Open minimal strings and open Gelfand-Dickey hierarchies

Abstract

We study the connection between minimal Liouville string theory and generalized open KdV hierarchies. We are interested in generalizing Douglas string equation formalism to the open topology case. We show that combining the results of the closed topology, based on the Frobenius manifold structure and resonance transformations, with the appropriate open case modification, which requires the insertion of macroscopic loop operators, we reproduce the well-known result for the expectation value of a bulk operator for the FZZT brane coupled to the general (q, p) minimal model. The matching of the results of the two setups gives new evidence of the connection between minimal Liouville gravity and the theory of Topological Gravity.

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.…”

Topological Two-Dimensional Gravity on Surfaces with Boundary

“…While aspects of gravity on Riemann surfaces with boundary were partially understood in terms of matrix models early on [9,10], a rigorous theory of topological gravity on Riemann surfaces with boundary was only recently established [11]. Since then, various perspectives on these theories have been developed [12][13][14][15][16][17]. The main approaches are through geometry and matrix models.…”

Topological open/closed string dualities: matrix models and wave functions

“…This gave rise to a flurry of mathematical activity which made earlier observations in the physics literature [13][14][15][16][17] precise [18][19][20][21]. See [22] for a partial review peppered with physical insight and [23][24][25] for more recent results.…”