Simon Dalley

Multicritical complex matrix models and non-perturbative two-dimensional quantum gravity

The one-complex-matrix model is analysed by orthogonal polynomials and saddle point methods. For the double scaled limit, multicritical matter theories arising from the perturbative phase have 12 the free energy of that derived from the hermitian matrices. However, certain other potentials yield a new hierarchy of “string equations”. These string equations are the most general equations compatible with the same operator content, KdV flows and asymptotic expansions of the m-critical points in hermitian matrix models. In particular, the non-perturbative solution of the m = 2 equation, corresponding to 2D quantum gravity, yields the same genus expansion as the cosmological constant μ tends to +∞, but tends to zero as μ → −∞. Semi-classical analysis shows that this solution is real and stable at this level.

Non-perturbative two-dimensional quantum gravity

Abstract The KdV flow structure of 2D quantum gravity, known to exist perturbatively, holds non-perturbatively if and only if the string susceptibility obeys a unique “string equation”. Physical boundary conditions then imply that there is at most a discrete set of acceptable solutions and a numerical study of pure gravity reveals only one. The non-perturbative solution differs from that of the one-dimensional superstring/stochastic quantization method, which violates the KdV flow structure non-perturbatively. Our Virasoro algebra is the same as in hermitian matrix models except for a missing translation constraint L −1 τ = 0. The string equation arises from the dilatation constraint L 0 τ = 0. We present the corresponding string equations for gravity coupled to any ( p , q ) minimal model.

THE WEINGARTEN MODEL À LA POLYAKOV

The Weingarten lattice gauge model of Nambu-Goto strings is generalized to allow for fluctuations of an intrinsic worldsheet metric through a dynamical quadrilation. The continuum limit is taken for c≤1 matter, reproducing the results of Hermitian matrix models to all orders in the genus expansion. For the compact c=1 case the vortices are Wilson lines, whose exclusion leads to the theory of non-interacting fermions. As a by-product of the analysis one finds the critical behavior of SOS and vertex models coupled to 2D quantum gravity.The Weingarten lattice gauge model of Nambu-Goto strings is generalised to allow for fluctuations of an intrinsic worldsheet metric through a dynamical quadrilation. The continuum limit is taken for

CLASSIFICATION OF CRITICAL HERMITIAN MATRIX MODELS

c\leq1

Nonperturbative two-dimensional quantum gravity, again

matter, reproducing the results of hermitian matrix models to all orders in the genus expansion. For the compact

Instability of even m multicritical matrix models of 2D gravity

c=1

String theory and quantum spin chains

case the vortices are Wilson lines, whose exclusion leads to the theory of non-interacting fermions. As a by-product of the analysis one finds the critical behaviour of SOS and vertex models coupled to 2D quantum gravity.

Factorization properties of critical matrix models

The critical properties of Hermitian matrix models in the one-arc phase may be simply understood and completely classified by the behavior of the eigenvalue distribution at its ends. The most general critical behavior involves two scaling functions naturally associated with each end of the distribution, and two KdV-type string equations with differing values of the critical index m. The critical conditions are shown to include previous discoveries as special cases.

PHASE STRUCTURE IN BOSONIC STRING THEORY

This is a talk given by S.D. at the the workshop on Random Surfaces and 2D Quantum Gravity, Barcelona 10-14 June 1991. It is an updated review of recent work done by the authors on a proposal for non-perturbatively stable 2D quantum gravity coupled to c<1 matter, based on the flows of the (generalised) KdV hierarchy.This is a talk given by S.D. at the the workshop on Random Surfaces and 2D Quantum Gravity, Barcelona 10-14 June 1991. It is an updated review of recent work done by the authors on a proposal for non-perturbatively stable 2D quantum gravity coupled to c<1 matter, based on the flows of the (generalised) KdV hierarchy.

NON-PERTURBATIVE DYNAMICS

Abstract The hermitian one-matrix model of two-dimensional gravity is studied in the large N limit by the saddle-point method for a general even matrix potential giving a well defined partition function. A simple demonstration of the connection between instability of m even models and the existence of multi-arc phases is given.