On the Jeffrey-Kirwan residue of BCD-instantons

TL;DR

This paper employs the Jeffrey-Kirwan residue technique to evaluate Nekrasov partition functions for BCD-type gauge theories, introducing a graphical rule for pole selection and computing instanton corrections for specific cases.

Contribution

It introduces a graphical distinction rule for pole determination in Jeffrey-Kirwan integrals and computes explicit instanton corrections for BCD-type theories, extending previous methods.

Findings

Derived a pole selection rule for Jeffrey-Kirwan integrals.

Computed instanton partition functions for $Sp(0)$ gauge theory.

Found a formula for $Z^{Sp(0)}_{k}$ resembling $U(1)$ gauge theory results.

Abstract

We apply the Jeffrey-Kirwan method to compute the multiple integrals for the $BCD$ type Nekrasov partition functions of four dimensional $\mathcal{N}=2$ supersymmetric gauge theories. We construct a graphical distinction rule to determine which poles are surrounded by their integration cycles. We compute the instanton correction of the "$Sp(0)$" pure super-Yang-Mills theory and find that $Z^{Sp(0)}_{k}=(-1)^{k}(2^{k}k!\varepsilon_{1}^{k}\varepsilon_{2}^{k})^{-1}$ for $k\le 8$, which resembles the formula $Z^{U(1)}_{k}=(k!\varepsilon_{1}^{k}\varepsilon_{2}^{k})^{-1}$ for the pure super-Yang-Mills theory with gauge group $U(1)$.