Abstract: “I will describe a technique for exact computation of a large class of correlation functions in 3d N=4 gauge theories placed on S^3. In general, N=4 theories on S^3 have interesting supersymmetric sectors, defined by passing to the cohomology of a certain supercharge, and described by an emergent 1d theory. There are two such 1d sectors that are exchanged by the mirror symmetry: one related to the Higgs branch and one related to the Coulomb branch. When the corresponding branch is not lifted, the 1d theory is a TQFT. This is a 3d analog of the chiral algebra in 4d N=2 SCFTs discovered by Beem et. al. Moreover, our construction provides a generalization away from the conformal point in 3d case. (Which is still missing in the 4d case!)
We use supersymmetric localization to access the 1d sectors. The final result can be encoded as a star-product on the corresponding branch.
One of the many applications is that the Coulomb branch result (so far, only for abelian theories) provides a proof of the construction of Coulomb branches due to Bullimore-Dimofte-Gaiotto. “