereas the coordination number of the vertices is O(N), the curvature assignments stay bounded. As N ~ 00 one may loosely describe this phase as a rough phase in which

< gab >== O.

(6.104)

We have no obvious continuum limit interpretation compatible with a finite continuum volume and a finite (Hausdorff) dimension: all points are at a finite distance from each other, (and the Hausdorff dimension of such a space tends to be infinite). The lack of extended configurations suggests that this phase may, in some sense, correspond to a topological phase of our model. In order to discuss these points of view, let us remark that the crumpling of the triangulations in this phase is basically associated with the fact that the set of dynamical triangulations with bone-wise negative curvature is closed under connected sum along isometric subspaces, (recall that this is the basic geometrical rationale underlying the onset of the exp[-m(b(n, n - 2))N 1/n H ] subleading asymptotics). This implies that the creation of a large triangulation with small diameter is entropically enhanced, (such triangulations are basically the ones known in PL geometry as neighborly triangulations, [111]). Often, it has been remarked that the behavior of dynamical triangulations in this phase do resemble, to some extent, the behavior of hyperbolic geometry. The exponential growth (with the radius) of the volume of geodesic balls, typical of hyperbolic geometry, is indeed observed in the numerical simulations. But, contrary to the situation for smooth manifolds with negative curvature, an exponential-like growth persists down to the smallest possible distances. Even if this numerical scenario is not consistent with the properties of smooth hyperbolic manifolds, it must be noted that it resembles what happens when describing Gromov's almost flat manifolds. Recall, (see 5.79) that a compact manifold M is almost flat if there is a sequence of Coo Riemannian metrics {gj} on M such that {(M.g j )} converges, in the Gromov- Hausdorff sense to a point in such a way that the metrics {gj} have a uniform bound on the absolute value of the sectional curvature. We may (tentatively) extend this characterization to Alexandrov spaces, which properly describe the geometry of dynamical triangulations, and correspondingly define an almost flat dynamical triangulation as a sequence of triangulations (Alexandrov spaces), with bounded curvature (in the comparison sense), Gromov-Hausdorff converging to a point. In the smooth case, by suitably scaling the metric, almost flat manifolds are equivalent to requiring that there is, for each € > 0, a metric g€ on M such that

ISec(M,9€)ldiam(M,9€)2 ::;

€,

(6.105)

where Sec(M,9€) denotes (all) sectional curvatures of (M,9€). A similar rescaling should be feasible also for an almost flat Alexandrov space, showing that an almost flat dynamical triangulations corresponds to a triangulation with very small diameter and bounded curvature. These remarks strongly

148

6. Entropy Estimates

suggests that almost flat triangulations, of the type just described, may be connected with the configurations characterizing the strong coupling regime of our model. They also suggest that the after all we may have also in this phase a possible continuum limit interpretation compatible with a finite continuum volume if we connect this limit with a rescaling such as (6.105). Recall that in order to define a continuum theory in simplicial quantum gravity the first necessary (but not sufficient) step is to seek for a critical line k~rit(kn_2) such that as (k n - k~rit) ~ 0+ the grand-canonical partition function Z[kn- 2, knl becomes singular and the corresponding average number of top- dimensional simplices < N >grand~ 00. In such a case, we may define a renormalized cosmological constant k~en as a-n(kn - k~rit), where a is the edge-length of the triangulations considered. Then, it makes sense to take a limit whereby a ~ 0, (k n - k~rit) ~ 0+ while keeping fixed the physical k~en. In this way, the average physical volume of the configurations dominating our statistical system is finite: < Vol(M) ><X an < N ><X (k~en)-l, [7], [41], and the statistical sum of the theory is dominated by physically extended configurations whose geometrical properties are parametrized by the inverse gravitational coupling kn - 2 • This picture, emphasizing the role of the volume as the only parameter characterizing the physical extended configurations, is the simplest coming to mind, and it works out nicely in the 2-dimensional case. Moreover, the volume-scaling is the natural one associated with an additive renormalization of kn , and geometrically the volume, as a Riemannian invariant, is quite effective in dimension 2 in controlling the geometry of the underlying manifolds, (also with strong topological implications, if we fix our attention on metrics of gaussian curvature = -1). However in higher dimensions, as we have seen, both the diameter and bounds on the sectional curvatures naturally come into play. In particular, the above remarks suggest that, in controlling the infinite volume limit of the theory, may be useful to blend the volume-scaling with the scaling properties of the diameter and of the bounds on curvatures. These remarks are very tentative, and much detailed work is necessary in order to establish their precise substance. Thus, it is difficult to discuss in what sense the stroung coupling phase of simplicial gravity may correspon~ to a topological phase of quantum gravity. Nonetheless a few gepmetrical properties related to the above discussion may point in the right directiQ~. First of all, if some form of hyperbolic geometry is involved in describing the strong coupling phase of quantum gravity, (in the form of a dominance of triangulations with negative curvature), then we should expect that the resulting triangulations are topologically complicated. Since we typically work with n-spheres, §n, this seems to exclude a priori such a dominance of hyperbolic geometry, (recall that, by Hadamard theorem, §n cannot carry any hyperbolic metric). However the fact that in the strong coupling phase we have, for §n, vertices of large coordination number, (of order O(N)), implies that, in the large N limit, the dominant triangulations for §n are singular. Taken

6.7 Distinct Asymptotic Regimes

149

at face value, such triangulations describe, in the large volume limit, metric spaces of infinite Hausdorff dimension rather than n- dimensional spheres. A way out, implicitly suggested by the proof of theorem 5.2.1, is to remove the singular vertices (and more generally, the singular subsimplices whose coordination number grows to infinity with some power of N). This removal gives rise to an incomplete metric space, (e.g., in dimension n == 3, to a 3- sphere minus a knot), because not all simplicial lines can be continued indefinitely. From a mathematical point of view, this is not a particular problem since the resulting space can be made complete with respect to the hyperbolic distance. However, the crux of the argument is that, from the point of view of statistical field theory, this completion must be realized not by hand, but rather when N ~ 00, by a suitable scaling of the cut-off a, in such a way that when N ~ 00 and a ~ 0, the physical extended configuration associated with such an incomplete §n is complete in the hyperbolic metric. As a trivial example of what we mean here, let us consider a smoothly triangulated 2-sphere minus a vertex. Assume that the triangulation in question is realized by equilateral (curvilinear) triangles with sides of length a. This space is incomplete with respect to spherical distance. However, by stereographic projection from the missing vertex onto a plane, we get that such an incomplete sphere, is actually complete with respect to Euclidean distance. What we actually get from such construction is the plane triangulated by triangles with ever increasing sides as we approach 00. Thus, in order to get a complete metric space out of the incomplete dynamically triangulated 2-sphere, the cut-off a has been non- trivially rescaled; a ~ f(a), where f can be explicitly computed by stereographic projection. Thus, as the number of triangles goes to infinity we have to let a ~ 0 in such a way that N . f(a) goes to infinity. Actually, this example does not involve hyperbolic geometry at all, but along its lines it is trivial to construct examples involving completion with respect to hyperbolic distance by considering, for instance, a 2-sphere minus two vertices, (which can also be completed with respect to the Euclidean distance), or minus three vertices. In such a case, as N ~ 00, we have to let a ~ 0 in such a way as to have N . f (a) finite, where now f (a) is the appropriate cut-off rescaling associated with hyperbolic completion. If the above rescaling is feasible, it is clear that the resulting space is parametrized by the removed singular set: typically a knot or a link in dimension n == 3, and a linked surface in dimension n == 4. Dimension n == 3 is particularly interesting in this connection since three-dimensional hyperbolic geometry, (as well as the geometry of Alexandrov spaces of negative curvature), has been thoroughly discussed, in connection with Thurston's geometrization program. One basic point is Mostow rigidity theorem, which implies that if a closed, orientable 3-manifold possesses a hyperbolic structure, then that structure is unique up to isometry. More precisely, let n 2:: 3 and let M1 and M2 denote n- dimensional closed orientable manifolds with an hyperbolic structure. If h: M1 ~ M2 is a homotopy equivalence, then

150

6. Entropy Estimates

there exists an isometry f: M 1 ~ M 2 homotopic to h. This theorem implies that invariants associated with the hyperbolic structure of a manifold (M, g) are topological invariants of the underlying M, (see e.g., [19] for an excellent introduction to modern hyperbolic geometry). Among the invariants associated with an hyperbolic manifold, (M,g), the most natural one is the minimal volume, MinVol(M), defined as the Riemannian volume of (M,g) when the sectional curvature of 9 is normalized to Sec(M,g) = -1. When dim(M) = 2, MinVol(M) = 121rx(M)I, when dim(M) ~ 3, Mostow's theorem implies that different minimal volumes are associated with non-homeomorphic manifolds. Note, however, that the volume is certainly not a complete invariant for hyperbolic manifolds since there are examples of topologically distinct 3-manifolds sharing the same minimal volume. It is important to recall that for n = 4 the minimal volume has basically the same properties as for the 2-dimensional case. In particular, when n ~ 4, for any real v there are only finitely many (isometry classes of) n-dimensional hyperbolic manifolds with MinVol(M) ~ v, (this is Wang's finiteness theorem, see e.g., [19]). This result implies that for n = 4 the values of MinV ol (M) form a discrete set on the real line, and more in particular (via Gauss-Bonnet theorem) MinVol(M) = CnX(M), where as usual X(M) denotes the Euler-Poincare characteristic of M and Cn is a universal constant, (see Gromov's review in [19]). The situation is quite different in dimension n = 3. In such a case JfZjrgensen and Thurston [19] proved that the values of MinV ol (M), for M ranging over all 3-dimensional hyperbolic manifolds with finite volume, form a closed non-discrete set on the real line. Moreover, there are only finitely many M's with a given minimal volume, and the set of minimal volumes is well-ordered, namely each subset in the image of MinVol(M) has a smallest element. These remarks on the properties of the minimal volume suggests that this topological invariant may be connected with the parameter m(b(n, n - 2), characterizing the strong coupling phase of simplicial quantum gravity. Such a parameter is indeed related to a volume function, (see the proof of theorem 5.2.1). This is not certainly a solid evidence for sustaining such a conjecture, however, it is tempting to relate the different behavior of the minimal volume, between dimensions n = 3 and n = 4, with the possible different nature of the phase transition between the strong and weak coupling phases, differences which are exhibited by Monte Carlo simulations, (in this connection, see the next chapter for caveats). At the moment of writing these notes, we have no further evidence for establishing a more explicit connection between the minimal volume and a possible topological interpretation of the strong coupling phase of simplicial quantum gravity. Clearly a similar tentative connection could call into play also other invariants of hyperbolic manifolds, such as the Chern-Simons invariant (for n = 3), but we leave any further speculation in this direction to the imaginative and optimistic reader.

