Dual Frobenius manifolds of minimal gravity on disk

A bstract 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 A q −1 Frobenius manifold, which may exhibit a similar duality p ↔ 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 ℳ( q , q + 1). However, for the Lee-Yang series ℳ(2, 2 q + 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.