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 $X$, 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 quantized Homological and K theoretic 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. 

Lastly, I will sketch a proof for a weaker version of the quantum Hikita conjecture by identifying these twisted traces with the vertex functions of quasimaps to the corresponding Higgs branch. This leads to the categorification of Jacobi-Trudi identity at principal specialization.