6.7 Distinct Asymptotic Regimes

151

6.7.2 Critical and Weak Coupling When (k n - 2 - k~~~) ~ 0+ we approach the critical coupling. The asymptotics for the effective entropy in this range of kn - 2 is, as far as the parabolic cylinder functions are concerned, just the same as in the previous case. What changes is the subleading behavior associated with < Card{TJi)}curv > which now is polynomial, thus we can write

W(N k ,

. exp

)

n-2 eff

=

en

((A(k n - 2 ) + 3A(kn - 2 )

2))

[[~n(n + 1) In A(kn~2) + 2]N]

-n

N T (7J*)-n-l.

(1 + O(N- 3 / 2)),

(6.106)

where T( TJ*) is the exponent characterizing the polynomial asymptotics of < Card{T~i)}curv(b) >, (evaluated for TJ = TJ*), according to theorem 5.2.1. As (k n- 2 - k~~~) ~ 0+ the average number < Card{T~i)}curv >, of distinct triangulations with given curvature assignements, increases. Since curvature is no longer bone-wise negative, the connected sum mechanism giving rise to the strong coupling phase cannot be activated, and < Card{T~i) }curv > is no longer entropically damped. In order to understand the geometry underlying this mechanism, let us discuss the geometrical meaning of the presence of distinct triangulations with the same curvature assignements {T~ i) } curv' In dimension 2, say for the 2-sphere, we have a deep result that may help in developing a geometrical rationale of what happens also in higher dimension. We are referring here to the geometric approach to the Riemann mapping theorem developed by Thurston, based on Andreev's theorem [108]. Recall that Riemann mapping theorem implies that for every open simply-connected region U C C, there is a conformal homeomorphism from the unit disk onto U. Andreev's theorem refers instead to circles packings, and states that any triangulation of the sphere is isomorphic to the triangulations associated to some circle packing, (note that the circles in question do not have necessarily all the same radius). Such an isomorphism can be required to preserve the orientation of the sphere and then the circle packing is unique up to Mobius transformations, (see [100] for a very nice discussion). The basic idea underlying Thurston's approach to a sort of finite approximation to the Riemann mapping theorem is roughly speaking the following [100]. Take a simply connected region R and fill it up with a regular exagonal circle packing, (with circles with the same radius). Then use Andreev's theorem in order to generate a combinatorially equivalent circle packing of the unit disk, (now the circles will have different radiuses, in general). The correspondence so generated between the circles of the two packings approximate the Riemann mapping. Indeed, [100], Rodin and Sullivan have explicitly shown that two

152

6. Entropy Estimates

circle packings with the same combinatorial triangulation approximate closer and closer, (as the radius of the circles goes to zero), a conformal mapping. Thus distinct triangulations of the 2-sphere are associated with distinct conformal mappings. Obviously, this observation is a restatement, in our setting, that distinct triangulations of a surface approximate distinct riemannian structures on the surface. Rather, what is very interesting here is the explicit connection that this approach exhibits between triangulations and conformal geometry. In order to (heuristically) extend these remarks to higher dimension, let us stress that the circle packings involved in Andreev's theorem are only those whose first order intersection pattern (defined by the tangency of the disks) is the I-skeleton of a triangulation of some open connected subset in the plane or the sphere, [100]. This remark immediately calls into play the role that geodesic balls covering and curvature assignements have in our geometrical approach to dynamical triangulations. The role of distinct geodesic ball coverings with the same first order intersection pattern, and thus of the distinct triangulations with the same curvature assignements, appears to correspond precisely to the role of circle packings and distinct triangulations in Andreev's theorem. Thus, one is very tempted to consider, as a solid conjecture, the possibility a direct correspondence between {T~i) }curv and finite approximations to conformal transformations . If such a conjecture could be proven, then we will have geometric evidence that the transition from the subexponential asymptotics to a polynomial asymptotics in < Card{T~i)}curv >, as (k n - 2 - k~~~) ~ 0+, is nothing but the liberation of the conformal mode. From a physical point of view this conjecture is quite reasonable since (k n - 2 > k~~~) characterizes a region which formally corresponds to small values of the gravitational coupling constant. This range k~~~ < kn - 2 < k~~2 is the weak coupling phase of simplicial quantum gravity. It is interesting to note that k~~~) is very near to k~~2' Thus in such a weak coupling phase, the corresponding values of the average incidence b(n, n - 2) are very near to the lower bound b(n, n - 2)min' As D. Gabrielli and G. Gionti have recently observed, this implies that stacked spheres tend to dominate such a phase. They have also remarked that, according to a theorem of Walkup [111], stacked spheres have a natural tree-like structure. A basic observation which provides the geometrical rationale for the natural tendency, observed in Monte Carlo simulations, to have triangulations which, in the large volume limit, tend to collapse to a branched polymer phase. This geometrical picture lends further belief to the interpretation of the region (k n - 2 - k~~~) ~ 0+ as the transition where excitations related to the conformal mode are liberated. It is also important to stress that from a physical point of view, the tendency to having stacked spheres as the dominating configurations is directly related to the proliferation of baby universes in this phase. Recall [7] that

6.7 Distinct Asymptotic Regimes

153

baby universes, (or MinBU's: Minimal Bottleneck Universes), are associated with triangulations Ta containing the boundary complex of an n-simplex, a"; but not a" itself. It follows, (see [111] for a mathematically detailed description), that T a can be cut along Ba", This surgical operation, opens up T a , and if we patch up the resulting complex with two n- simplices ul' and u2' we form a new complex which results in the disjoint union of two triangulations Ta(N - V) and Ta(V), where N and V are the number of simplices in the two disconnected pieces, (we are here excluding the possibility that such a surgery along Ba" corresponds to handle removal-see [111] for details). If V << N, we call Ta(V) the baby universe, and Ta(N - V) the parent universe. Such a baby universe surgery provides a very efficient computer algorithm which supplements the ordinary moves (see the Appendix) in Monte Carlo simulations. What we want to stress here is that stacked spheres are nothing but (particular) examples of triangulations plagued by baby universes. Recall that a stacked sphere is by definition a triangulation of a manifold which can be constructed from the boundary of a simplex by the successive adding of pyramids over some facets. Then it is clear that the boundary of the faces, along which the pyramids are glued, provides the disconnecting necks Bo" separating a parent universe from the babies (the boundaries of the added pyramids). Note also, that this baby universe surgery is quite different from the surgery dominating the strong coupling phase. The baby universes carry along with them positive curvature, (typically, as large as possible). The connecting neck gives rise to negative curvature, but when baby universes proliferate, their positive curvature contribution is the dominating one. It is possible that the transition from strong coupling to weak coupling as (kn - 2 - k~~~) ~ 0+ is marked by a favourable competition between the contribution of the negative curvature necks and the positive curvature baby universe (the pyramids of the staking), namely it may be seen as the critical competition between the relative concentration of two gas of curvature defects.

6.7.3 Weak Coupling and Complete Polymerization This corresponds to the range k:;:~~ ~ kn - 2 < +00. In this range, according to theorem 5.2.1 the behavior of < Card{TJi)}eurv > remains polynomial, however the above asymptotics for the parabolic cylinder functions is no longer appropriate when, for a given (large) value of N, 'l/J is so small that (6.107)

In such a regime, the parabolic cylinder function is more correctly described by

154

6. Entropy Estimates

v:rr

----

Dr(z) ~ 2- r/ 2r((l _ r)/2) exp[-v-r - 1/2z + VI], where VI ~

-sgnz

«

!Z)3

2J3-r2 -

(this holds for -r - ~ 'l/Jmax

(2J -r -

1/2

»

Z2,

1/2)2

(6.108)

(6.109)

... ,

see[l] pp.689). Setting r = -1 and z

J N + 1, we get for the canonical partition function the expression

~(.I.

~ o/max

Ji/ + 1) [In 77max(1277max)2 _ k _ ] (1 _ 3"lmax)3

n 2

-1 .

-1/2(1 - 2"lmax )-n "lmax x J(l - 3"lmax)(1 - 2"lmax)

--;:~===;:::::::::===::;:

(6.110) where ... stands for higher order correction terms (which can be obtained by the appropriate substitutions from (6.109)). It is interesting to note that in this latter regime, the subleading asymptoties of (6.110) appears again exponential in 'l/JmaxVN. However,this is a transient subexponential behavior, which disappears as N -+ 00, necessary for taking care of the fact that as k n - 2 -+ k~~2 the term (-1 / f 17) tends to di-

J

verge. This divergence is compensated for by the vanishing of 'l/Jmax N + 1, and indeed, the singularity does not manifest itself in the exponential leading behavior of the effective entropy W(N, kn - 2 )e! ! which goes sufficiently smootly from e(N+l)!('Thn a x ) into e(N+l)!(17*), but rather in a change of the polynomial part NT(n)-n-3/2 of the subleading asymptotics, which changes by a factor i/l/2. The fact that this subexponential asymptotics is not relevant is confirmed by noticing that the range of kn - 2 , for which the exponential subleading asymptotics is active depends on the actual value of N, and as N -+ 00, the range k~~2 - e < k n- 2 < k~~2 + f becomes smaller and smaller since e = f(N) is characterized by the condition 2'l/J2(k~~2 ±f)N ~ 1 which implies that limN~cx> f(N) = o. In this case we have (6.111) and by setting 'l/Jmax = 0 in (6.76) we get that the leading asymptotics for I is

I ~ 'TJ *-1/2 (1 - 2'TJ *)-n J(1 - 3'TJ*)(1 - 2'TJ*)

J _

6.7 Distinct Asymptotic Regimes

1f (N -2frJrJ('TJ*)

+ 1)-1/2 e(N+1)f(rJ*).

155

(6.112)

The corresponding expression for the canonical partition function is easily seen to be

W(N k

, n-2

)

eff

==

Cn

2

((A(k n - 2 ) + 3A(kn - 2 )

2))

-n NT(n)-n-l.

(6.113) It remains to discuss the case for which we have For such range of values of kn- 2 , 'TJ* > 'TJmax and

'l/Jmax(kn -2)

k~2

< kn - 2 < +00.

== sgn('TJ* - 'TJmax)J2[f('TJ*) - f('TJmax)] J2[f('TJ*) -:- f('TJmax)] > o.

(6.114)

In such a case, and as long as 'l/J2 N >> 1, the appropriate asymptotic expansion for the parabolic cylinder function is

D ( ) r

rv

z - e

-z2/4 r

z

(1- r(r -1) r(r -1)(r - 2)(r - 3) _ 2z 2 + 2 . 4z 4

(which holds for all complex z such that

Izi »

1,

Izi »

. ..

) (6115) ,

.

r, and larg(z)1 <

~1f). Setting z == 'l/JmaxJN + 1 and r == -1, -2 in (6.76) we get that the leading asymptotics for I is I

~

-1/2(1 2 )-n x 'TJmax - 'TJmax J(l - 3'TJmax) (1 - 2'TJmax) (6.116)

which provides the following expression for the canonical partition function in this regime

W(N, kn -

.

