Coulomb branches of noncotangent type (with appendices by Gurbir Dhillon and Theo Johnson-Freyd)

We propose a construction of the Coulomb branch of a 3d N = 4 gauge theory corresponding to a choice of a connected reductive group G and a symplectic finite-dimensional reprsentation M of G, satisfying certain anomaly cancellation condition. This extends the construction of [BFN1] (where it was assumed that M = N ⊕N∗ for some representation N of G). Our construction goes through certain “universal” ring object in the twisted derived Satake category of the symplectic group Sp(2n). The construction of this object uses a categorical version of the Weil representation; we also compute the image of this object under the (twisted) derived Satake equivalence and show that it can be obtained from the theta-sheaf [Ly, LL] on BunSp(2n)(P) via certain Radon transform. We also discuss applications of our construction to a potential mathematical construction of S-duality for super-symmetric boundary conditions in 4-dimensional gauge theory and to (some extension of) the conjectures of Ben-Zvi, Sakellaridis and Venkatesh.

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