$\mathcal{N}=2$ supersymmetric field theories on 3-manifolds with A-type boundaries

TL;DR

This paper develops A-type boundary conditions for N=2 supersymmetric theories on 3-manifolds, revealing geometric structures like contact structures and foliations, and explores their implications for supersymmetric field theories with boundaries.

Contribution

It introduces a geometric framework for A-type boundary conditions on 3-manifolds, connecting supersymmetry, contact geometry, and foliations, and applies this to various supersymmetric theories.

Findings

Manifolds with A-type structures admit supersymmetric boundary conditions.

Boundaries can be placed along leaves of a toric foliation, forming solid tori.

Supersymmetric theories can be consistently defined with these boundary conditions.

Abstract

General half-BPS A-type boundary conditions are formulated for N=2 supersymmetric field theories on compact 3-manifolds with boundary. We observe that under suitable conditions manifolds of the real A-type admitting two complex supersymmetries (related by charge conjugation) possess, besides a contact structure, a natural integrable toric foliation. A boundary, or a general co-dimension-1 defect, can be inserted along any leaf of this preferred foliation to produce manifolds with boundary that have the topology of a solid torus. We show that supersymmetric field theories on such manifolds can be endowed with half-BPS A-type boundary conditions. We specify the natural curved space generalization of the A-type projection of bulk supersymmetries and analyze the resulting A-type boundary conditions in generic 3d non-linear sigma models and YM/CS-matter theories.