Omega-deformed SYM on a Gibbons-Hawking Space

TL;DR

This paper explores an Omega-deformed N=2 U(1) SYM theory on a Gibbons-Hawking space, revealing a hyperkähler sigma model with a deformed complex structure and a NUT center contribution expressed as a Jacobi theta function.

Contribution

It introduces a novel Omega-deformation of N=2 SYM on Gibbons-Hawking spaces and reformulates it as a generalized hyperkähler sigma model with explicit dependence on the deformation parameter.

Findings

The deformed theory differs from the undeformed at NUT centers.

The target space's complex structure depends on the Omega-deformation parameter.

NUT center contribution is a Jacobi theta function.

Abstract

We study an $\mathcal{N}=2$, pure $U(1)$ SYM theory on a Gibbons-Hawking space $\Omega$-deformed using the $U(1)$ isometry. The resultant 3D theory, after an appropriate "Nekrasov-Witten" change of variables, is asymptotically equivalent to the undeformed theory at spatial infinity but differs from it as one approaches the NUT centers which are fixed points under the $U(1)$ action. The 3D theory may be recast in the form of a generalized hyperk\"ahler sigma model introduced in \cite{Dey:2014lja} where the target space is a one-parameter family of hyperk\"ahler spaces. The hyperk\"ahler fibers have a preferred complex structure which for the deformed theory depends on the parameter of $\Omega$-deformation. The metric on the hyperk\"ahler fiber can be reduced to a standard metric on $\mathbb{C} \times T^2$ with the modular parameter of the torus depending explicitly on the $\Omega$-deformation parameter. The contribution of the NUT center to the sigma model path integral, expected to be a holomorphic section of a holomorphic line bundle over the target space on grounds of supersymmetry, turns out to be a Jacobi theta function in terms of certain "deformed" variables.