Nekrasov and Argyres-Douglas theories in spherical Hecke algebra representation

TL;DR

This paper explores the connection between Nekrasov partition functions and Liouville conformal blocks using spherical Hecke algebra, constructing gauge conformal states and deriving irregular states for Argyres-Douglas theories.

Contribution

It introduces a novel approach using the central extension of spherical Hecke algebra in q-coordinate representation to relate gauge and conformal states, including irregular states.

Findings

Constructed gauge conformal states matching Liouville states

Derived formal irregular conformal states for Argyres-Douglas theories

Established the role of Young diagrams in irregular state expressions

Abstract

AGT conjecture connects Nekrasov instanton partition function of 4D quiver gauge theory with 2D Liouville conformal blocks. We re-investigate this connection using the central extension of spherical Hecke algebra in q-coordinate representation, q being the instanton expansion parameter. Based on AFLT basis together with interwiners we construct gauge conformal state and demonstrate its equivalence to the Liouville conformal state, with careful attention to the proper scaling behavior of the state. Using the colliding limit of regular states, we obtain the formal expression of irregular conformal states corresponding to Argyres-Douglas theory, which involves summation of functions over Young diagrams.