${\mathcal N} = 2$ SUPERSYMMETRIC THEORIES ON SQUASHED THREE-SPHERE

Abstract

We argue a squashing deformation of 3D N = 2 supersymmetric theories on round S 3 , which preserves SU (2) L × U (1)r isometry and the supersymmetries. We stress that the choice of a supersymmetry for the localization affects whether the partition function calculated using the supersymmetry depends on the squashing parameter or not.

The geometry of supersymmetric partition functions

“…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].…”

“…They can be realized by reducing the complex structure and the Hermitian metric (4.7) on primary Hopf surfaces M p,q 4 along ∂ x . (Various reductions from S 3 × S 1 to squashed three-spheres were discussed in [44,45,54,73,74].) Since this family of THFs is one-dimensional, the corresponding partition functions Z S 3 should only depend on the geometry through a single complex parameter -the squashing parameter b.…”

“…If γ i = 0, then (E.1) is isometric to the squashed sphere with SU(2) × U(1) isometry studied in [44,45], which preserves four supercharges. This corresponds to a deformation of the first type with γ = −2iγ r , which may be real or purely imaginary.…”