$SU(2)_k\times SU(2)_l/SU(2)_{k+l}$ Coset Conformal Field Theory and Topological Minimal Model on Higher Genus Riemann Surface

TL;DR

This paper extends the coset minimal conformal field theory to higher genus Riemann surfaces, developing a BRST formalism and calculating correlation functions, with applications to topological minimal models.

Contribution

It introduces a BRST-invariant formalism for the $SU(2)_k\times SU(2)_l/SU(2)_{k+l}$ coset theory on higher genus surfaces and connects it to topological minimal models.

Findings

Derived BRST invariant screened $g$-loop operator for higher genus

Extended vertex operator formalism to higher genus surfaces

Calculated correlation functions on higher genus surfaces

Abstract

We consider the Feigin-Fuchs-Felder formalism of the $SU(2)_k\times SU(2)_l/SU(2)_{k+l}$ coset minimal conformal field theory and extend it to higher genus. We investigate a double BRST complex with respect to two compatible BRST charges, one associated with the parafermion sector and the other associated with the minimal sector in the theory. The usual screened vertex operator is extended to the BRST invariant screened three string vertex. We carry out a sewing operation of these string vertices and derive the BRST invariant screened $g$-loop operator. The latter operator characterizes the higher genus structure of the theory. An analogous operator formalism for the topological minimal model is obtained as the limit $ l=0$ of the coset theory. We give some calculations of correlation functions on higher genus.