  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, 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.
