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 O for symplectic singularities. Letting O Coulomb and O Higgs be these categories for the Coulomb and Higgs branches associated to V and G, we obtain a functor O ! Coulomb → O 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 * V by G is a Nakajima quiver variety or smooth hypertoric variety. This includes as special cases the parabolic-singular Koszul duality of category 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. * (X σ,w × V X σ ′ ,w ′ ), with composition given by convolution. The Steinberg algebra is simply the sum of all the morphisms in this category; modules over the Steinberg algebra are naturally

Coulomb branches of quiver gauge theories with symmetrizers

“…One can apply techniques of the localization theorem in equivariant (K)-homology groups of affine Steinberg varieties in [VV10] to study the category C. This theory, for the ordinary Coulomb branch, will be explained elsewhere [Nak19]. (See also [Web16,Web19] for another approach.) It also works in our current setting.…”

Symplectic resolutions, symplectic duality, and Coulomb branches

“…Webster [Web2] proved a general Koszul duality result for category O for Higgs and Coulomb branches (the Koszul duality described in section 6.5 is a special case of this result). This essentially establishes (O).…”

Some Open Questions in Quiver Gauge Theory

“…The phase diagrams for the full moduli spaces (including mixed branches) were studied in [31], and it was found that certain theories have Higgs and Coulomb branches which are related by inversion of their phase diagrams. On the other hand, a duality known as symplectic duality between Higgs and Coulomb branches of a 3d N = 4 theory was conjectured in [34] and has been extensively studied in [35][36][37][38][39][40][41][42]. Therefore, for phase diagrams, another question could be raised.…”