We provide an elementary proof of the exponential bound to the number of distinct dynamical triangulations of an n‐dimensional manifold M (n≥2), of given volume and fixed topology. The resulting entropy estimates emphasize the basic role, in simplicial quantum gravity, of the moduli spaces Hom(π1(M),G)/G associated with the representations of the fundamental group of the manifold, π1(M), into a Lie group G.
REFERENCES
1.
J. Ambjo/rn, “Quantization of Geometry,” Lecture given at Les Houches Nato A.S.I., Fluctuating Geometries in Statistical Mechanics and Field Theory, Session LXII, 1994;
F. David, “Simplicial quantum gravity and random lattices,” Lectures given at Les Houches Nato A.S.I. Gravitation and Quantizations, Saclay Prep. T93/028, 1992.
2.
T.
Regge
, “General relativity without coordinates
,” Nuovo Cimento
19
, 558
–571
(1961
).3.
R.
Williams
and P. A.
Tuckey
, “Regge calculus: A brief review and bibliography
,” Class. Quantum Grav.
9
, 1409
–1422
(1992
);H. W.
Hamber
and R. M.
Williams
, “Simplicial quantum gravity in three dimensions: Analytical and numerical results
,” Phys. Rev. D
47
, 510
–532
(1993
).4.
P. Menotti and P. Peirano, “Faddeev-Popov determinant in 2-dimensional Regge gravity,” Prep. IFUP-TH-12/95.
5.
D.
Weingarten
, “Euclidean quantum gravity on a lattice
,” Nucl. Phys. B
210
, 229
(1982
).6.
F.
David
, “Planar diagrams, two-dimensional lattice gravity and surface models
,” Nucl. Phys. B
257
, 45
–48
(1985
);B.
Durhuus
, J.
Fröhlich
and T.
Jónsson
, “Critical behaviour in a model of planar random surfaces
,” Nucl. Phys. B
240
, 453
–480
(1984
);R. Fernandez, J. Fröhlich, and A. Sokal, Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory (Springer-Verlag, Berlin, 1992);
V. A.
Kazakov
, “The appearance of matter fields from quantum fluctuations of 2D gravity
,” Mod. Phys. Lett. A
4
, 2125
–2139
(1989
);C. Itzykson and J.-M. Drouffe, Statistical Field Theory: 2 (Cambridge University Press, Cambridge, 1989).
7.
J.
Ambjo/rn
, “Regularized quantum gravity
,” Nucl. Phys. B (Proc. Suppl.) A
25
, 8
–24
(1992
);J.
Ambjo/rn
, B.
Durhuus
, and J.
Fröhlich
, “Diseases of triangulated random surface models, and possible cures
,” Nucl. Phys. B
257
, 433
–449
(1985
).8.
E.
Moise
, “Affine structures in 3-manifolds I,II,III,IV,V
,” Ann. Math.
54
, 506
–533
(1951
);9.
S. C. Ferry, “Finiteness theorems for manifolds in Gromov-Hausdorff space,” Preprint SUNY at Binghamton, 1993;
S. C. Ferry, “Counting simple homotopy types in Gromov-Hausdorff space,” Preprint SUNY at Binghamton, 1991.
10.
K.
Grove
and P. V.
Petersen
, “Bounding homotopy types by geometry
,” Ann. Math.
128
, 195
–206
(1988
);K.
Grove
, P. V.
Petersen
, and J. Y.
Wu
, “Controlled topology in geometry
,” Bull. Am. Math. Soc.
20
, 181
–183
(1989
);K.
Grove
, P. V.
Petersen
, and J. Y.
Wu
, “Geometric finiteness theorems via controlled topology
,” Invent. Math.
99
, 205
–213
(1990
);erratum,
K.
Grove
, P. V.
Petersen
, and J. Y.
Wu
, Invent. Math.
104
, 221
–222
(1991
).11.