2 )eff

=

Cn

rn= In V 21f [

'TJmax ( 1 - 2'TJmax ) 2 ( )3 - kn 1 - 3'TJmax

-1/2(1 2 )-n 'TJmax - 'TJmax (N J(l - 3'TJmax)(1 - 2'TJmax)

] -1 2

+ 1)T(n)-n-3/2 e(N+1)f(rJ

·

Tnax ) ,

(6.117)

which clearly exhibits a different subleading asymptotic behavior, NT(n)-n-3/2, with respect to the previous case.

156

6. Entropy Estimates

This difference in the polynomial subleading asymptotics will be explicitly related in Sect. 7.2 to the polymerization mechanism which is observed in the weak coupled phase of simplicial quantum gravity during Monte Carlo simulations. Here, let us note that in this region of the weak coupling phase of simplicial quantum gravity, the stacking of triangulations is complete. For kn - 2 = k~~2' Gabrielli and Gionti have observed that the configurations sampled are exactly stacked spheres, hence tree-like triangulations. Moreover, the above asymptotics clearly shows that as; kn - 2 > k~~2 the contribution to triangulations counting coming from the enumeration of distinct curvature assignements decreases with respect to the contribution given by < Card{TJi}}curv >. This seems to indicate that the conformal mode becomes dominant in this phase. The tendency to stacking is so pronounced in this phase that the geometry tends to degenerates to the lowest dimensional tree-like structure possible, that of branched polymers. Obviously, these results could have been obtained by standard Laplace approximation working in the appropriate range of values of the inverse gravitational coupling, but the expression (6.89), (and its related counterpart (6.49)), has the advantage of describing in a uniform way all possible regimes for kn - 2 in an open neighborhood of k~~2. Roughly speaking, the asymptoties (6.102) corresponds to the location of maximum of f('fJ) well inside the integration interval 'fJmin < 'fJ < 'fJmax, whereas the asymptotics (6.117) occurs when 'fJ* is larger than 'fJmax, (but sufficiently near to it). Notice also that A(kn - 2 ) > 1, \/kn - 2 E JR. With these elements in our hands, we can compare our analytical estimates with some of the known numerical data coming from Monte Carlo simulations. We start with the 4-dimensional case.

7. Analytical

V8.

N urnerical Data

The interplay between the asymptotic estimates arising from (6.102) and the results coming from Monte Carlo simulations requires some care, since there are manifolds which are algorithmically unrecognizable. Denote such a manifold by M o, assume that n = 4, and let M o be finitely described by a dynamical triangulation T(Mo). Monte Carlo simulations make use of two kinds of moves altering the initial triangulation T(Mo): a finite set of local moves [2, 96, 97, 67], and global baby universe surgery moves. The local moves are ergodic, since given any two distinct triangulations of the manifold it is possible in a finite number of moves, carried out successively, to change on triangulation into the other. The algorithmic unrecognizability of Mo means that there exists no algorithm which allows us to decide wether another manifold M, again finitely described by a triangulation T(M), is combinatorially equivalent (in the PL-sense), to Mo. It has been proved[95] that the number of ergodic moves[13] needed to connect two triangulations of M o, T and T*, with N4(T) = N 4(T*), cannot be bounded by any recursive function of N 4 . This result implies that there can be very large barriers between some classes of triangulations of M o and that there would be triangulations which can never be reached in any reasonable number of steps, to the effect that the local moves, although ergodic, can be computationally non-ergodic. The manifold used in actual Monte Carlo simulations of 4-dimensional simplicial gravity is mainly the four- sphere §4, (see [26] for other topologies), and it is not presently known if it is algorithmically recognizable, and no large barriers have been observed yet [11]. However, simulations have also been performed in the case of §5 which is known to be algorithmically non- recognizable[45], but no barriers have been observed here either. This implies that in the case of'§5 either the class of configurations separated by high barriers contains only very special configurations which are of measure zero and not important for numerical simulations, or the barriers expected are so high that they have not be encountered yet in the actual simulations. It is important to have these remarks clearly in mind when discussing the comparision with the analytical data.

158

7. Analytical

VB.

Numerical Data

7.1 The 4-Dimensional Case In order to discuss how the asymptotics associated with (6.102) compares with Monte Carlo simulations, let us rewrite W(N, k2 )eff as

W(N, k2 )eff = 10en·

::Cek4rit(k2)N N"Y s (k 2 )-3(1 y21r

+ ...),

(7.1)

where according to theorem 6.6.1 and lemma 6.6.1, the critical k4rit is provided by:

(7.2) for k2

::;

rit

k4

k'2 ax

- €,

and

( k 2 ) ==

(1 -

21Jmax)(1-2'11rnax ) ] )(1-3) 31Jmax 'Y/rnax

10 { log [ 1Jmax 1/Tnax ( 1for k2

> k'2 ax + € and 1Jmax

==

•

+ k2 'TJmax + Nhm

-+-00

In Cn } -N

(7.3)

i, with

(7.4) and where the factor Cn is the normalizing factor of lemma 5.3.1. The critical exponent fs(k 2 ) is also provided by lemma 6.89, (see the discussion on the distinct asymptotic regimes), according to

(7.5) for k 2 ::; k 2rit - €, (recall that in this range we have a non-polynomial subleading asymptotics which formally corresponds to such a critical exponent).

(7.6) for k2rit

::;

k2

::;

k'2 ax , and finally

fs(k 2) == 7(n)

for k'2 ax < k2 •

3

+ 2- n

(7.7)

7.1 The 4-Dimensional Case

159

Notice that in these critical exponents we have introduced the additive normalization by the term r(n), associated with the polynomial asymptotics of < Card[{TJi)}] > entering in the rescaling factor en, (see theorem 5.2.1). This remark reminds us that we still have to fix the remaining part of the scale factor Cn, affecting the canonical partition function W(M; N, k2 )ejj. According to (5.94) and lemma 5.3.1 this factor normalizes the counting of the partitions P-x to the partitions p~urv coming from actual curvature assignments. In dimension n == 4, if we choose r(n) - ~, then a simple exponential (down)shift is all is needed, and if we set e

== (2.066)

Cn

-lON

(

7.8

)

then we get an extremely good agreement with Monte Carlo simulations. It is interesting to note that the exponential scaling factor (2.~66) -lON is a very close approximation to (7.9) suggesting that the ratio p~urv /p-x is, as expected, of geometrical origin. We wish to stress that we do not have yet a convincing geometrical proof that the scale factor Cn is actually given by (7.9), and leave this as a conjecture to be proven by the interested reader. The above choice (7.8) for the rescaling factor Cn provides

k crit == 10 I A(k2 ) + 2 with k2

(7.10)

n 3e/2.066

4

< k'2 ax

-

€,

and

(1 - 21]~ax)(l-21]m.ax) ] 10 { log [ 'T/::h~';,x (1 _ 3'T/max)(1-31j",ax)

+ k 2 'T/max -log

(

e )} 2.066

(7.11)

(with k2 > k'2 ax + f). (Note that the subleading correction N5/2 is irrelevant for the characterization of (7.10) and (7.11)). The rescaled critical value of k4 so obtained gives the critical k4 for any simply-connected closed 4-manifold, in particular for the 4-sphere §4. Indeed, it provides an extremely good fit with the numerical data relative to the characterization of the co-volume line in the (k 4 , k2 )- couplings space. Explicitly, if we denote by K 4rit(Mont.Car) the critical value of k4 obtained from Monte Carlo simulations [12] for dynamically triangulated four-spheres, then the following table demonstrates how well this agreement works out

160

7. Analytical

k2

= 0.2

k 2 = 0.3 k 2 = 0.4 k2 k2 k2 k2 k2

= 0.5 = 0.6 = 0.7 = 0.8 = 0.9

k 2 = 1.0 k 2 = 1.1 k 2 = 1.3 k 2 = 1.5 k 2 = 1.8

VB.

Numerical Data

K 4Tit (Mont.Car)

= 1.45

K 4Tit (Mont.Car) = 1.65

K 4Tit (Mont.Car) = 1.89 K 4 (Mont.Car) = 2.07 Tit

K 4Tit (Mont.Car) = 2.30 K 4Tit (Mont.Car) = 2.54 K 4Tit (Mont.Car) K 4 (Mont.Car) Tit

= 2.78 = 3.02

K 4Tit (Mont.Car) = 3.25

K 4Tit (Mont.Car) = 3.49

K 4Tit (Mont.Car) = 3.97 K 4Tit (Mont.Car) = 4.47 K 4Tit (Mont.Car) = 5.20

K4Tit = 1.47 K4Tit = 1.67

K4Tit = 1.87 K4Tit = 2.07 K4Tit = 2.33 K4Tit = 2.55 K4Tit = 2.78 K4Tit = 3.01 K4Tit = 3.25 K4Tit = 3.49 K4Tit = 3.97 K4Tit = 4.46 K4Tit = 5.19 (7.12)

(note that the numerical data are affected by an error of the order ±O.02, and the analitical data are rounded to the largest second significant figure). According to the distinct asymptotic regimes provided by theorem 6.6.1 and lemma 6.89, we have used (7.11), (evaluated for TJmin = 1/5), in the range for o ~ k2 ~ 0.5 , (however we stress that in such region, and in particular near k 2 ~ 0, the Monte Carlo data are rather unreliable being strongly polluted by finite size effects). For 0.6 ::; k2 ::; k-g: ax ~ 1.38 we have used (7.10). Incidentally, it is not difficult to check that the use of (7.10) works quite well also for 0 ::; k2 ::; 0.5, with just a sligth discrepancy between the Monte Carlo simulations and the analytical data. Finally, for k2 > k'2 ax we have used (7.11). The numerical data in (7.12) come from simulations [12] which are already a few years old, however we wish to stress that the agreement with the analytical estimates holds beautifully also when using more recent and accurate Monte Carlo data, a fact that confirms that our simple counting strategy based on the use of the generating function of partitions of integers p).. has a sound foundation in the geometry of dynamical triangulations.

7.2 Polymerization A well known consequence of the numerical simulations in 4- dimensional simplicial quantum gravity is the onset of a branched polymer phase[10] in the weak coupling regime, i.e., for k 2 > k 2Tit . As we have already pointed out, this polymerization is very effectively described by the asymptotics of theorem 6.6.1 and lemma 6.89, and it is generated again by a mechanism of entropic origin that we want to describe in detail here.

7.2 Polymerization

Fig. 7.1. Polymerization of a sphere.

161

162

7. Analytical

V8.

