Construction of Gaiotto states with fundamental multiplets through Degenerate DAHA

TL;DR

This paper constructs Gaiotto states with fundamental multiplets for $SU(N)$ gauge theories using the spherical degenerate double affine Hecke algebra, connecting it to $W_n$ algebra representations and generalized Whittaker conditions.

Contribution

It introduces a novel method to construct Gaiotto states via SH algebra, extending the approach to general $SU(N)$ cases and linking to $W_n$ algebra structures.

Findings

Construction of Gaiotto states with fundamental multiplets using SH algebra

Demonstration of generalized Whittaker conditions in this framework

Extension of the method to arbitrary $SU(N)$ gauge theories

Abstract

We construct Gaiotto states with fundamental multiplets in $SU(N)$ gauge theories, in terms of the orthonormal basis of spherical degenerate double affine Hecke algebra (SH in short), the representations of which are equivalent to those of $W_n$ algebra with additional $U(1)$ current. The generalized Whittaker conditions are demonstrated under the action of SH, and further rewritten in terms of $W_n$ algebra. Our approach not only consists with the existing literature but also holds for general $SU(N)$ case.