Dual Frobenius manifolds of minimal gravity on disk

Abstract

Liouville field theory approach to 2-dimensional gravity possesses the duality (b ↔ b−1). The matrix counterpart of minimal gravity ℳ(q, p) (q < p co-prime) is effectively described on Aq−1 Frobenius manifold, which may exhibit a similar duality p ↔ q, and allow a description on Ap−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 ℳ(q, q + 1). However, for the Lee-Yang series 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.