The rôle of Coulomb branches in 2D gauge theory

Source abstract

I give a simple construction of the Coulomb branches {\mathscr{C}_{3,4}(G;E)} of gauge theory in three and four dimensions, defined by H. Nakajima [Adv. Theor. Math. Phys. 20 (2016)] and A. Braverman, M. Finkelberg and H. Nakajima [Adv. Theor. Math. Phys. 22 (2018)] for a compact Lie group G and a polarizable quaternionic representation E . The manifolds {\mathscr{C}(G;\mathbf{0})} are abelian group schemes over the bases of regular adjoint {G_\mathbb{C}} -orbits, respectively conjugacy classes, and {\mathscr{C}(G;E)} is glued together over the base from two copies of {\mathscr{C}(G;\mathbf{0})} shifted by a rational Lagrangian section {\varepsilon_V} , representing the Euler class of the index bundle of a polarization {V} of {E} . Extending the interpretation of {\mathscr{C}_3(G;\mathbf{0})} as “classifying space” for topological 2D gauge theories, I characterize functions on {\mathscr{C}_3(G;E)} as operators on the equivariant quantum cohomologies of {M\times V} , for compact symplectic {G} -manifolds {M} . The non-commutative version has a similar description in terms of the {\Gamma} -class of {V} .