Twisted Traces on Abelian Quantum Higgs and Coulomb Branches

We study twisted traces on the quantum Higgs branches \(A_{\operatorname {Higgs}}\) of \(3d, \mathcal {N}=4\) gauge theories, that is, the quantum Hamiltonian reductions of Weyl algebras. In theories which are good or ugly, we define a twisted trace that arises naturally from the correlation functions of the gauge theory. We show that this trace induces an inner product and a short star product on \(A_{\operatorname {Higgs}}\). We analyze this trace in the case of an abelian gauge group and show that it has a natural expansion in terms of the twisted traces of Verma modules, confirming a conjecture of the first author and Okazaki. This expansion has a natural interpretation in terms of 3-d mirror symmetry, and we predict that it can be interpreted as an Atiyah-Bott fixed-point formula under the quantum Hikita isomorphism.