Numerical Data

Let us start by noticing that if in the expression yielding for the canonical partition function (6.42) we replace the upper integration limit TJmax == 1/4 with the limit i} == 1/3, associated with the algebraic singularity of the integrand at TJ == 1/3 (and corresponding to the unphysical average incidence b(n, n - 2) == 3), then we would get a canonical partition function W(N, kn - 2 )poly whose large N limit is provided by the asymptotics (6.102), viz., W(N k

)

, n-2 poly

==

en

((A(kn-2)+2))-nN'T(n)-n-1. 3A(kn - 2 ) (7.13)

for all k 2rit + 8 :s; k 2 < +00, where k 2rit corresponds to the transition from the strong coupled phase to the weak coupled phase, and 8 E jR+ is a suitably small number. .In the range k 2rit + 8 < k 2 < kr;:ax this is exactly the expression for the asymptotics of the canonical partition function W(N, kn - 2 )ejj, but this identification breaks down as soon as k 2 approaches kr;:ax. The geometrical reason for such a behavior is very simple: as k2 ~ kr;:ax, the possible maximum value of the (positive) average curvature for a given volume, (vol(M) ex N), is saturated: no more positive average curvature is allowed for that volume. Think of the average curvature on a smooth sphere of given volume: it is maximized for the round metric corresponding to that volume. As remarked by Gabrielli and Gionti, for a dynamical triangulation positive curvature saturation occurs for stacked spheres, which corresponds to proliferation of baby universes. On the other hand, (7.13) as it stands, would allow, as k 2 grows, for more and more (positive) average curvature at fixed volume, and in order to comply with the constraint relating volume to positive curvature we have to turn to the correct weak coupling description of the partition function, viz., (6.117). However, (7.13) can still be interpreted as the partition function of a large positive curvature PL manifold of given volume, but now the manifold in question is disconnected. Such a disconnected manifold is generated by means of a polymerization mechanism, namely it results as the (GromovHausdorff) limit of a network of many smaller manifolds (stacked spheres) carrying large curvature. Such curvature blobs are connected to each other by thin tubes of negligible volume so that the sum of the volumes of the costituents manifolds adds up to the given total volume, (the standard example in the smooth case is afforded by a collection of highly curved small spheres connected by tubules of negligible area, vs. the standard round sphere of the some total area). The basic observation is that for kr;:ax + € < k 2 < +00, the effective entropy (7.13) is larger than (6.117): thus in the weak coupling region, for a dynamical triangulation is energetically favourable to polymer-

7.2 Polymerization

163

ize rather than to comply to the curvature-volume constraint with a large connected manifold. The above remarks can be easily formalized. According to (3.15), the average curvature (around the collection of bones {B}) on a dynamically triangulated manifold is proportional to

< K(B)

>~ ~ L K(B)q(B) oc ~ 1=2 [211" - q(B) cos~l .!.]. N

N

B

n

B

(7.14)

Namely

< K(B) >oc 211" - b(n, n ~ 2) cos-l(~) .

(7.15)

The canonical average, « K(B) », of < K(B) > over the ensemble of dynamical triangulation considered is, up to an inessential positive constant, provided by (see (6.42)),

1

"'max

X [

"'min

-1 2

-1

-n

dTJ TJ / (1 - 2TJ) e(N+1)!(1/) ] J(l - 31])(1 - 21])

.

(7.16)

For k2 = k'2 ax - €, with € E jR+ sufficiently small, the maximum for !(1]), (see (6.42)), is attained for an 1]* which uniformly approaches 1] = i as € ~ o. Correspondingly, the integrands in (7.16) are peaked around 1] ~ 1/4 and « K(B) » is positive, (explicit expression can be easily obtained from the uniform asymptotics of the previous paragraphs). Let us now extend (7.16) to 1]max = namely consider the ensemble average, « K(B) »poly of < K(B) > over W(N, kn - 2 )poly. It is trivially checked that « K(B) »poly formally remains positive for all k'2 ax < k2 < +00, (corresponding to the maximum 1]* ranging from t ~ 1]* ~ But according to lemma 2.1.1, there cannot be connected four-manifolds (regardless of topology) with b < 4, (i.e., 1] > i), because for any dynamically triangulated 4-manifold

!'

!).

5N4

~

2N2

-

5X.

(7.17)

An inequality which implies that as 1] ~ i,·the average curvature < K(B) > attains its maximum ex: [21r-4 cos- 1 ( )]/4 compatibly with the given volume, (ex: N). There is a natural way of circumventing the implications of the constraint (7.17): we have to remove the connectivity requirement for M. Let us assume that we have a collection {Ma } of dynamically triangulatedmanifolds of fixed volume ex: N 4 (Ma ) = N, and where each M a is

i

164

7. Analytical vs. Numerical Data

generated by a large number m of dynamically triangulated 4-manifolds {5(i)}i=1, ... ,m(a) C Mo., connected to each other by tubes of negligible volume. Such a sequence of manifolds naturally converges, in the GromovHausdorff topology, to the disjoint union of a set of m 4-manifolds, viz., m

{Mo.} ~ M ~ IlS(i),

(7.18)

such that N4(M) = N 4(Mo.) = N. The constraint (7.17), which holds for each connected manifold Mo. of the sequence, carries over to the disconnected limit space M and implies that m

5

m

m

L N (S(i)) 2:: 2 L N (S(i)) - 5 L X(3(i)), 2

4

(7.19)

namely,

b ~ lO L

:n N (E(i)) > 4 _ 10 L:n X(E(i))

L:n N (5(i)) -

M

4

2

L:n N (5(i))

.

(7.20)

2

Thus, if each S(i) is topologically a 4-sphere, §4, and N 2 (5(i)) is 0(1), we can make bM take any value in the range 3 :::; b :::; 4. In particular, if (7.21 )

and (7.22)

then for such M bM

~

3.

= U5(i),

we get (7.23)

Such an M is the disjoint union of m ~ N4~M) sma1l4-dimensional spheres. Recall that the standard triangulation of a 4-sphere obtained as the boundary of the standard 5-simplex a 5 is characterized by the f-vector (7.24)

It is trivially checked that for k'2 ax + f < k2 < +00, the effective entropy (7.13) is larger than (6.117). Thus in the weak coupling region, it is energetically favourable to generate dynamical triangulations associated with a disconnected manifold M = U5(i), rather than to comply with the curvature-volume constraint on a large connected manifold. From the asymptoties (6.117) and (6.102) it follows that the critical exponent associated with

7.2 Polymerization

such a polymer phase, by the relation

,poly

,poly,

1

== ,s(k2 ) + 2·

is related to ,s(k2 ) == T(n)

+2 -

165

n, (see (7.6)),

(7.25)

This jump of ~ has been observed in the numerical simulation [10] exactly ax ~ 1.3. In such simulations there is also some indication that around rit ,s(k2 ~ k 2 ) ~ 0, so that ,poly is exactly the critical exponent of branched polymers, but this evidence is not yet conclusive due to critical slowing down near the transition point. According to our counting strategy, we still have to characterize the exponent T(n) associated with the count of distinct dynamical triangulations with the same curvature assignments. Since the critical exponent of a branched polymer is exactly ,poly == ~, and as we have shown (7.13) seems to geometrically describe a polymer phase, we can use this result to fix the value of T(n). Indeed, if at least provisionally, we accept this heuristic argument, from (7.5) we find that the critical exponent scaling factor T(n) must satisfy the relation

kr

2 - n + T(n)ln=4 ==

1

2'

(7.26)

namely, T(n)ln=4 == ~, a result already anticipated in the expression (7.8) of the normalizing factor en' Not surprisingly, this position provides also the correct critical exponent near criticality, viz., (7.27) These latter results, although not fully rigorous, seem to be conforted by the numerical simulations. We wish to stress that the polimerization phase is a real effect generated on real manifolds. Thus it gives us confidence in our model the possibility of providing a simple geometrical and entropical understanding of the fact that the weak coupling phase of simplicial gravity seems to consist of manifolds which degenerate to branched polymer-like structures. Few remarks are also in order for what concerns the transition between the various phases: strong coupling, vs. weak coupling, and then polymerization. The transition threshold towards a polymer phase occurs, in our modelling, somewhere around the value of k2 such that 1]* == 1]max. Explicitly, for a k2 ~ kr;:ax such that A(k2 ) -1 A(k2 )

3

4·

(7.28)

kr

ax ~ 1.387 which appears quite in a good agreeThis condition provides ment with the value indicated by the numerical simulations. For instance in [10], the polymerization point can be located around krax(Mont.Carl) ~ 1.336(±0.006). This value is quite near to the value k 2rit corresponding to

166

7. Analytical vs. Numerical Data

Kinf n-2

crit

Kn-2

Fig. 7.2. The entropic mechanism for the onset of polymerization. On the geometrical side, D. Gabrielli and G. Gionti have recently observed that for kn - 2 = k~~2 the triangulations dominating the canonical partition function are stacked spheres which, according to D. Walkup, have a natural tree-like structure.

7.3 Summing over Simply-Connected

4-Dimensional Manifolds

167

which the transition between a strong coupled phase to a weak coupled phase occurs. Recent simulations [45] tend to move this point towards k'2 ax ~ 1.24. As we have seen, there is a simple geometrical rationale for having k2rit quite near to k'2 ax . Moreover a further accidental explanation of this fact is related to the observation that 'f}* == 'f}max is a theoretical lower bound, and ax changes very rapidly with small variations of the ratio A~~~2) 1 . that Thus for instance, if we set 'f}max == 1/4.1 we would get k'2 ax ~ 1.2.

kr

7.3 Summing over Simply-Connected 4-Dimensional Manifolds It is important to stress that the asymptotic estimates of theorem 6.6.1 refers to all simply-connected 4-manifolds, not just to the 4-sphere which is the only simply-connected 4-manifold for which numerical data are currently available. However, there exists now numerical results for nonsimply-connected manifolds like y4 [26]. These results indicate that the critical line k4rit(k2) is independent of the topology of the 4-manifold. We will exploit this in the following. Besides §4, well-known example of closed simply-connected topological 4-manifolds are provided by K, where <, > K is the Kronecker pairing between the cohomology group and the homology group. The case of 2-dimensional (connected) surfaces E provides a direct illustration of what is meant by this construction. Here the relevant homology group is HI (E, 'Z2), every homology class in HI (E, Z2) can be represented by a simple closed curve 0 C E, and if 0 1 , O2 are any two such curves, their intersection number 0 1 . O2 == O2 . 0 1 E 'Z2 is defined. Note that the self-intersection number 0 . 0 is zero as soon as the curve 0 1 has a neighborhood which is orientable, (by deforming the curve 0 ~ 0, while remaining in the same homology class, we can make 0 disjoint from 0), while if such a neighborhood is non-orientable, the self-intersection number will be odd, (think of a deformation of the curve on a Mobius band). Actually, by Poincare duality we can make H 1 (E,Z2) an inner product space over Z2 if we use the intersection number 0 1 . O2 as inner product. It can be proven that any two closed connected surfaces E I and E 2 are homeomorphic if and only if the associated inner product spaces H I (E 1 ,Z2) and H 1 (E 2,Z2) are isomorphic (see Milnor-Husemoller's book[91]), and that every inner product

zt zi

168

7. Analytical vs. Numerical Data

space over Z2 is isomorphic to HI (E, Z2) for some closed connected surface

E.

