Physical States in G/G Models and 2d Gravity

TL;DR

This paper analyzes the BRST cohomology of G/G topological models, computes physical states, and establishes a connection between SL(2)/SL(2) models and minimal models coupled to 2D gravity.

Contribution

It provides a detailed cohomological analysis of G/G models and reveals a novel correspondence with minimal models coupled to gravity.

Findings

Computed characters and partition functions of the models.

Established a correspondence between SL(2)/SL(2) models and (p,q) gravity models.

Analyzed the cohomology structure using singular vectors and fusion rules.

Abstract

An analysis of the BRST cohomology of the G/G topological models is performed for the case of $A_1^{(1)}$. Invoking a special free field parametrization of the various currents, the cohomology on the corresponding Fock space is extracted. We employ the singular vector structure and fusion rules to translate the latter into the cohomology on the space of irreducible representations. Using the physical states we calculate the characters and partition function, and verify the index interpretation. We twist the energy-momentum tensor to establish an intriguing correspondence between the ${SL(2)\over SL(2)}$ model with level $k={p\over q}-2$ and $(p,q)$ models coupled to gravity.