SUSY gauge theories on squashed three-spheres

Abstract

We study Euclidean 3D N = 2 supersymmetric gauge theories on squashed three-spheres preserving isometries SU (2) × U (1) or U (1) × U (1). We show that, when a suitable background U (1) gauge field is turned on, these squashed spheres support charged Killing spinors and therefore N = 2 supersymmetric gauge theories. We present the Lagrangian and supersymmetry rules for general gauge theories. The partition functions are computed using localization principle, and are expressed as integrals over Coulomb branch. For the squashed sphere with U (1) × U (1) isometry, its measure and integrand are identified with the building blocks of structure constants in Liouville or Toda conformal field theories with b = 1.

The geometry of supersymmetric partition functions

“…If γ r = −i γ r is purely imaginary and γ r = γ i , then (E.1) is isometric to the ellipsoid with U(1) × U(1) isometry studied in [43]. According to the preceding discussion, this is a deformation of the first type with real deformation parameter γ = 4 γ r .…”

“…In special cases, they are equivalent to the ellipsoid squashings with U(1) × U(1) isometry studied in [43], which only give rise to real γ, and the squashings of [44,45]. Some of the squashings discussed in [43,46] give γ = 0, i.e. they do not change the THF, even though they deform the transversely Hermitian metric, and this explains why they do not affect the partition function.…”

“…The squashed spheres studied in [48], which are reviewed in appendix E, realize deformations with complex γ. In special cases, they are equivalent to the ellipsoid squashings with U(1) × U(1) isometry studied in [43], which only give rise to real γ, and the squashings of [44,45]. Some of the squashings discussed in [43,46] give γ = 0, i.e.…”

“…Following [40][41][42], who studied partition functions of three-dimensional N = 2 theories with a U(1) R symmetry on round spheres preserving four supercharges, much recent work has focused on squashed spheres [43][44][45][46][47][48][49][50][51], i.e. three-manifolds that are diffeomorphic to S 3 but carry a more general metric with less symmetry.…”

“…The partition function on a round S 3 with four supercharges residing in SU(2|1) was computed in [40][41][42] using localization. This was subsequently generalized to various squashed spheres [43][44][45][46][47][48][49][50][51], all of which preserve at least two supercharges. 29 By looking at the explicit expressions for the partition functions in these examples, one is lead to the following observations: a) Z S 3 only depends on the squashing through a single complex parameter, usually called b, where b = 1 corresponds to the round S 3 of [40][41][42].…”