  We consider generalizations of equivariant volumes of abelian GIT quotients
obtained as partition functions of 1d, 2d, and 3d supersymmetric GLSM on $S^1$,
$D^2$ and $D^2 \times S^1$, respectively. We define these objects and study
their dependence on equivariant parameters for non-compact toric K\"ahler
quotients. We generalize the finite-difference equations (shift equations)
obeyed by equivariant volumes to these partition functions. The partition
functions are annihilated by differential/difference operators that represent
equivariant quantum cohomology/K-theory relations of the target and the
appearance of compact divisors in these relations plays a crucial role in the
analysis of the non-equivariant limit. We show that the expansion in
equivariant parameters contains information about genus-zero Gromov-Witten
invariants of the target.
