Minimal gravity and Frobenius manifolds: bulk correlation on sphere and disk

TL;DR

This paper extends the matrix model approach to minimal gravity by analyzing bulk correlation functions on the disk topology, utilizing Frobenius manifold structures to connect with Liouville gravity results.

Contribution

It develops the Frobenius manifold framework for gravitating disk topology and demonstrates its consistency with Liouville gravity resonance transformations.

Findings

Explicit form of resonance transformations for disk topology.

Generation of bulk correlators using flat coordinates on Frobenius manifolds.

Reproduction of FZZ results within the matrix model approach.

Abstract

There are two alternative approaches to the minimal gravity - direct Liouville approach and matrix models. Recently there has been a certain progress in the matrix model approach, growing out of presence of a Frobenius manifold (FM) structure embedded in the theory. The previous studies were mainly focused on the spherical topology. Essentially, it was shown that the action principle of Douglas equation allows to define the free energy and to compute the correlation numbers if the resonance transformations are properly incorporated. The FM structure allows to find the explicit form of the resonance transformation as well as the closed expression for the partition function. In this paper we elaborate on the case of gravitating disk. We focus on the bulk correlators and show that in the similar way as in the closed topology the generating function can be formulated using the set of flat coordinates on the corresponding FM. Moreover, the resonance transformations, which follow from the spherical topology consideration, are exactly those needed to reproduce FZZ result of the Liouville gravity approach.