An $$ \mathcal{N}=1 $$ 3d-3d correspondence

Abstract

Abstract M5-branes on an associative three-cycle M 3 in a G 2-holonomy manifold give rise to a 3d $$ \mathcal{N}=1 $$ N = 1 supersymmetric gauge theory, $$ {T}_{\mathcal{N}=1}\left[{M}_3\right] $$ T N = 1 M 3 . We propose an $$ \mathcal{N}=1 $$ N = 1 3d-3d correspondence, based on two observables of these theories: the Witten index and the S 3-partition function. The Witten index of a 3d $$ \mathcal{N}=1 $$ N = 1 theory $$ {T}_{\mathcal{N}=1}\left[{M}_3\right] $$ T N = 1 M 3 is shown to be computed in terms of the partition function of a topological field theory, a super-BF-model coupled to a spinorial hypermultiplet (BFH), on M 3. The BFH-model localizes on solutions to a generalized set of 3d Seiberg-Witten equations on M 3. Evidence to support this correspondence is provided in the abelian case, as well as in terms of a direct derivation of the topological field theory by twisted dimensional reduction of the 6d (2, 0) theory. We also consider a correspondence for the S 3-partition function of the $$ {T}_{\mathcal{N}=1}\left[{M}_3\right] $$ T N = 1 M 3 theories, by determining the dimensional reduction of the M5-brane theory on S 3. The resulting topological theory is Chern-Simons-Dirac theory, for a gauge field and a twisted harmonic spinor on M 3, whose equations of motion are the generalized 3d Seiberg-Witten equations. For generic G 2-manifolds the theory reduces to real Chern-Simons theory, in which case we conjecture that the S 3-partition function of $$ {T}_{\mathcal{N}=1}\left[{M}_3\right] $$ T N = 1 M 3 is given by the Witten-Reshetikhin-Turaev invariant of M 3.