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

We propose a construction of the Coulomb branch of a $3d\mathcal{N}=4$ gauge theory corresponding to a choice of a connected reductive group $G$ and a symplectic finite-dimensional reprsentation $\mathbf{M}$ of $G$, satisfying certain anomaly cancellation condition. This extends the construction of arXiv:1601.03586 (where it was assumed that $\mathbf{M}=\mathbf{N}\oplus {\mathbf{N}}^{\ast }$ for some representation $\mathbf{N}$ 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 introduced by S.Lysenko on ${\mathrm{Bun}}_{Sp(2n)}({\mathbb{P}}^1)$ 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 D.Ben-Zvi, Y.Sakellaridis and A.Venkatesh.