Generalized cohomologies and the physical subspace of the $SU(2)$ WZNW model

TL;DR

This paper explores the structure of the physical subspace in the SU(2) WZNW model using generalized cohomologies, revealing a finite-dimensional gauge theory with a novel BRS operator and invariant vectors.

Contribution

It introduces a generalized BRS operator with a finite order, analyzing its cohomologies to characterize the physical subspace in the SU(2) WZNW model.

Findings

Generalized cohomologies are 1-dimensional.

The physical subspace is spanned by these cohomologies.

A finite-dimensional gauge theory structure emerges from the zero modes.

Abstract

The zero modes of the monodromy extended SU(2) WZNW model give rise to a gauge theory with a finite dimensional state space. A generalized BRS operator $A$ such that $A^h=0 (h=k+2=3,4,...$ being the height of the current algebra representation) acts in a (2h-1)-dimensional indefinite metric space $H_I$ of quantum group invariant vectors. The generalized cohomologies $Ker A^n/ Im A^{h-n} (n=1,..., h-1)$ are 1-dimensional. Their direct sum spans the physical subquotient of $H_I$.