In a similar fashion, the intersection form wM(a, (3) on H2(M, Z) may be thought of as representing the transversal intersection of two oriented surfaces a and (3 in the compact, simply-connected 4-manifold M, a and (3 being cycles representing elements in H2 (M, Z). Again, we can think of H 2 (M, Z) as an inner product space over Z if we use the intersection number W M (a, (3) as inner product. Moreover, it can be proven that any two closed oriented simply- connected 4-dimensional manifolds M 1 and M 2 have the same homotopy type, (i. e., there exists an orientation preserving homotopy equivalence M 1 ~ M 2 ), if and only if the associated symmetric inner product spaces H 2 (M1 , Z) and H2 (M2 , Z) are isomorphic, (Whitehead's theorem [55, 56,48]). Actually, as in the case of surfaces more can be stated in terms of this isomorphism, since the set of simply-connected oriented closed 4-manifolds is in correspondence[55 , 56,48] with the set of all unimodular, (any matrix representing w(,) has determinant ±1), bilinear symmetric form, (the manifold is unique when the form is even, and there are exactly two distinct manifolds when the form is odd), (Freedman's theorem [55, 56, 48]). However, if the 4-manifolds in question carry differentiable structures, the correspondence is' with positive-definite intersection forms, (Donaldson's theorem[55, 48]). Let us denote by W(M; N, k2 )eff the canonical partition function enumerating the distinct triangulations on a 4-dimensional simply-connected manifold M of given topology and given volume (N). The set of such M is formally labelled by the set of all positive-definite, (we are dealing with PL 4-manifods), unimodular bilinear symmetric forms WM. According to theorem 6.6.1, the asymptotics (6.49) refers to all simply-connected 4-dimensional PL- manifolds of fixed volume, thus we can write

W(§4; N, k 2 )eff

+ W(§2

x

§2;

N, k 2 )eff

+ W(K 3 ; N, k 2 )eff +.... (7.29)

For N large but fixed, the dynamical triangulations entering (7.29) are of bounded geometry, thus according to theorem 3.4.1, there are only a finite number of distinct homotopy types of 4-manifolds contributing to the above sum, and Whitehead's theorem mentioned above implies that likewise finite is the number of distinct types of (isomorphism classes of) intersection forms W entering (7.29). Thus from (7.29) we get

7.3 Summing over Simply-Connected

4-Dimensional Manifolds

169

= W(§4; N, k2 )ejj + W(§2 x §2; N, k2 )ejj + W(K 3 ; N, k2 )ejj + .. (7.30) where Ts(k 2 ) is the scaled critical exponent. The identity (7.30) makes clear that the expression (6.89) for the canonical partition function being common to all dynamically triangulated 4-manifold (closed) with trivial fundamental group, tends to overestimate the canonical partition function of any 4-manifold M in (7.30). However, guided by the analogy with the 2dimensional case, and by the agreement between our analytical results and the Monte Carlo simulations referred to above [26], we may tentatively assume that all simply-connected DT manifolds contribute in (7.30) to the total entropy W(N, k2 )ejj with the some exponential behavior and that the topological differences come about only in the subleading asymptotics. Since (7.29) is a finite sum, (7.30) implies that the subleading asymptotics in each entropy function W(M; N, k2 )ejj is of polynomial type, (at least in the range of k 2 considered). Thus, we are led to the following ansatz:

W(M; N, k2 )eff

= lOen'

::Cesk~rit(k2)N N'Y s (w,k 2 )-3(1 + ...),

y27r

(7.31)

where TS(W, k2) is the critical exponent for the topological 4-manifold characterized by the intersection form w. Since by hypothesis the exponential leading term does not depend on topology, the above ansatz allows us to rewrite (7.30) as

L

NI's(w,k2)-3

= NI's(k2)-3.

(7.32)

w

The above hypothesis is fully consistent with the finiteness result for the topology of dynamical triangulations of bounded geometry discussed in Section V, (theorem 3.4.2), according to which the number of distinct homotopy types for such triangulations grows at most polynomially with N. In order to be more specific, let us recall that the rank r(w) of a unimodular symmetric bilinear form is the dimension of the space on which w is defined, namely r(w) = dim[H2 (M, Z)]. The signature s(w) of w is defined by choosing an orthogonal basis bl , . .. ,bn for the (rational) inner product space Q~H2(M,Z), then[55] s(w) = S(Q~H2(M,Z)) E Z is the difference between the number of basis elements bi with w(bi , bi ) > 0 and the number of basis elements bj with w(b j , bj ) < O. Roughly speaking, s(w) is the number of positive eigenvalues minus the number of negative eigenvalues of w (thought of as a real form). Finally, the type of w is defined by stating that w is even (or of type II) ifw(a,a) E 2Z for all Q E H2 (M,Z), (i.e., the diagonal entries of ware even), otherwise w is odd, (or of type 1[55]). The relevance of such definitions lies in the fact that any two (indefinite) inner product spaces (H2 (M I , Z), WI) and (H2 (M2 , Z), W2) are isomorphic if WI and W2 have the same rank, type, and signature[55, 56, 48, 91]. According to this observation, we can replace the sum over the possible intersection forms in (7.32) with

170

7. Analytical

V8.

Numerical Data

a sum over the possible ranks, signatures and types. To this end, let us introduce the functions A±(r, s) enumerating the distinct (isomorphism classes of) intersection forms w of rank r(w) == r, signature s(w) == s of even type, (A+(r, s)), and of odd type, (A_(r, s)) on a 4-manifold of bounded geometry. Then, we can write (7.32) as r s L L[A+(r, s)

+ A_(r, S)]NTS(w,k

2)

== NTS(k 2 ),

(7.33)

r=Os=O

where f < +00 and S < +00 are the largest value for the rank and signature allowed on a 4-manifold of bounded geometry. In analogy with the 2-dimensional case, (where the critical exponent depend linearly on topology), let us make the further natural ansatz (7.34) where (l, (3, 8 are constants possibly depending on the inverse gravitational coupling k 2 . Notice that since H 2 (§4, Z) == 0, 8 is the critical exponent for the 4-sphere. The ansatz concerning the structure of the effective entropy of each simply- connected 4-manifold M, implies that r

s

L L[A+(r, s)

+ A_(r, s)]Nar+,Bs+8 == NTS(k

2 ),

(7.35)

r=Os=O

which is a very strong bound on the set of possible intersection form for 4manifolds of bounded geometry. Indeed, without the control on geometry, the number of inequivalent intersection forms, say of given rank r, grows with r at an extremely fast rate. To make a rather well-known example, (see MilnorHusemoller book[91]), consider the subcase of positive-definite intersection forms of odd type. The number of such forms is known to grow with the rank r at least as C

__ r_ ( 21l"eJe, )

r2 /4 (

81l"e r

) r/ 4

r-1/24.

(7.36)

Thus, the interplay between our analytical approach and Monte Carlo simulations seems to indicate results of great potential interests in the borderline between 4D-simplicial quantum gravity and the topology of 4-manifolds.

7.4 The 3-Dimensional Case The confrontation between Monte Carlo simulations and the analytical estimates is quite satisfactory also in the 3-dimensional case.

7.4 The 3-Dimensional Case

171

Numerical data in dimension n == 3 often employ[13],[30],[8] the pair of variables N 3 and No rather than N 3 and Nl , thus we have to adapt our asymptotic estimates to this new pair of variables before comparing with the simulations. This can be trivially done by noticing that the Dehn-Sommerville relations No - N l + N 2 - N 3 == 0 and N 2 == 2N3 imply N l == N 3 + No, so that the partition function can be written as Z[kl(k o), k3 ] ==

L( L N

==

L( L N

W(A == N

+ No, N)ek1().=N+NO))

e- k3N

A=N+No

W(A == N

+ No, N)e k1No )

e(k 1-k 3)N,

(7.37)

)..=N+No

where N-l=N3 , and A == N l . In this case, it is better to use the results of theorem 6.6.1 and lemma 6.6.1 in a slightly more general setting. Again, the function !(TJ), (now expressed in terms of the variable No), is, in the interval TJmin ~ TJ ~ TJmax, a concave function symmetric with respect to the point TJ* where it attains its maximum, viz., (7.38) Since the parameters driving the uniform asymptotics of the saddle-point evaluation of the canonical partition function are, (see (6.45) and (6.46)), (7.39) and (7.40) then (7.38) implies that the correct asymptotics of the canonical partition function, in the range TJmin ~ TJ ::; TJmax is provided by a linear combination of the two uniform asymptotics obtained by developing the integral (6.52) in terms of both 1/Jmin ( kl ) and 1/Jmax ( kl ). Moreover, in this case in order to fit with the Monte Carlo data, it is better to allow for {}n-2 # 0, (see lemma 5.3.1). In this way we obtain the following Theorem 7.4.1. Let us consider the set of all simply-connected 3-dimensional dynamically triangulated manifolds. Let k~nf, and k ax respectively denote the unique solutions of the equations

o

1

1

1

"3(1 - A(ko)) + 6 = rymin

1

= 6'

(7.41)

172

7. Analytical

VB.

Numerical Data

111 2 3(1 - A(k )) + "6 = 1/max = g. o

(7.42)

Let 0 < € < 1 small enough, then for all values of the inverse gravitational coupling k o such that k~nf + € < k < k ax - €, (7.43)

o

o

the large N-behavior of the canonical partition function W(N, kO)eff for 3dimensional simplicial quantum gravity on a simply connected manifold is given by the uniform asymptotics ~NT-7/2e(N+l)f(TJTnin)-m('f/*)Nl/nHx

V2i

+~NT-7/2e(N+l)f(TJTnax)-m('f/*)Nl/nH

V2i

x

(7.44) where the constants ao, aI, and eo,

el are given by (7.45)

.

al

ao

1] *-1/2 (1 - 21]*)-n [

1

]

= 'ljJmin(ko) - J(1 - 31/*)(1 - 21/*) 'ljJmin(ko)J- 1,.,,.,(1/*) ,

'0 ~ _

1/;;;'~~?(1 - 21/max)-n

'ljJmax{ko) , J(l - 3"7max) (1 - 2"7max) f TJ ("7max)

el ~

(7.46)

(7.47)

C 1 ] ~o _ "7 *-1/2 (1 - 2"7*)-n [ . (7.48) 1Pmax(ko) J(l - 31]*)(1 - 21]*) 1Pmax(ko)J- fTJ'f/(1]*)

From this result it follows that k 3rit (k o) can be formally obtained from an expression similar to (6.91) by the substitution k 1 ---7 ako and k3rit(kl) ---7 k3rit(kl ---7 ako) - ako, where 0 < a :::; 1 is a suitable constant, viz.,

7.5 Concluding Remarks: the Order of the Phase Transition

A(ako)+2_ k 61 In en k 3crit (k)-61 0 n 3 a 0 + nN-+oo N '

173

(7.49)

where

[22

+ 7 eo:ko + 1-

V(2;

eo: ko

+ 1)2 -1] 1/3 -1.

