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arXiv · 2211.13269

From equivariant volumes to equivariant periods

Luca Cassia·Nicolo Piazzalunga·Maxim Zabzine
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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 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.

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Luca Cassia, Nicolo Piazzalunga, Maxim Zabzine. 2022-11-23. From equivariant volumes to equivariant periods. https://doi.org/10.4310/atmp.2023.v27.n4.a1

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