The rôle of Coulomb branches in 2D gauge theory

Abstract

I give a simple construction of the Coulomb branches C3;4.G I 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 C.G I 0/ are abelian group schemes over the bases of regular adjoint GC-orbits, respectively conjugacy classes, and C.G I E / is glued together over the base from two copies of C.G I 0/ shifted by a rational Lagrangian section "V , representing the Euler class of the index bundle of a polarization V of E. Extending the interpretation of C3.G I 0/ as “classifying space” for topological 2D gauge theories, I characterize functions on C3.G I E / as operators on the equivariant quantum cohomologies of M X V, for compact symplectic G -manifolds M . The non-commutative version has a similar description in terms of the Γ -class of V .