Generalized Whittaker states for instanton counting with fundamental hypermultiplets

TL;DR

This paper characterizes generalized Whittaker states in conformal field theory that correspond to instanton partition functions of N=2 gauge theories with fundamental hypermultiplets, extending previous constructions.

Contribution

It introduces new conditions for Whittaker states associated with SU(3) and SU(2) theories with surface operators, incorporating lower level annihilation operators and zero modes.

Findings

States are not strict coherent states but are characterized by specific annihilation operators.

Provides explicit conditions for states corresponding to theories with hypermultiplets.

Extends the understanding of instanton partition functions in the context of conformal field theory.

Abstract

M-theoretic construction of N=2 gauge theories implies that the instanton partition function is expressed as the scalar product of coherent states (Whittaker states) in the Verma module of an appropriate two dimensional conformal field theory. We present the characterizing conditions for such states that give the partition function with fundamental hypermultiplets for SU(3) theory and SU(2) theory with a surface operator. We find the states are no longer the coherent states in the strict sense but we can characterize them in terms of a few annihilation operators of lower levels combined with the zero mode (Cartan part) of the Virasoro algebra L_0 or the sl(2) current algebra J_0^0.