$(1,q=-1)$ Model as a Topological Description of $2d$ String Theory

TL;DR

This paper explores the $(1,q=-1)$ topological gravity model as a potential description of 2D string theory at the self-dual radius, establishing recursion relations and Ward identities that connect to $c=1$ string correlators.

Contribution

It introduces an analytical continuation of topological recursion relations to $q=-1$, linking them to $W_{1+ obreak ext{-}infinity}$ Ward identities and proposing a scheme for computing correlators of discrete states.

Findings

Recursion relations at genus zero yield $W_{1+ obreak ext{-}infinity}$ Ward identities.

A scheme for computing correlators of gravitational descendants is proposed.

Genus one recursion relations match Ward identities, suggesting a universal structure.

Abstract

We study the $(1,q=-1)$ model coupled to topological gravity as a candidate to describing $2d$ string theory at the self-dual radius. We define the model by analytical continuation of $q>1$ topological recursion relations to $q=-1$. We show that at genus zero the $q=-1$ recursion relations yield the $W_{1+\infty}$ Ward identities for tachyon correlators on the sphere. A scheme for computing correlation functions of $q=-1$ gravitational descendants is proposed and applied for the computation of several correlators. It is suggested that the latter correspond to correlators of discrete states of the $c=1$ string. In a similar manner to the $q>1$ models, we show that there exist topological recursion relations for the correlators in the $q=-1$ theory that consist of only one and two splittings of the Riemann surface. Using a postulated regularized contact, we prove that the genus one $q=-1$ recursion relations for tachyon correlators coincide with the $W_{1+\infty}$ Ward identities on the torus. We argue that the structure of these recursion relations coincides with that of the $W_{1+\infty}$ Ward identities for any genus.