Let $X$ and $X^{!}$be a pair of dual conical symplectic singularities, and let $\tilde{X}^{!} \rightarrow X^{!}$be a symplectic resolution. The quantum Hikita conjecture states that the D-module of graded traces for $X$, denoted $M$ reg, is isomorphic to the specialized quantum D-module of $\tilde{X}^{!}$after localization. This talk introduces an operator-theoretic framework to construct these traces and apply them to the conjecture.

We discuss representation of Coulomb branches as operators acting on a two-parameter function space. This concrete representation provides a direct path to constructing integral form twisted traces for conical theories. This also gives an explanation of why the quantum Hikita conjecture fails at “bad” cases like the pure gauge theory.

Lastly, I will discuss how to give a proof of a weaker version of the quantum Hikita conjecture by identifying these twisted traces with the vertex functions of quasimaps to the corresponding Higgs branch.