Supersymmetric field theories on three-manifolds

Abstract

We construct supersymmetric field theories on Riemannian three-manifolds M, focusing on N = 2 theories with a U (1) R symmetry. Our approach is based on the rigid limit of new minimal supergravity in three dimensions, which couples to the flat-space supermultiplet containing the R-current and the energy-momentum tensor. The field theory on M possesses a single supercharge if and only if M admits an almost contact metric structure that satisfies a certain integrability condition. This may lead to global restrictions on M, even though we can always construct one supercharge on any given patch. We also analyze the conditions for the presence of additional supercharges. In particular, two supercharges of opposite R-charge exist on every Seifert manifold. We present general supersymmetric Lagrangians on M and discuss their flat-space limit, which can be analyzed using the Rcurrent supermultiplet. As an application, we show how the flat-space two-point function of the energy-momentum tensor in N = 2 superconformal theories can be calculated using localization on a squashed sphere.

The geometry of supersymmetric partition functions

“…The corresponding supergravity theory was recently studied in superspace [19][20][21], as well as in components [3,4,16]. The supergravity background fields that couple to the operators in (1.15) are given by 16 16) 14 Under these conditions the flat-space theory does not admit a Ferrara-Zumino (FZ) supercurrent multiplet [6,18]. 15 By a D-term we mean a term in the Lagrangian that is constructed from the top component of a well-defined superfield.…”

“…This does not apply to FI-terms, which reside in the top components of gauge non-invariant vector superfields. 16 Again, our A (R) µ is related to the R-symmetry gauge field Aµ used in [16] by A (R)…”

“…This was used in [16] to obtain the non-linear completion of (1.18) from new minimal supergravity in four dimensions [12,13]. The variations of Ψ µ and Ψ µ lead to generalized Killing spinor equations in three dimensions.…”

“…These equations were analyzed in [16] (see also [14,22]). A solution of the first equation exists if and only if M 3 admits a certain mathematical structure -a transversely holomorphic foliation (THF) with a compatible transversely Hermitian metric -whose properties closely parallel those of integrable complex structures and Hermitian metrics.…”

“…The configurations that are consistent with the preserved supercharge Q on a three-manifold with a THF must satisfy δ Q λ = 0. It was shown in [16] that…”