An N= 1 3d-3d correspondence

Abstract

M5-branes on an associative three-cycle M3 in a G2-holonomy manifold give rise to a 3d N= 1 supersymmetric gauge theory, TN=1[M3]. We propose an N= 1 3d-3d correspondence, based on two observables of these theories: the Witten index and the S3-partition function. The Witten index of a 3d N= 1 theory TN=1[M3] 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 M3. The BFH-model localizes on solutions to a generalized set of 3d Seiberg-Witten equations on M3. 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 S3-partition function of the TN=1[M3] theories, by determining the dimensional reduction of the M5-brane theory on S3. The resulting topological theory is Chern-Simons-Dirac theory, for a gauge field and a twisted harmonic spinor on M3, whose equations of motion are the generalized 3d Seiberg-Witten equations. For generic G2-manifolds the theory reduces to real Chern-Simons theory, in which case we conjecture that the S3-partition function of TN=1[M3] is given by the Witten-Reshetikhin-Turaev invariant of M3.