Algebraic-geometrical formulation of two-dimensional quantum gravity

TL;DR

This paper develops an algebraic-geometrical framework for two-dimensional quantum gravity, connecting moduli space integrals with Painlevé I recursion relations and proposing a conjectural derivation using the Duistermaat-Heckman theorem.

Contribution

It introduces a volume form on moduli space satisfying Painlevé I relations and links the asymptotic expansion of specific heat to integrals over infinite-dimensional moduli space, offering a new geometric perspective.

Findings

Integral on moduli space satisfies Painlevé I recursion relations.

Asymptotic expansion expressed as an integral over infinite-dimensional moduli space.

Proposed conjectural derivation using Duistermaat-Heckman theorem.

Abstract

We find a volume form on moduli space of double punctured Riemann surfaces whose integral satisfies the Painlev\'e I recursion relations of the genus expansion of the specific heat of 2D gravity. This allows us to express the asymptotic expansion of the specific heat as an integral on an infinite dimensional moduli space in the spirit of Friedan-Shenker approach. We outline a conjectural derivation of such recursion relations using the Duistermaat-Heckman theorem.