Quiver gauge theories and symplectic singularities

Abstract

Braverman, Finkelberg and Nakajima have recently given a mathematical construction of the Coulomb branches of a large class of 3d N = 4 gauge theories, as algebraic varieties with Poisson structure. They conjecture that these varieties have symplectic singularities. We confirm this conjecture for all quiver gauge theories without loops or multiple edges, which in particular implies that the corresponding Coulomb branches have finitely many symplectic leaves and rational Gorenstein singularities . We also give a criterion for proving that any particular Coulomb branch has symplectic singularities, and discuss the possible extension of our results to quivers with loops and/or multiple edges.

Note on some properties of generalized affine Grassmannian slices

“…Note that this theorem is known in [BFN16a] for two special cases: dominant µ and µ ≤ w 0 λ. For gauge theories defined by simple quivers, this conjecture has been proven by Weekes using Coulomb branch techniques [Wee20]. In particular, this implies that W λ µ has symplectic singularities when G is of ADE-type.…”

“…In this section we prove Conjecture 1.4 for W λ µ . Note that our main result (Theorem 4.7) can be deduced using Coulomb branch techniques, see [Wee20] for ADE-type and [NW19] for extension to non-simply laced cases. Here we provide an elementary approach.…”

The Geometry of the affine closure of $T^*(\mathrm{SL}_n/U)$

“…There are many interesting and important examples of symplectic singularities. These include finite quotient singularities [Bea00], normal closures of nilpotent coadjoint orbits [Pan91], Nakajima quiver varieties [BS16], and more recently Coulomb branches associated to simple quivers [Wee20].…”