From equivariant volumes to equivariant periods

Authors: Luca Cassia, Nicolo Piazzalunga and Maxim Zabzine Preprint number: UUITP-56/22 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×S^1, respectively. We define these objects and study their dependence on equivariant parameters for non-compact toric Kahler 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 appearence 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.