Koszul duality between Higgs and Coulomb categories $\mathcal{O}$

Abstract

We prove a Koszul duality theorem between the category of weight modules over the quantized Coulomb branch (as defined by Braverman, Finkelberg and Nakajima) attached to a group $G$ and representation $V$ and a category of $G$-equivariant D-modules on the vector space $V$. This is proven by relating both categories to an explicit, combinatorially presented category. These categories are related to generalized categories $\mathcal{O}$ for symplectic singularities. Letting ${\mathcal{O}}_{\mathrm{Coulomb}}$ and ${\mathcal{O}}_{\mathrm{Higgs}}$ be these categories for the Coulomb and Higgs branches associated to $V$ and $G$, we obtain a functor ${\mathcal{O}}_{\mathrm{Coulomb}}^{!}\to {\mathcal{O}}_{\mathrm{Higgs}}$ from the Koszul dual of one to the other. This functor is an equivalence in the special cases where the hyperkähler quotient of $T^{\ast }V$ by $G$ is a Nakajima quiver variety or smooth hypertoric variety. This includes as special cases the parabolic-singular Koszul duality of category $\mathcal{O}$ in type A, the categorified rank-level duality proposed by Chuang and Miyachi and proven by Shan, Vasserot and Varagnolo, and the hypertoric Koszul duality proven by Braden, Licata, Proudfoot and the author. We also show that this equivalence intertwines so-called twisting and shuffling functors. This together with the duality discussed confirms the most important components of the symplectic duality conjecture of Braden, Licata, Proudfoot and the author in this case.

87 pages, major revision and expansion

References in corpus (4)

• 2-Kac-Moody algebras
• Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories and slices in the affine Grassmannian (with appendices by Alexander Braverman, Michael Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Hiraku Nakajima, Ben Webster, and Alex Weekes)
• Representation theory of the cyclotomic Cherednik algebra via the Dunkl-Opdam subalgebra
• A quantum Mirković-Vybornov isomorphism

Cited by in corpus (13)

• Coulomb branches of quiver gauge theories with symmetrizers
• Generators for Coulomb branches of quiver gauge theories
• BFN Springer Theory
• Symmetries of categorical representations and the quantum Ngô action
• Gelfand-Tsetlin modules in the Coulomb context
• Coherent sheaves and quantum Coulomb branches I: tilting bundles from integrable systems
• Coulomb Branches of Star-Shaped Quivers
• Some Open Questions in Quiver Gauge Theory
• On modular categories O for quantized symplectic resolutions
• Note on some properties of generalized affine Grassmannian slices
• Gelfand-Tsetlin modules: canonicity and calculations
• On category $\mathcal{O}$ for affine Grassmannian slices and categorified tensor products
• Derived equivalences for trigonometric double affine Hecke algebras