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Lecture Notes in Physics New Series m: Monographs Editorial Board H. Araki, Kyoto, Japan E. Brezin, Paris, France J. Ehlers, Potsdam, Germany U. Frisch, Nice, France K. Hepp, Zurich, Switzerland R. L. Jaffe, Cambridge, MA, USA R. Kippenhahn, Gottingen, Germany H. A. Weidenmuller, Heidelberg, Germany J. Wess, Munchen, Germany J. Zittartz, Koln, Germany
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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
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement,that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready by authors Cover design: design & production GmbH, Heidelberg SPIN: 10550667 55/3144-543210 - Printed on acid-free paper
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
np-l > ... > n r 2:: r an integer in this way), then s
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('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~ 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. 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 = 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
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)
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 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
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