Abstract
We consider mass-deformed theories with \mathcal{N}\geq2𝒩≥2 supersymmetry on round and squashed three-spheres. By embedding the supersymmetric backgrounds in extended supergravity we show that at special values of mass deformations the supersymmetry is enhanced on the squashed spheres. When the 3d partition function can be obtained by a limit of a 4d index we also show that for these special mass deformations only the states annihilated by extra supercharges contribute to the index. By using an equivalence between partition functions on squashed spheres and ellipsoids, we explain the recently observed squashing independence of the partition function of mass-deformed ABJ(M) theory on the ellipsoid. We provide further examples of such simplification for various 3d supersymmetric theories.
Highlights
Introduction and summaryLocalization is a powerful tool with a broad range of applications in supersymmetric field theories [1]
It is still possible to localize the supersymmetric theory after squashing the sphere, that is deforming the metric by a parameter b, where b=1 corresponds to the round sphere [3,4,5,6]
For the first kind of squashing independence, it is sufficient to consider the transversely holomorphic foliation (THF) that determines the partition functions of the supersymmetric theories. We show that this structure is the same for the round sphere and the SU(2) × U(1)-deformed sphere of [3]
Summary
Localization is a powerful tool with a broad range of applications in supersymmetric field theories [1]. For the first kind of squashing independence, it is sufficient to consider the transversely holomorphic foliation (THF) that determines the partition functions of the supersymmetric theories. We show that this structure is the same for the round sphere and the SU(2) × U(1)-deformed sphere of [3]. The rest of this paper is structured as follows: In section 2 we review general features of supersymmetric partition functions on three manifolds and analyze the THF on the round sphere, the U(1)× U(1) symmetric ellipsoid and the SU(2)× U(1) symmetric squashed sphere. Infinitesimal deformations of the THF away from the round sphere were already analyzed in [20]
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