(7.50)

If we fix the leading exponential part of the rescaling factor en according to 1

[cos- 1 ( "3)r<6/5)N,

(7.51)

and set the constant a 0.16, then it is not difficult to check that the rit rescaled ka (k o) has a distribution of values, as ko increases, which is in very good agreement with the corresponding values determined by Monte Carlo simulations. Explicitly we get

ko == 0 ko == 1 ko == 2 ko == 3 ko == 4

K 3rit (Mont.Car) == 2.050 K 3rit (Mont.Car) == 2.055 K 3rit (Mont.Car) == 2.070 K~rit(Mont.Car) == 2.100 K~rit(Mont.Car) == 2.220

K 3rit == 2.04 K 3rit == 2.07 K 3rit == 2.11 K~rit == 2.15 K 3rit == 2.20.

(7.52) rit The critical point k8 corresponding to the transition from a strong coupling to a weak coupling phase of 3D- simplicial gravity, and associated to the asymptotic behavior of < Card{T~i)}curv >, is, according to theorem 5.2.1, located near k'Qax, viz., k rit ~ klr ax ~ 3.845, (7.53)

o

which agrees well with the location of the critical point 3.9 obtained by the Monte Carlo simulations.

k8rit (Mont.Car)

~

7.5 Concluding Remarks: the Order of the Phase Transition The agreement between the analytical methods we developed here and the Monte Carlo data indicates that the picture of 4-dimensional simplicial quantum gravity which emerges from our model is similar to that one already

174

7. Analytical vs. Numerical Data

elaborated on the basis of computer simulations. In particular, our analytical results predict a phase transition between a weak coupling and a strong coupling phase of gravity. This result strongly suggests that the ensemble of 4-manifolds may develop a genuine extension when k 2 ~ k 2 crit, and make this transition point a potential candidate for a non-perturbative theory of quantum gravity. However, recent Monte Carlo simulations have cast doubts to the standard wisdom concerning the second order nature of this transition.It has been observed [46],[24] a double peak in the distribution of order of vertices, and this has been interpreted of some indication of a first order nature of the transition, similarly to what happens in three dimensions. From the numerical simulations point of view the situation might be quite similar to the one we encounter for compact U(l) in four dimensions: we have a strong coupling phase were the string tension is different from zero (to be compared with the strong coupling phase in 4-dimensional gravity). The string tension scales to zero at the critical point (in 4D-gravity this would correspond to a mass parameter scaling to zero at the critical point). At this critical point it is unclear if there is a two-phase signal, and if the transition is first or higher order. A two-phase signal has first been observed, then it has disappeared in a sort of see-saw game, and in a way akin to the situation we are experiencing in 4D-gravity. This similarity persists in the weak coupling phase: for small coupling constant we have the coulomb phase in U(l), viz., for all coupling constants smaller than the critical one we have a continuum massless phase, and in 4D-gravity for all (bare) gravitational coupling constants smaller than the critical one we have branched polymers, which indeed admit a continuum limit. Since, as we have shown, there is good agreement between Monte Carlo simulations and our analytical approach, one wonders if there is a way of understanding the nature of the transition by exploring further the geometrical arguments exploited here. Such an insight will constitute a significant improvement over our analytical model, and will be a foundation to build upon a more realistic model of simplicial quantum gravity. Acknowledgement. We extend special gratitude to B. Briigmann, U. Bruzzo, L. Crane, G.M.D'Ariano, G. Jug, R. Loll, R. Penrose, C. Rovelli, and L. Smolin for many interesting discussions and critical remarks which motivated several improvements. We especially single out D.Gabrielli,G.Gionti, C.F.Kristjansen, J.Jurkiewicz, J. Lewandowski, and B. Petersson for detailed commentary on many subtle issues and inspiring conversations. The final version of this work owes a lot to the critique by A. Krzywicki. This work has been supported by the National Institute for Nuclear Physics (INFN).

A. Appendix

A.I Pachner Moves Even if we have not used in any significant way the properties of Pachner moves in our geometrical approach to dynamical triangulations, we think that these notes are an appropriate place for a brief description of this topic. As a matter of fact, besides playing an important role in computer simulations of dynamical triangulations, Pachner theory is a key ingredient in many analytical developments of simplicial gravity. In this appendix we let M denote as usual a simplicial PL-manifold of dimension nand T ~ M ~ ITI a particular triangulation of M, (sometimes T will denote the underlying abstract simplicial complex as well). Recall that two PL-manifolds M 1 , M 2 are PL- homeomorphic if there exists a map f: M 1 ~ M 2 which is both a homeomorphism and a simplicial isomorphism. This definition of PL- equivalence turns out to be quite difficult to be handled with in practice, since one should go over and over through subdivisions in order to find out (eventually) isomorphic triangulations. The issue of combinatorial equivalence was first addressed by Alexander [3] who proved the following

Theorem A.I.I. For any polyhedron P which is dimensionally homogeneous (viz., is the union of some collection of closed simplices of the same dimension) any two triangulations of P can be trasformed one into the other by a finite sequence of stellar subdivisions and their inverse transformations. (This theorem actually holds for more general complexes). The stellar (or star) subdivisions are typically known as Alexander moves (or transformations). In order to discuss some general features of these transformations, let us anticipate here that an Alexander move can be performed along each simplex a (of any dimension r ~ n) belonging to a triangulation T of dimension n. Such a transformation involves all the simplices in T which share the simplex a, namely the closed star of a. However, Alexander himself showed that, at least for closed polyhedra, it is sufficient to consider just those moves which are performed along the I-simplices of T. Notwithstanding this simplification, it turns out that even these reduced moves are not elementary

176

A. Appendix

in the sense that each one of them may involve, at least in principle, a different number of n-simplices of T, (the number of n-simplices which share the particular I-simplex along which the transformation occurs). Thus, being interested in transformations between triangulations of a PL-manifold M, one should implement Alexander moves over a lot of different local arrangements of simplices which cannot be factorized into simpler blocks. On the other hand, the Alexander theorem provides us with a powerful tool since it ensures that PL-manifolds are mapped homeomorphically into PL-manifolds and moreover that all the admissible triangulations of a given M are related to each other by suitable sets of Alexander moves. The way out from this situation is to looking for a different set of moves which are both elementary (i. e., they involve just a fixed number of simplices in any dimension n) and equivalent to the Alexander moves, that is topology-preserving and ergodic, (i. e., they must span all over the possible triangulations of a given M). A set of moves sharing all the above requirements was found by Pachner in 1987[96]. These moves were named bistellar elementary operations (Pachner moves, for short) and they can be used for dealing with closed PL-manifolds. Later on, the same author has found an enlarged set of elementary operations which allows one to dealing with PL-manifolds with boundary as well[97]. In the meanwhile, being aware of Pachner's work, people involved in computer simulations on simplicial quantum gravity models have worked out a set of suitable ergodic, elementary moves, the so called (k, l)-moves (see e.g., the paper of M.Gross and S.Varsted[67] for a complete account), which turns out to be the same as the former Pachner moves for closed PL-manifolds. From now on, the symbol ~ will always denote PL-homoemorphism, whereas the symbol cx, equipped with some subscript, will denote isomorphic simplicial complexes. Thus, for instance, the Alexander theorem A.l.l can be rephrased in the concise form: (A.I) where CXst stands for equivalence under stellar subdivisions and their inverse transformations. Following Pachner [96, 97] with some minor changes in order to fit with our previous notations we introduce the following definitions.

Definition A.I.I. The join of two simplicial complexes T 1 and T 2 is the simplicial complex defined by

(A.2) where 0"1·0"2 ~ 0"1 U0"2 if the complexes are considered as abstract complexes. A realization in Euclidean space is given by the convex hull 0"1·0"2 ~ conv( 0"1 U0"2) This definition is extended in an obvious way to the join of subsets of complexes. The join of a complex T with a simplex r will be denoted by T . r for short (instead of T· {r}). Notice however that it is necessary to distinguish

A.l Pachner Moves

177

between the join of T with the empty simplex, T· {0} == T, and the join with the empty complex T ·0 == 0.

Definition A.l.2. Let T be a triangulation of a simplicial PL-manifold M of dimension n, T ~ ITI ~ M, and let 0i= a E T be a simplex such that

link(a)

~

8(star(a)) == 8T· L,

(A.3)

where T i= 0 is a simplex not belonging to T but such that its boundary 8T is contained in link(a), and L is a complex. Then the transformation S(u,T):T~Tl ~

(T\(a·8·L))U(8a·T·L)

(A.4)

is a stellar exchange. Stellar exchanges are indeed generalizations of Alexander moves, for:

Definition A.l.3. If, in the former definition, we restrict ourselves to the case in which dim(T) == 0, namely T == {p} is just a (new) vertex, then

S(U,T)

~

S(u):T

~

Tl

~

(T\(a· L)) U (aa· {p}. L),

(A.5)

is a stellar subdivision along a E T. (Here we have used the fafct that, since aT == 0, a . {0} . L == a . L). Coming back to definition A.l.2, we notice that T l = S(u,T)T is again a triangulation with ITll ~ ITI, (owing to the fact that for a simplicial manifold link(a r ) ~ sn-r-l for each r-simplex). Moreover S~~T) = S(U,T) holds. Thus, a stellar subdivision is topology-preserving, and S~~T) ~ S0-~ when dim(a) =

o.

Finally, a stellar exchange is a generalization of a single Alexander move also in the sense that it is possible to show that S(U,T) == S(T)S0~. This remark should make it clear that stellar exchanges are not elementary in general, as well as the Alexander moves are not. In order to find elementary operations it is necessary to put a restriction on the possible dimensions of the simplices a, T which appear in definition A.l.2 according to the following

Definition A.l.4. If dim(a)

S(U,T) ~ B(u,r) ~ B(u)

+ dim(T) =

n in definition A.l.2 then (A.6)

is called a bistellar operation, or a bistellar k-operation when dim(a) == k. Being a particular case of a stellar exchange, a bistellar operation is topology-preserving and B~~r) = B(r,u). The equivalence of triangulations of PL-manifolds under bistellar operations and their inverse transformation is denoted by T l ~bst T. Unlike general stellar exchanges, bistellar operations and their inverses (Pachner moves for short) turn out to be elementary in any dimension n. Pachner proved the following theorem[96, 97]

178

A. Appendix

Theorem A.I.2 (Pachner). Let T, T1 be triangulations of closed simplicial PL-manifolds. Then

(A.7) In what follows we discuss the complete list of Pachner moves in dimension n == 2, n == 3, n == 4. Notice that we shall consider just the portions of triangulations involved by the move under discussion. The remaining part of each triangulation has to be left unchanged.

