The role of Coulomb branches in 2D gauge theory

I give a simple construction of certain Coulomb branches $C_{3,4}(G;E)$ of gauge theory in 3 and 4 dimensions defined by Nakajima et al. for a compact Lie group $G$ and a polarisable quaternionic representation $E$. The manifolds $C(G;0)$ are abelian group schemes (over the bases of regular adjoint $G_c$-orbits, respectively conjugacy classes), and $C(G;E)$ is glued together from two copies of $C(G;0)$ shifted by a rational Lagrangian section ${\epsilon }_V$, the Euler class of the index bundle of a polarisation $V$ of $E$. Extending the interpretation of $C_3(G;0)$ as “classifying space” for topological 2D gauge theories, I characterise functions on $C_3(G;E)$ as operators on the equivariant quantum cohomologies of $M\times V$, for all compact symplectic $G$-manifolds $M$. The non-commutative version has an analogous description in terms of the $\mathrm{\Gamma }$-function of $V$, appearing to play the role of Fourier transformed J-function of the gauged linear Sigma-model $V\mathrm{/}G$.