# The rôle of Coulomb branches in 2D gauge theory

### Constantin Teleman

UC Berkeley, USA

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## 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 ${V}$.

## Cite this article

Constantin Teleman, The rôle of Coulomb branches in 2D gauge theory. J. Eur. Math. Soc. 23 (2021), no. 11, pp. 3497–3520

DOI 10.4171/JEMS/1071