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

Twisted traces on abelian quantum Higgs and Coulomb branches