Stacky GKM Graphs and Orbifold Gromov-Witten Theory

TL;DR

This paper introduces the concept of stacky GKM graphs for smooth GKM stacks, providing a combinatorial framework to compute equivariant orbifold Gromov-Witten invariants via localization, and generalizes the notion to abstract graphs.

Contribution

It defines and axiomatizes stacky GKM graphs, constructs formal GKM stacks from these graphs, and develops a method to compute orbifold Gromov-Witten invariants using these combinatorial structures.

Findings

Stacky GKM graphs encode neighborhoods of the 1-skeleton in smooth GKM stacks.

Abstract stacky GKM graphs generalize the original concept and allow construction of formal GKM stacks.

Orbifold Gromov-Witten invariants can be computed from the GKM graph data using virtual localization.

Abstract

A smooth GKM stack is a smooth Deligne-Mumford stack equipped with an action of an algebraic torus $T$, with only finitely many zero-dimensional and one-dimensional orbits. (i) We define the stacky GKM graph of a smooth GKM stack, under the mild assumption that any one-dimensional $T$-orbit closure contains at least one $T$ fixed point. The stacky GKM graph is a decorated graph which contains enough information to reconstruct the $T$-equivariant formal neighborhood of the 1-skeleton (union of zero-dimensional and one-dimensional $T$-orbits) as a formal smooth DM stack equipped with a $T$-action. (ii) We axiomize the definition of a stacky GKM graph and introduce abstract stacky GKM graphs which are more general than stacky GKM graphs of honest smooth GKM stacks. From an abstract GKM graph we construct a formal smooth GKM stack. (iii) We define equivariant orbifold Gromov-Witten invariants of smooth GKM stacks, as well as formal equivariant orbifold Gromov-Witten invariants of formal smooth GKM stacks. These invariants can be computed by virtual localization and depend only the stacky GKM graph or the abstract stacky GKM graph. Formal equivariant orbifold Gromov-Witten invariants of the stacky GKM graph of a smooth GKM stack $\mathcal{X}$ are refinements of equivariant orbifold Gromov-Witten invariants of $\mathcal{X}$.