Open KdV hierarchy and minimal gravity on disk

Abstract

We show that the minimal gravity of Lee-Yang series on disk is a solution to the open KdV hierarchy proposed for the intersection theory on the moduli space of Riemann surfaces with boundary.

From minimal gravity to open intersection theory

“…Similar interplay is noted for the models on the Riemann surfaces with boundaries [4,5]. The GF of the MG with boundaries (BMG) satisfies the relations, similar to the ones known for the open intersection theory (OIT) [6].…”

“…Here v(x) is a function of x, which is governed by the string equation of the polynomial form (2.7), with τ 0 substituted by x. The GF in (2.30) satisfies the following equation [4]:…”

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

“…One may conjecture that the free energy F disk on a disk [21,22] can be a solution of the hierarchy (3.5) [15]. It is noted that the free energy F disk corresponds to the continuum limit of the trace of log(M + µ B ) where the one-matrix element is replaced by Q and µ B is the boundary cosmological constant.…”

“…It is naturally expected that the open KdV hierarchy also describes minimal gravity of Lee-Yang series on a disk. In our previous paper [15], we check this expectation using the free energy on a disk, where the boundary cosmological constant µ B is introduced as an additional boundary parameter that has a similar role of the boundary flow parameter s in the intersection theory [16,17,18,19].…”

Open minimal strings and open Gelfand-Dickey hierarchies

“…In this paper we continue our study [15] (see also [16][17][18], focused on (2, 2k + 1) Lee-Yang series) of the boundary version of minimal Liouville gravity (BMLG). In the worldsheet approach, constructing correlation numbers, apart from the moduli integration problems [19], needs a set of new ingredients [20][21][22][23][24][25], connected with the boundary, and represents quite a non-trivial problem.…”