Coulomb branches of noncotangent type (with appendices by Gurbir Dhillon and Theo Johnson-Freyd)
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866909700201644032 |
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| author | Braverman, Alexander Dhillon, Gurbir Finkelberg, Michael Raskin, Sam Travkin, Roman |
| author_facet | Braverman, Alexander Dhillon, Gurbir Finkelberg, Michael Raskin, Sam Travkin, Roman |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_09475 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Coulomb branches of noncotangent type (with appendices by Gurbir Dhillon and Theo Johnson-Freyd) Braverman, Alexander Dhillon, Gurbir Finkelberg, Michael Raskin, Sam Travkin, Roman Algebraic Geometry High Energy Physics - Theory Mathematical Physics Representation Theory 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. |
| title | Coulomb branches of noncotangent type (with appendices by Gurbir Dhillon and Theo Johnson-Freyd) |
| topic | Algebraic Geometry High Energy Physics - Theory Mathematical Physics Representation Theory |
| url | https://arxiv.org/abs/2201.09475 |