Dual Frobenius manifolds of minimal gravity on disk

TL;DR

This paper investigates dual Frobenius manifold descriptions of minimal gravity on disks, confirming duality for unitary series and exploring partial evidence for Lee-Yang series, highlighting challenges in continuum trace definitions.

Contribution

It demonstrates the duality between $A_{q-1}$ and $A_{p-1}$ Frobenius manifolds in minimal gravity on disks, especially for unitary series, and discusses partial results for Lee-Yang series.

Findings

Dual Frobenius manifold description confirmed for unitary series.

Partial evidence of duality for Lee-Yang series.

Challenges identified in continuum trace formulation.

Abstract

Liouville field theory approach to 2-dimensional gravity possesses the duality ($b \leftrightarrow b^{-1}$). The matrix counterpart of minimal gravity $\mathcal{M}(q,p)$ ($q<p$ co-prime) is effectively described on $A_{q-1}$ Frobenius manifold, which may exhibit a similar duality $p\leftrightarrow q$, and allow a description on $A_{p-1}$ Frobenius manifold. We have positive results from the bulk one-point and the bulk-boundary two-point correlations on disk that the dual description of the Frobenius manifold works for the unitary series $\mathcal{M}(q, q+1)$. However, for the Lee-Yang series $\mathcal{M}(2, 2q+1)$ on disk the duality is checked only partially. The main difficulty lies in the absence of a canonical description of trace in the continuum limit.