From equivariant volumes to equivariant periods

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.

v2: typos fixed and references added, version to appear in Adv.Theor.Math.Phys

References in corpus (3)

• Enumerative geometry of Calabi-Yau 4-folds
• Wall-crossing for K-theoretic quasimap invariants I
• Quantum K-invariants and Gopakumar--Vafa invariants I. The quintic threefold

Cited by in corpus (3)

• Constant maps in equivariant topological strings and geometric modeling of fluxes
• Symplectic cuts and open/closed strings II
• Symplectic cuts and open/closed strings I