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

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

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Abstract M5-branes on an associative three-cycle M 3 in a G 2-holonomy manifold give rise to a 3d N=1 $$ mathcal{N}=1 $$ supersymmetric gauge theory, TN=1M3 $$ {T}_{mathcal{N}=1}left[{M}_3 ight] $$.We propose an N=1 $$ mathcal{N}=1 $$ 3d-3d correspondence, based on two observables of these theories: goblin toe d2r the Witten index and the S 3-partition function.The Witten index of a 3d N=1 $$ mathcal{N}=1 $$ theory TN=1M3 $$ {T}_{mathcal{N}=1}left[{M}_3 ight] $$ 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 TN=1M3 $$ {T}_{mathcal{N}=1}left[{M}_3 ight] $$ 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 blues cube function of TN=1M3 $$ {T}_{mathcal{N}=1}left[{M}_3 ight] $$ is given by the Witten-Reshetikhin-Turaev invariant of M 3.