Fermionic Quantum Gravity

TL;DR

This paper investigates fermionic matrix models related to quantum gravity, revealing their combinatorial structure, integrability properties, and continuum limits, thus connecting discrete models with continuum two-dimensional quantum gravity.

Contribution

It introduces fermionic matrix models equivalent to unitary generalized Penner models, solves their combinatorics, and links them to integrable hierarchies and continuum quantum gravity.

Findings

Genus expansion is Borel summable and alternating.

Partition functions belong to the relativistic Toda chain hierarchy.

Continuum limit relates to two-dimensional quantum gravity.

Abstract

We study the statistical mechanics of random surfaces generated by NxN one-matrix integrals over anti-commuting variables. These Grassmann-valued matrix models are shown to be equivalent to NxN unitary versions of generalized Penner matrix models. We explicitly solve for the combinatorics of 't Hooft diagrams of the matrix integral and develop an orthogonal polynomial formulation of the statistical theory. An examination of the large N and double scaling limits of the theory shows that the genus expansion is a Borel summable alternating series which otherwise coincides with two-dimensional quantum gravity in the continuum limit. We demonstrate that the partition functions of these matrix models belong to the relativistic Toda chain integrable hierarchy. The corresponding string equations and Virasoro constraints are derived and used to analyse the generalized KdV flow structure of the continuum limit.