Residue construction of quantized Coulomb branches

Abstract

We describe the image of a quantized BFN Coulomb branch $\mathcal{A}_{G,N}^{\hbar=1}$ under localization (abelianization) map for any $G,N$. In most cases, such as quiver gauge theories without loops, this description works for any flavors, sometimes we need to take generic or formal flavor parameters. The answer is given by a roots and residue condition similar to Ginzburg---Kapranov---Vasserot construction of DAHA arXiv:alg-geom/9512017. As a corollary, for any quantized conical Coulomb branch (and zero or small flavors) we provide a mathematical construction of the sphere trace introduced by Gaiotto and Okazaki arXiv:1911.11126.