Computations in equivariant Gromov-Witten theory of GKM spaces

TL;DR

The paper refines equivariant Gromov–Witten theory on GKM spaces by proving that localization data and GW invariants depend only on the GKM graph, not on any compatible connection, and by providing a connection-free reformulation. It introduces GKMtools.jl, a Julia-based tool that computes equivariant GW invariants and quantum products directly from GKM graphs, enabling explicit computations across diverse geometries. The authors apply this framework to Calabi–Yau local models, twisted flag varieties, and realizability problems, deriving new connections to Donaldson–Thomas theory, and exploring enumerative questions such as curves in hyperplanes and quantum Schubert calculus with infinitely many $q$-terms. Collectively, the work demonstrates both the theoretical flexibility of GKM methods and the practical power of computational tools for exploring novel phenomena in symplectic and algebraic geometry, with implications for enumerative geometry and mathematical physics.

Abstract

We study equivariant Gromov-Witten invariants and quantum cohomology in GKM theory. Building on the localization formula, we prove that the resulting expression is independent of the choice of compatible connection, and provide an equivalent formulation without auxiliary choices. Motivated by this theoretical refinement, we develop a software package, $\mathtt{GKMtools . jl}$, that implements the computation of equivariant GW invariants and quantum products directly from the GKM graph. We apply our framework to several geometric settings: Calabi-Yau rank two vector bundles on the projective line, where we obtain a new proof for a recent connection to Donaldson-Thomas theory of Kronecker quivers; twisted flag manifolds, which give symplectic but non-algebraic examples of GKM spaces; realizability questions for abstract GKM graphs; classical enumerative problems involving curves in hyperplanes; and quantum Schubert calculus for smooth Schubert varieties. These results demonstrate both the theoretical flexibility of GKM methods and the effectiveness of computational tools in exploring new phenomena.

Figures (8)

• Figure 2.1: The underlying graphs of the GKM graphs of $\mathbb{P}^1$ (left) and $\mathbb{A}^1$ (right).
• Figure 2.2: Two ways of depicting the GKM graph of the full flag variety for $\mathbb{C}^3$ acted on by $T=(\mathbb{C}^\times)^2$. It has six vertices, labeled by the elements of $S_3$. See also Guillemin_Holm_Zara_2006.
• Figure 2.3: Illustration of Proposition \ref{['prop:curve_classes']} for the GKM graph of $\mathbb{P}^2$ with the diagonal action of $(\mathbb{C}^\times)^3$. In the equations on the right hand side, we draw an edge to indicate its class in $H_2(\mathbb{P}^2)$.
• Figure 5.1: The GKM graph of $X_k$. Here, $t_1$ is the weight of $T_{\mathbb{P}^1,[1:0]}$, and $t_2,t_3$ are the weights of the fibers at $[1:0]$. The weights at $[0:1]$ are automatically determined by the degrees $a_1=k-1$ and $a_2=-k-1$ (see Section \ref{['sec:connections']}).
• Figure 5.2: GKM graph of a compact Hamiltonian GKM space $X$ (see GKZ20), illustrated following Example \ref{['ex:GKM_illustration']} (2). Each edge $e$ is labeled with $\mathcal{C}_1(e)$. By Proposition \ref{['prop:curve_classes']}, $H_2(X)$ is the free $\mathbb{Z}$-module generated by $\beta_1,\beta_2$, and $\gamma$.

Theorems & Definitions (84)

• Theorem 1.1: Theorem \ref{['cor:GKM_determines_GW']}
• Theorem 1.2: Theorem \ref{['thm:gw_equivariant_CY']}, §\ref{['sec:BPS']}
• Theorem 1.3: Theorem \ref{['thm:Schubert']}
• Definition 2.1
• Definition 2.2: Guillemin_Guillemin_Ohsawa_Ginzburg_Karshon_2014
• Remark 2.3
• Definition 2.4
• Remark 2.5
• Remark 2.6
• Example 2.7