On Integrable c<1 Open--Closed String Theory

TL;DR

This paper explores the integrable structure of open-closed string theories within minimal models, revealing their connection to $sl(q,C)$ KdV hierarchies and providing tools for deriving loop equations and correlation functions.

Contribution

It uncovers the relation between open-closed and closed string tau-functions and embeds the open-closed string theory into the $sl(q,C)$ KdV hierarchy framework.

Findings

The tau-function of open-closed string theory relates to that of closed strings.

The structure fits into the $sl(q,C)$ KdV hierarchy.

Twisted bosons facilitate derivation of loop equations and recursion relations.

Abstract

The integrable structure of open--closed string theories in the $(p,q)$ conformal minimal model backgrounds is presented. The relation between the $\tau$--function of the closed string theory and that of the open--closed string theory is uncovered. The resulting description of the open--closed string theory is shown to fit very naturally into the framework of the $sl(q,{\rm C})$ KdV hierarchies. In particular, the twisted bosons which underlie and organise the structure of the closed string theory play a similar role here and may be employed to derive loop equations and correlation function recursion relations for the open--closed strings in a simple way.