Advances in Theoretical and Mathematical Physics

Volume 27 (2023)

Number 4

From equivariant volumes to equivariant periods

Pages: 961 – 1064

DOI: https://dx.doi.org/10.4310/ATMP.2023.v27.n4.a1

Authors

Luca Cassia (Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden)

Nicolò Piazzalunga (Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden; and NHETC and Department of Physics and Astronomy, Rutgers University, New Jersey, U.S.A.)

Maxim Zabzine (Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden)

Abstract

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 Kähler 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.

Published 6 June 2024