Publication : t90/178

Non-perturbative effects in 2D gravity and matrix models

David F. ()
Two dimensional Euclidean quantum gravity may be formulated as a functional integral over 2--dimensional Riemannian manifolds. This infinite dimensional integral may be discretized in such a way that the topological expansion in terms of the genus of the manifold is mapped onto the $1/N$ expansion of some zero--dimensional matrix model \refs{\Planar}. The $N=\infty$ limit exhibits critical points which can be shown to describe the continuum limit of 2--dimensional gravity on a genus zero manifold, eventually coupled to some matter fields. Recently it was shown that a scaling limit can be constructed \ref\Scaling{E. Br\'ezin and V. A. Kazakov, Phys. Lett. 236B (1990) 2125. M. R. Douglas and S. H. Shenker, Nucl. Phys. B335 (1990) 635. D. J. Gross and A. A. Migdal, Phys. rev. Lett. 64 (1990) 27.} . In this limit all the terms of the topological expansion survive and thus one obtains a fully non--perturbative solution for two dimensional gravity. However in the most interesting cases, in particular for pure gravity, the solution is defined as a solution of a non--linear differential equation of the Painlev\'e type and presents some non--perturbative ambiguities, related to the delicate issue of boundary conditions, which are usually attributed to some ``non--perturbative effects" of the theory. In this talk I shall review some attempts to get a better understanding of these effects. For simplicity and shortness I shall mainly deal with the case of pure gravity, which seems to embody the main problems. The approach that I have followed consists in trying to relate those non--perturbative issues to the non--perturbative effects which are present in the original matrix models.
Année de publication : 1990
Chapitre de livre : Random Surfaces and Quantum Gravity, volume 262 in NATO ASI series B
Maison d'édition : Plenum Press, 1990
Pages : 21-34
Ecole - Communication invitée : Random Surfaces and Quantum Gravity ; Cargèse, France ; 1990-05-28 / 1990-06-01
Langue : Anglais
NB : pp. 21-34
Editeurs : Alvarez O., Marinari E., Windey P.

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