Operator Representations and Twisted Traces of Cohomological and K-Theoretic Coulomb Branches

Abstract

This thesis establishes a rigorous mathematical framework for twisted traces on quantized Coulomb branches in 3D \(\mathcal{N}=4\) supersymmetric gauge theories. By developing an explicit operator representation of the Coulomb branch algebra \(\mathcal{A}_{\hbar}(G, \bN)\) for conical theories, generalized shift operators are constructed to act on a specifically calibrated function space. For non-abelian groups, localization to torus fixed points enables the formulation of monopole operators as symmetrized sums of abelian shift operators. We derive explicit integral formulas for twisted traces, providing a concrete realization of correlation functions previously assumed in the physics literature. These constructions are also extended to K-theoretic Coulomb branches via representations by $q$-difference operators, offering analytical tools to investigate the quantum Hikita conjecture.