Abstract
M5-branes on an associative three-cycle M3 in a G2-holonomy manifold give rise to a 3d \( \mathcal{N}=1 \) supersymmetric gauge theory, \( {T}_{\mathcal{N}=1}\left[{M}_3\right] \). We propose an \( \mathcal{N}=1 \) 3d-3d correspondence, based on two observables of these theories: the Witten index and the S3-partition function. The Witten index of a 3d \( \mathcal{N}=1 \) theory \( {T}_{\mathcal{N}=1}\left[{M}_3\right] \) 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 \( {T}_{\mathcal{N}=1}\left[{M}_3\right] \) 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 \( {T}_{\mathcal{N}=1}\left[{M}_3\right] \) is given by the Witten-Reshetikhin-Turaev invariant of M3.
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Eckhard, J., Schäfer-Nameki, S. & Wong, JM. An \( \mathcal{N}=1 \) 3d-3d correspondence. J. High Energ. Phys. 2018, 52 (2018). https://doi.org/10.1007/JHEP07(2018)052
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DOI: https://doi.org/10.1007/JHEP07(2018)052