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 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, 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.

Open KdV hierarchy of 2d minimal gravity of Lee-Yang series

“…This is because the KdV hierarchy of the one-matrix model in this dual picture does not work anymore and is to be replaced by the Gelfand-Dickey hierarchy. In addition, the flow parameter of the dual theory coincides with the conformal parameter in the Liouville minimal gravity [24], which is in contrast with the original A 1 description since the resonance transformation among those parameters plays a central role in general [20,25,26]. As a result, the open hierarchy in the dual picture will show very different behavior unlike the open KdV hierarchy.…”

Open KdV hierarchy and minimal gravity on disk

“…It is known that its dual A 2p Frobenius manifold (which has 2p number of coordinates {u a }) can describe the same correlation on sphere. However, the correlation on disk is not fully understood in terms of the dual manifold due to the non-canonical nature of the integral representation of the free energy [21,22]. The open KdV hierarchy can be a guide to define the free energy on disk whose details will be considered in a separate paper.…”

From minimal gravity to open intersection theory

“…Finally, one may expect that the idea of this paper can be extended to the case of the so-called r-spin open intersection numbers [41]. One may relate this theory with M (q, p) series of minimal gravity in terms of the A q−1 Frobenius manifolds [22,23,[42][43][44]. It would be interesting to apply the Laplace transform to investigation of the open p − q duality.…”