A.I.I The 2-Dimensional Case The condition dim( a) + dim(r) == 2 implies that there are three distinct case to be considered:

dim((J) == 2 and dim(r) == o. This latter equality tells us that we are actually dealing with an Alexander move (see Fig. A.I). Since, from the point of view of 2-simplices in T, this move transforms one 2-simplex into three 2-simplices, we shall denote this move by (1 -+ 3). (This notation, focusing on the number of simplices of maximal dimension involved by the move, will be adopted henceforth). II dim((J) == 1 and dim(r) == 1. This is a true stellar exchange and the simplex (J E T is shared exactly by two 2 - simplices. This move is depicted in Fig. A.2 and is denoted by (2 -+ 2). III dim((J) == 0 and dim(r) == 2. It is easy to realize that this move is nothing but the inverse of the transformation I. It can be performed on Fig. A.l by going from the right to the left. This is the move (3 ---+ 1). I

A.I.2 The 3-Dimensional Case The condition dim( a) to be considered:

+ dim(r) == 3 implies that there are four

distinct case

dim((J) == 3 and dim(r) == O. This is an Alexander move (see Fig. A.3), and it is denoted by (1 -+ 4). II dim((J) == 2 and dim(r) == 1. This is a stellar exchange, (see Fig. A.4) and is denoted by (2 ~ 3). Notice that a simplex in T is always shared exactly by two 3-simplices. III dim((J) == 1 and dim(r) == 2. As can be easily seen, this move is just the inverse of the move (2 -+ 3) depicted in Fig. A.4 by imagining the transformation acting from the right to the left. Denote this move by (3 -+ 2). IV dim((J) == 0 and dim(r) == 3. Again this case represents the inverse of the move I, and consequently is denoted by (4 -+ 1). I

A.l Pachner Moves

179

Fig. A.1. The move (1 ~ 3): on the left there is a 2-simplex a E T; in the middle there appears 'T (a new vertex not belonging to T); on the right there is the final configuration

Fig. A.2. The move (2 ~ 2): on the left there is a I-simplex a E T; in the middle the I-simplex'T is represented by a dashed segment (note that 'T is not in T any longer but a'T E link(a)); on the right there is the final configuration

180

A. Appendix

Fig. A.3. The move (1 --+ 4): on the left there is a 3-simplex a E T; in the middle there appears T (a new vertex not belonging to T); on the right there is the final configuration

A.l.3 The 4-Dimensional Case

The condition dim(a) to be considered:

+ dim(r)

= 4 implies that there are five distinct case

I dim(a) = 4 and dim(r) = O. This is an Alexander move involving just one 4-simplex. We shall denote this move by (1 ~ 5), (see Fig. A.5). Notice that in the figures, illustrating this and the following moves, the vertices of the simplices are labelled to keep track more easily of the transformations involved. II dim(a) = 3 and dim(r) = 1. Since each 3-simplex in T is shared exactly by tow 4-simplices, we pick up such a configuration as the starting point. The move is depicted in Fig. A.6 and is denoted by (2 ~ 3). Notice that the subsimplices which are common to the simplices of the configuration are denoted by boldface lines from now on. III dim(a) = 2 and dim(r) = 2. The starting configuration consists of three 4-simplices which share a common 2-simplex. This move is represented in A.7, and is denoted by (3 ~ 3). IV dim(a) = 1 and dim(r) = 3. This case represents the inverse 9fthe move (2 ~ 3) described in II. It is denoted by (3 ~ 2), (see Fig.'A.6 from right to left).

A.l Pachner Moves

181

Fig. A.4. The move (2 ~ 3): on the left there is a 2-simplex a which is the common face shared by two 3-simplices in T; in the middle the I-simplex T is represented by a dashed segment (T is not in T but aT E link(a)); on the right there is the final configuration

182

A. Appendix

V dim(a) = 0 and dim(r) = 4. This case represents the inverse of the Alexander move (1 ~ 5) described in I. It is denoted by (5 ~ 1), (see Fig. A.5 from right to left).

o 8 E

o E Fig. A.5. The move (5 -4 1): on the left there is a 4-simplex (7 = (ABCDE); in the middle there appears T = (F) (a new vertex not belonging to T); on the right there is the final configuration made up by five 4-simplices (ABCDF), (ABCEF), (ABDEF) , (ACDEF) , (BCDEF). Notice that the original a does not belong to the triangulation anymore

A.2 The Tangent Space to the Representation Variety We summarize here some basic facts about the representation variety of the fundamental group of a manifold M. Given the representation 8: 1rl (M) ~ G let us consider the associated representation () on the Lie algebra 9 generated by composing with the adjoint action of G on g, viz., ():1rl(M) ~e G ~Ad End(g), (henceforth we will always refer to this representation ). The representation () defines a flat bundle, go, over the underlying space M of the dynamical triangulation considered. This bundle is costructed by[85] exploiting the adjoint representation of G on its Lie algebra g, i.e., Ad: G ~ End(g), and by considering the action of 1rl (M) on 9 generated by composing the adjoint action and the representation 8:

e

A.2 The Tangent Space to the Representation Variety

183

A Fig. A.6. The move (2 ~ 3). (a): two 4-simplices, (ACDEF) , (BCDEF) , are joined in a common 3-simplex (CDEF) = (I. (b): The I-simplex r = (AB) joins the vertices. (c): the final configuration is made up by three 4-simplices (ABCDE), (ABCDF), (ABDEF) which share the edge r = (AB)

go ==

M X g/1f1 ® [Ad(8(.))]-1,

(A.8)

where 1f1 ® [Ad(8(·))]-1 acts, through 1f1(M), by deck transformations on M and by [Ad(8(·))]-1 on the Lie algebra g. More explicitly, if 0-1, 0-2 are simplices in M, and 91,92 are elements of g, then 0- 1,91) rv (0- 2 ,92) if and only if 0-2 == ala, and 92 == [Ad(8(a))]-1 91 , for some a E 1r1(M). Given a dynamical triangulation Ta and a flat bundle go over M == ITa I, we can define homology and cohomology groups with value in go, (namely homology and cohomology with values in the sheaf of Killing vectors of M riem ). On T a we define a p- chain with values in go to be a formal sum 2:::1 giaf where af E T a is an ordered p-simplex and gi is an element of the fiber of go over the first vertex of af. Namely, first we consider chains with coefficients in the Lie algebra g, viz., E j 9jo-j with 9j E g, and then quotient the resulting chain complex C(M) ® g by the action of 1r1 ® [Ad(O(·))]-l. We denote the group of such chains by Cp(M, go). There is an action of 1f1 (M) on the above chains expressed by

a(L9jO-;) j

-7

L([Ad(O(a))]-1 9j )a(0-;),

(A.9)

j

for any a E 1r1(M), (i.e., we are considering g as a 1r1(M)-module). This action commutes with the boundary operator, and we may define homology groups H*(M, go) with local coefficients in the flat bundle go. By dualizing one defines the cohomology H*(M, go), which enjoys the usual properties of a cohomology theory. These homology groups are naturally related to the geometry of the infinitesimal deformations of representations of the fundamental group of M.

184

A. Appendix

A

B

c A

~---------....B

c Fig. A.7. The move (3 ~ 3). (a): three 4-simplices, (ABDEF) , (ACDEF) , (BODEF), are joined through the 2-simplex (DEF) == 0". (b): The 2-simplex r == (ABC). (c): the final configuration is made up by three 4-simplices (ABODE), (ABCDF), (ABCEF) sharing the triangle r = (ABC)

A.2 The Tangent Space to the Representation Variety

185

Given the representation 0: 7fl (M) ~ G let us consider the associated representation on the Lie algebra 9 generated by composing 0 with the adjoint action of G on g, viz., 0: 7fl (M) ~ G ~ End(g), (henceforth we will always refer to this representation). Then a I-cocycle on 7fl(M) with coefficients in o is a map c: 7fl (M) ~ 9 such that for " 8 E 7fl (M) we have (A.IO) The space of such cocycles (often called crossed homomorphisms ) is denoted by Zl(7fl(M), g). We denote by B l (7fl(M),g) c Zl(7fl(M), g), the subspace generated by the cocycles which are I-coboundaries, namely the set of C E Zl(7fl(M),g) for which there exists agE 9 such that (A.II) for all , E 7fl (M). The first cohomology group of 7fl (M) with values in 9 is then defined by 1

Zl(7fl(M),g)

H (11"1 (M), g) = B1(1I"1(M),g)"

(A.12)

The definition of the higher cohomology groups is somewhat more complicated, and interested readers may consult[50]. In order to relate H l ( 7fl (M), g) to the infinitesimal deformations of the representations of 7fl (M), let

Hom(7fl(M), G)

(A.I3) G denote the set of all conjugacy classes of representations of the fundamental group 7fl(M) into the Lie group G. Notice that if 0 and FOF- l are two conjugate representations of 7t"l(M) in G, then through the map Ad(F): 9 ~ 9 we get a natural isomorphism between the groups Hi(7fl(M),g) and H i (7t"l(M),FgF- l ). We are interested in understanding the structure of Hom( 7rb(M),G) when deforming a particular representation 0: 7fl (M) ~ G through a differentiable one-parameter family of representations Ot with 00 == 0 which are not tangent to the G-orbit of (} E Hom(7fl(M), G). To this end, let us rewrite, for t near 0, the given one-parameter family of representations Ot as[80],[IIO]

Ot == exp[tu(a) + O(t 2 )]O(a),

(A.I4)

where a E 7fl (M), and where u: 7fl (M) ~ g. In particular, given a and b in 7fl (M), if we differentiate the homomorphism condition ()t (ab) == ()t (a )()t ( b), we get that u actually is a one-cocycle of 1fl (M) with coefficients in the 7fl(M)-module g(h viz.,

u(ab) == u(a) + [Ad(()(a»]u(b).

(A.I5)

Moreover, any u verifying the above cocycle condition leads to a map Ot: 7fl (M) ~ G which, to first order in t, satisfies the homomorphism condition. This remark implies that the (Zariski) tangent space to H om( 7fl (M), G)

186

A. Appendix

at (), can be identified with Zl(1rl (M), g). For this reason, space Zl( 1rl (M), g) is called the space of infinitesimal deformation of the representation o. In a similar way, it can be shown that the tangent space to the Ad-orbit through () is B 1 (1rl (M), g). Thus, the (Zariski) tangent space to Hom( 1fb(M),G) corresponding to the conjugacy class of representations [0] is H 1 ( 1rl (M), g) which, therefore, can be identified with the formal tangent space to the representation space. It is not difficult to establish an explicit isomorphism between the tangent space H1(1rl(M), g) and the first cohomology group with coefficient in the flat bundle go, H1(M, go), associated with the underlying space M of the dynamical triangulation Ta . We choose a base Xo E M, (say a vertex of the base simplex a[)), and a corresponding basepoint i o E M, the universal cover of M, such that 1r(io) == Xo, 1r: M ~ M being the covering map. If go is the fiber of go over Xo, then go is also the fiber of 1r*go over i o. Since parallel transport (for the flat connection) is independent from the path on M, we can parallel transport the fibers of 1r*go to go, and this generates a map €: CO(M, 1r*go) ~ go, where CO(M, 1r*go) is the space of functions assigning to each vertex of M an element of the fiber of 1r* go over the vertex considered. By exploiting this map, we can define, for every l-cocycle Q on M with values in go, a function cPa: 1rl (M, Xo) ~ go by setting

(A.16) where, E 1rl (M, xo), and;Y is a simplicial lift of, to M (starting at xo). Such cPa is a l-cocycle on 1rl(M,xo) with values in g, and the mapping Q ~ cPa establishes the seeked isomorphism from the cohomology group H1(M, go) to the cohomology group H1(1rl(M),g).

