Ground Rings and Their Modules in 2D Gravity with $c\le 1$ Matter

TL;DR

This paper explores the algebraic structure of 2D quantum gravity models with matter central charge less than or equal to one, revealing rings and modules that constrain correlation functions and aid in solving these theories.

Contribution

It introduces the concept of rings and modules formed by BRST cohomology states, providing a new algebraic framework for understanding 2D gravity models.

Findings

Identification of a ring structure in BRST cohomology

Modules of states with physical interpretation

Constraints on correlation functions derived

Abstract

All solvable two-dimensional quantum gravity models have non-trivial BRST cohomology with vanishing ghost number. These states form a ring and all the other states in the theory fall into modules of this ring. The relations in the ring and in the modules have a physical interpretation. The existence of these rings and modules leads to nontrivial constraints on the correlation functions and goes a long way toward solving these theories in the continuum approach.