On Higgs Branch Localization of Seiberg-Witten Theories on Ellipsoid

TL;DR

This paper explores Higgs branch localization in 4d N=2 theories on ellipsoids, revealing new saddle points, factorization properties of the partition function, and connections to 2d vortex theories.

Contribution

It introduces new saddle point equations from Higgs branch deformations and demonstrates the factorization of the ellipsoid partition function into vortex-related components.

Findings

Partition function factorizes into b and 1/b dependent parts.

Identifies vortex world volume theory as 2d N=(2,2) SQCDA.

Includes solutions with Higgs and instanton-vortex configurations.

Abstract

In this note, we consider so-called "Higgs Branch Localization" for four dimensional N=2 field theories on 4d ellipsoid. We find a new set of saddle point equations arising from additional Higgs branch deformation term, whose solutions include both Higgs branch and BPS instanton-vortex mixed configurations. By evaluating the contour integral, we also demonstrate the ellipsoid partition almost factorizes into purely b and 1/b dependent parts, using SQCD as an explicit example. We identify various factorized parts with the ellipsoid partition function of two dimensional N=(2,2) SQCDA, which is precisely the vortex world volume theory. We also give physical interpretation for the non-factorizable parts and discuss future directions.