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Index

Action - combinatorial Einstein-Hilbert 8 - Einstein- Hilbert 3 - Regge action 46 Adjoint representation - of Lie algebra 71 Alexander 175 Alexander moves 175 Alexandrov 49 - comparision triangle 51 - space with bounded curvature 54 Alexandrov spaces 51 almost flat manifolds 115 Analytic continuation 84 - for locally homogeneous geometries 77 Analytic subvariety 86 Andreev's theorem 151 Approximation property - for dynamical triangulations 57 Approximations - of smooth manifolds 49 Auslander- Kuranishi 72 Automorphism group - of a triangulation 48 Average degree 59 Average number of simplices 25 Baby universe surgery 157 Baby universes 152 Bare - coupling constants 8 - gravitational coupling 8 Best filling 58 Betti numbers - for dynamical triangulations - polynomial growth 67 Bistellar operations 176 Bones - of a triangualtion 25 Boundary 125

67

Boundary terms 3 Bounded geometry 55 - PL metric space of 49 Cartan-Killing metric 91 Cerf theorem 8 Characters 123 Chern class 128 circle packings 151 Class functions of the holonomy 123 Cocompact subgroups 86 Cohomology - of the fundamental group 71 - with local coefficients 71, 183 Collapse of simplices 68 Combinatorial equivalence 175 Combinatorial properties 19 Comparision techniques 51 Cone 20 conformal mode 152 Conical neighborhood 40 Connection - 8U(2) connection 122 Connectivity - in polymerization 163 Continuum limit 9 Contractibility function - for compact metric spaces 66 Controlled topology 2 Counting problem 49 Counting strategy 121 Couplings - running 48 Covering - minimal geodesic covering for dynamical triangulations 60 Critical exponent 158 Critical value - rescaled 159 Crossed homomorphisms 185 Curvature 44

194

Index

-

simplicial 42 Alexandrov curvature 50 average 48 average curvature for a small volume 45 - Gauss curvature 51 - of a bone 45 Curvature assignments 46 Curve - in dynamical triangulations 40 Cut-off 40 - short distance 5 Deck transformation group 86 Deck trasformation group 72 Deficit angle 44 Deformation - geometry of 75 - infinitesimal deformations 75 - non-trivial 70 - theory 71 Deformation of representations 183 Deformation space - of a locally homogeneous manifold 81 Dehn- Sommerville equations 25 Delzant polytope 127 Developing map 77, 85 Development pair 79 Diameter 57 Differentiable structures 5 Dimensionality - effective 32 Directed edges 39 Distance - in dynamical triangulations 40 Donaldson-Freedman theory 6 double scaling limit 6 Dynamical triangulations 1, 39 Effective dimension 94 Eigenvectors - of the rotation matrix 42 Energy - versus entropy 5 Entropy - effective 9,49,133 - the entropy problem 35 Entropy function 49 Enumeration - of triangulations 68 Enumeration of distinct curvature assignments 95

Enumeration of distinct triangulations with given curvature assignments 95 Enumeration strategy 102 Ergodic moves 104, 176 Euler- Poincare equation 25 Expansion - of simplices 68 Exponential bound 2 f-vector 25 Face 18 Fake three-sphere 68 Filling function 58 Finite size effects 144 Finitely generated group 86 Flat bundle - defined by a representation of 7rl (M) 182 Flat riemannian manifolds 72 Flatness constraint 73 Formal tangent space 89 Foundamental group 11 Frame - orthonormal 39 Freedman theorem 168 Fundamental group - of a dynamical triangulation 69 G-structures 11 Gabrielli 152 Gauge fixing 88 Gauss polynomials 126 Generating function 126 Genus - infinite genus 63 Geodesic balls 58 Geodesic rays 49 Geodesic space _. dynamical triangulations as a 41 Geodesics - of a dynamical triangulation 40 Geometric structure 73 Gionti 135, 152 Glueings 22 Goldman 75 Goldman-Thurston deformation theorem 84 Grid - of reference manifolds 63 Gromov- Hausdorff - topology 45 Gromov-Hausdorff limit 63

Index Gromov-Hausdorff approximation Gromov-Hausdorff closure 57 Gromov-Hausdorff distance 55 Group of diffeomorphisms 3 Grove 67

55

Haar measure 89 Hamber 44 Hamming distance 99 Hard Hexagon Model 127 Hard sphere - gas 10 Hauptvermutung 35 Hausdorff distance 55 Heat kernel asymptotics 91 Heat kernel on the representation variety 90 Holonomy - homomorphism 72, 79, 84 - of a connection 68 - of a locally homogeneous manifold 79 - of riemannian manifolds 69 Holonomy group - of a dynamical triangulation 69 - restricted holonomy group of a dynamical triangulation 69 Holonomy matrix 123 Holonomy representation 69 Homeomorphism type 5 - finiteness results 68 Homogeneous geometries 76 Homology - with values in a sheaf of Killing vectors 183 Homology manifold 35 Homotopy type 5 - definition 65 - estimate of distinct homotopy types 66 - finiteness for dynamical triangulations 65 - polynomial growth 67 - wild 65 hyperbolic completion 149 Incidence matrix 42, 68 Incidence numbers - actual distribution of 29 Independence - statistical 5 Infinite volume limit 9, 142 Interior geometry

- of dynamical triangulations Intersection form 167 Intersection pattern 58, 59 Isometries - group of 71 Join of a complex

195 40

176

Killing vector fields - sheaf of 71 Krzywicki 120 Lasso lemma 122 Lassos 122 Lattice - infinite dimensional 33 Lebesgue covering property 57 Length - of a curve 40 Length space - dynamical triangulations as a 41 Levi-Civita connection 42, 68 - flat 71 - of a dynamical triangulation 42 Lewandowski 122 Link - of a simplex 20 - spherical link 53 Link length 7 Local moves 157 Locally homogeneous space 69 Loop - homotopic to the trivial loop 69 - simplicial 69 - winding around a bone 42 Markings 84 Mean field theory 120 Measure - a priori 4 Metric - fixed edge- length 39 - interior metric of a dynamical triangulation 41 - piecewise-flat 39 Metric geometry 40 Metrics - equivalence class of 8 minbus 109 Minimal net 58 minimal volume 150 Minimizing geodesic - in a dynamical triangulation 41 Minimum incidence 25,96

196

Index

Model surface 51 Moduli of constant curvature metrics 70 Moduli space of dynamical triangulations 86 Monte Carlo simulations 132, 157 - for the 4-sphere 159 Mostow rigidity theorem 71,149 Moves - (k, l) moves 176 - elementary 175 Multiplicity of intersection pattern 59 n-bein 39 neighborly triangulations 147 Number of generators - of the fundamental group 86 Orbifolds 86 Overcounting measure

121

Pachner 176 Pachner moves 105, 176, 177 Packings in spheres 10 Parabolic cylinder function 136 - definition 139 Parallel transporter 42 Parent universe 153 Partition function 8 - canonical 48,49,133 - grand-canonical 48 - microcanonical 48, 49 Partition of integers 95-97 - unrestricted 103 Path integral 3 - formal 48 Path-ordered - product 42 Petersen 67 Piecewise-Linear(PL) - distinct PL structures 35 - geometry 7 - manifold 20 - non- standard structures 35 - properties 19 - topology 17, 34 Poincare conjecture 6 Polyhedral subsets - of Euclidean space 63 Polyhedron 20 - convex 35 - underlying 18 Polymer - branched polymer phase 160

Polymer phase - critical exponents in Polymerization 160 Projective limit 65 Pseudo-manifolds 23

165

Romer 70 Romer-Zahringer deformation 88 - definition of 75 - of a dynamical triangulation 70 Reducible representations 93 Regge 7 Regge calculus 8, 39, 42 Regge curvature 53 Regular point - of a polyhedron 24 Reidemeister torsion 92 Representation - of the fundamental group 69,182 Representation variety 11, 82 Ricci curvature 57 Riemann mapping 151 Riemann-Roch number 128 Riemannian metrics - space of 3 Rigidity of lattices 86 Root blob of curvature 95 Rooted enumeration 88 Rooted traingulations - definition of 85 Rotation angle 44 Rotation matrix 42 Rough isometry 55 Scale factor 159 Simple homotopy types 68 Simplex 18 Simplices - incident on a bone 44 Simplicial complex 19 Simplicial division 19 Simplicial isomorphism 19 Simplicial loop 42 Simplicial manifolds 20 Simplicial map 19 Simplicial path 42 Simply-connected - 4-manifolds 167 Singular component of the representation variety 93 Singular point - of a polyhedron 24 Skeleton 18

Index Slice theorem 81 Smooth component of the representation variety 93 Smooth type 5 Space forms 59 Stable representation 82 Stacked sphere 26, 152 Stacked spheres and baby universes 153 Stanley's theorem 35 Star - of a simplex 18 - open 19 Stellar exchanges 177 Stellar subdivisions 175 Stirling formula 133 string exponent 9 Strong coupling regime 145 Subadditivity 102 Subdivision 19,20 Subdivisions - isomorphic 34 Subexponential growth 98 Subleading asymptotics - exponential 146 Subleading behavior 146 Symmetry factor - of the triangulation 48 Symplectic manifold - associated with triangulation enumeration 128

-

Three-sphere - fake 6 Thurston 22, 75 - geometrization program 76 THings 73 Todd class 128 Topological complexity 67 topological phase 147 Topologies - sum over 5 Topology - control of 63, 65 - finiteness theorems 67 Topology-preserving moves 176 Torus - definition of flat tori 72 Tree-like structure 152 Triangle - comparision 51 Triangle inequality 53 Triangulations 8, 19

Yamaguchi

197

combinatorially equivalent 34 deforming triangulations 73 distinct 8 distinct triangulations of the same PL-manifolds 33 - fixed edge-length 39 - isomorphic 175 - rooted 11 - small 30 - variable edge-length 39 Tutte 33, 131 Uniform asymptotic estimation 137 Uniform Laplace estimation 136 Universal cover 79 Unrolling 85 Unrooted curvature assignments 102

Volume - of an equilateral simplex 45 Volume density - of a bone 45 Volume element of the moduli space 89 Volume of moduli space 130 Walkup's theorem 26 Whitehead theorem 168 Williams 44 Wilson loop 68 Wu 67

Zahringer

66 70