Chiral Rings and Physical States in c<1 String Theory

TL;DR

This paper explores the algebraic structure of chiral rings in c<1 string theory using double cohomology of BRST charges, revealing polynomial relations and connecting to matrix model observables.

Contribution

It introduces a cohomological framework for understanding the chiral ring structure in c<1 string theory and identifies operators related to matrix model observables.

Findings

Chiral ring is polynomials modulo x^p ≃ y^{p+1} relation.

States at the edges of the conformal grid are essential for ring closure.

Candidate operators are proposed as observables linked to matrix models.

Abstract

We show how the double cohomology of the String and Felder BRST charges naturally leads to the ring structure of $c<1$ strings. The chiral ring is a ring of polynomials in two variables modulo an equivalence relation of the form $x^p \simeq y^{p+1}$ for the (p+1,p) model. We also study the states corresponding to the edges of the conformal grid whose inclusion is crucial for the closure of the ring. We introduce candidate operators that correspond to the observables of the matrix models. Their existence is motivated by the relation of one of the screening operators of the minimal model to the zero momentum dilaton.