D. S. Freed and K. K. Uhlenbeck, Instantons and Four-Manifolds, 2nd. ed. (Springer-Verlag, New York, 1991).
12.
J. Fröhlich, “Regge calculus and discretized gravitational functional integrals,” Preprint IHES, 1981;
reprinted in Non-Perturbative Quantum Field Theory—Mathematical Aspects and Applications, Selected Papers of J. Fröhlich (World Scientific, Singapore, 1992);
H.
Römer
and M.
Zähringer
, “Functional integration and the diffeomorphism group in Euclidean lattice quantum gravity
,” Class. Quantum Grav.
3
, 876
–910
(1986
);W. Kühnel, “Triangulations of manifolds with few vertices,” in Advances in differential geometry and topology, edited by I. S. I.-F. Tricerri (World Scientific, Singapore, 1990).
13.
D.
Bessis
, C.
Itzykson
, and J. B.
Zuber
, “Quantum field theory techniques in graphical enumeration
,” Adv. Appl. Math.
1
, 109
–157
(1980
);E.
Brézin
, C.
Itzykson
, G.
Parisi
, and J. B.
Zuber
, “Planar diagrams
,” Commun. Math. Phys.
59
, 35
–51
(1978
);D.
Bessis
, “A new method in the combinatorics of the topological expansion
,” Commun. Math. Phys.
69
, 147
–163
(1979
);J. Ambjo/rn and J. Jurkiewicz, “On the exponential bound in four dimensional simplicial gravity,” Preprint NBI-HE-94-29 (1994);
D. V. Boulatov, “On entropy of 3-dimensional simplicial complexes,” preprint NBI-HE-94-37 (1994).
14.
C. Bartocci, U. Bruzzo, M. Carfora, and A. Marzuoli, “Entropy of random coverings and 4-D quantum gravity,” SISSA Prep. 97/94/FM, to appear in J. Geom. Phys.;
M. Carfora and A. Marzuoli, “Entropy estimates for simplicial quantum gravity,” preprint NSF-ITP-93-59;
to appear in J. Geom. Phys.
15.
W.
Goldman
, “Geometric structures on manifolds and varieties of representations
,” in AMS-IMS-SIAM Joint Summer Research Conference Geometry of Group Representations, edited by W. Goldman and A. R. Magid, Contemp. Math.
74
, 169
–198
(1988
);W. M. Goldman and J. J. Millson, “Deformations of flat bundles over Kähler manifolds,” in Geometry and Topology, edited by C. McCrory and T. Shifrin, Lecture Notes in Pure Applied Math (Markel Dekker, New York, 1987) Vol. 105, pp. 129–145);
N. J.
Hitchin
, “Lie groups and Teichmm̈uller space
,” Topology
31
, 449
–473
(1992
);K. Morrison, “Connected components of representation varieties,” in Ref. 15, pp. 255–269;
K. Walker, An Extension of Casson’s Invariant (Princeton Univ. Press, Princeton, NJ, 1992).
16.
B. Bollobás, Graph theory: an introductory course, GTM 63 (Springer-Verlag, New York, 1979).
17.
C. P. Rourke and B. J. Sanderson, Introduction to Piecewise-Linear Topology (Springer-Verlag, New York, 1982).
18.
A. Besse, Einstein Manifolds (Springer-Verlag, New York, 1986).
19.
E.
Witten
, “Two-dimensional gravity and intersection theory on moduli space
,” Surveys Diff. Geom.
1
, 243
–310
(1991
). (Lehigh University, Bethlehem, PA);E.
Witten
, “Two-dimensional gauge theories revisited
,” J. Geom. Phys.
9
, 303
–368
(1992
);L. C. Jeffrey and J. Weitsman, “Half density quantization of the moduli space of flat connections and Witten’s semiclassical manifold invariants,” preprint IASSNS-HEP-91/94;
M.
Kontsevitch
, “Intersection theory on the moduli space of curves and the matrix Airy functions
,” Commun. Math. Phys.
147
, 1
–23
(1992
).
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