On the Nekrasov Partition Function of Gauged Argyres-Douglas Theories

TL;DR

This paper extends the calculation of Nekrasov partition functions for $SU(2)$ gauge theories coupled to $(A_1,D_N)$ theories, providing a new formula for odd $N$ and revealing a relation between the $(A_2,A_5)$ theory and $SU(2)$ with four flavors.

Contribution

We generalize the Nekrasov partition function formula for odd $N$ in $(A_1,D_N)$ theories and connect the $(A_2,A_5)$ theory's prepotential to a known gauge theory.

Findings

Derived a formula for odd $N$ in the classical limit.

Connected the $(A_2,A_5)$ prepotential to $SU(2)$ with four flavors.

Extended previous results for even $N$ to odd $N$ cases.

Abstract

We study $SU(2)$ gauge theories coupled to $(A_1,D_N)$ theories with or without a fundamental hypermultiplet. For even $N$, a formula for the contribution of $(A_1,D_N)$ to the Nekrasov partition function was recently obtained by us with Y.~Sugawara and T.~Uetoko. In this paper, we generalize it to the case of odd $N$ in the classical limit, under the condition that the relevant couplings and vacuum expectation values of Coulomb branch operators of $(A_1,D_N)$ are all turned off. We apply our formula to the $(A_2,A_5)$ theory to find that its prepotential is related to that of the $SU(2)$ gauge theory with four fundamental flavors by a simple change of variables.