Flavor deformations and supersymmetry enhancement in $4d$ $\mathcal{N}=2$ theories

TL;DR

This paper explores how deforming 4d $ abla=2$ theories with background fields at fixed points can lead to supersymmetry enhancement and independence of certain parameters, revealing new insights into their structure and dualities.

Contribution

It introduces a method to deform $ abla=2$ theories on four-manifolds that results in supersymmetry enhancement and parameter independence, connecting to AGT correspondence and Liouville/Toda CFT.

Findings

Partition function depends only on background scalar values at fixed points.

Supersymmetry enhances from Donaldson-Witten to Marcus or Vafa-Witten twists at special scalar values.

Identifies a new squashing independent point in $ abla=2^*$ theory.

Abstract

We study $\mathcal{N}=2$ theories on four-dimensional manifolds that admit a Killing vector $v$ with isolated fixed points. It is possible to deform these theories by coupling position-dependent background fields to the flavor current multiplet. The partition function of the deformed theory only depends on the value of the background scalar fields at the fixed points of $v$. For a single adjoint hypermultiplet, the partition function becomes independent of the supergravity as well as the flavor background if the scalars attain special values at the fixed points. For these special values, supersymmetry at the fixed points enhances from the Donaldson-Witten twist to the Marcus twist or the Vafa-Witten twist of $\mathcal{N}=4$ SYM. Our results explain the recently observed squashing independence of $\mathcal{N}=2^*$ theory on the squashed sphere and provide a new squashing independent point. Interpreted through the AGT-correspondence, this implies the $b$-independence of torus one-point functions of certain local operators in Liouville/Toda CFT. The position-dependent deformations imply relations between correlators of partially integrated operators in any $\mathcal{N}=2$ SCFT with flavor symmetries.