Perturbing the Ground Ring of 2-D String Theory

TL;DR

This paper investigates how the algebraic structure of 2D string theory's ground ring changes under perturbations by the cosmological constant, revealing a geometric transition to a hyperboloid shape.

Contribution

It computes the perturbed ground ring algebra in 2D string theory and confirms the geometric conjecture of a hyperboloid shape, also analyzing the moduli deformation algebra.

Findings

Ground ring preserved as a hyperboloid under perturbation

Explicit calculation of the (1,1) current algebra of moduli deformations

Discussion of Liouville/matrix model dictionary in this context

Abstract

We use free field techniques in D=2 string theory to calculate the perturbation of the special state algebras when the cosmologi- cal constant is turned on. In particular, we find that the "ground cone" preserved by the ring structure is promoted to a three dimen- sional hyperboloid as conjectured by Witten. On the other hand, the perturbed (1,1) a three dimensional hyperboloid as conjectured by Witten. On the other hand, the perturbed (1,1) current algebra of moduli deformations is computed completely, and no simple geometrical inter- pretation is found. We also quote some facts concerning the Liouville/matrix model dictio- nary in this class of theories.