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arXiv · 2201.09475

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

Alexander Braverman·Gurbir Dhillon·Michael Finkelberg·Sam Raskin·Roman Travkin
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Abstract

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 ${\mathbf M}={\mathbf N}\oplus{\mathbf N}^*$ 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 $\operatorname{Bun}_{Sp(2n)}({\mathbb P}^1)$ 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.

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BibTeXRIS

Alexander Braverman, Gurbir Dhillon, Michael Finkelberg, Sam Raskin, Roman Travkin. 2022-01-24. Coulomb branches of noncotangent type (with appendices by Gurbir Dhillon and Theo Johnson-Freyd). https://arxiv.org/abs/2201.09475

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