BRST Cohomology in Quantum Affine Algebra $U_q(\widehat{sl_2})$

TL;DR

This paper explores the structure of Fock modules over the quantum affine algebra $U_q( ext{sl}_2)$ using a $q$-deformed BRST formalism, constructing singular vectors and analyzing cohomology to understand irreducible representations.

Contribution

It introduces a $q$-analog of the BRST formalism for quantum affine algebras and explicitly constructs singular vectors within this framework.

Findings

Explicit construction of singular vectors using BRST charge

Identification of irreducible highest weight modules as cohomology groups

Calculation of traces of $q$-vertex operators on these modules

Abstract

Using free field representation of quantum affine algebra $U_q(\widehat{sl_2})$, we investigate the structure of the Fock modules over $U_q(\widehat{sl_2})$. The analisys is based on a $q$-analog of the BRST formalism given by Bernard and Felder in the affine Kac-Moody algebra $\widehat {sl_2}$. We give an explicit construction of the singular vectors using the BRST charge. By the same cohomology analysis as the classical case ($q=1$), we obtain the irreducible highest weight representation space as a nontrivial cohomology group. This enables us to calculate a trace of the $q$-vertex operators over this space.