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Tropical super Gromov-Witten invariants

Artan Sheshmani, Shing-Tung Yau, Benjamin Zhou

TL;DR

The paper develops a tropical framework to define and compute super Gromov-Witten invariants, translating the problem into tropical geometry to handle both point targets and convex toric targets $X$ via a tropical inverse Euler class of the SUSY normal bundle. Central to the approach is an enlarged, decorated dual-graph moduli space that records powers of psi-classes, yielding a tropical formula for SGW$_{0,n}$(pt) as a sum over decorated graphs and enabling a tropical interpretation for toric targets through the extended cone complex $TSM_{0,n}(ar\Sigma_X,\beta)$. The SUSY normal bundle $\overline{N}_{n,\beta}$ is shown to be a vector bundle for convex $X$, and its tropical inverse equivariant Euler class $e^{K,\text{trop}}(\overline{N}_{n,\beta})^{-1}$ is defined via tropical Chern classes of summands, allowing the definition of SGW$^{\text{trop}}_{0,n}(X,\beta)$. The work connects tropical psi-class machinery with super GW theory and provides computationally tractable tropical formulas, demonstrated through examples and consistency with known point-target results. Overall, it builds a bridge between super GW invariants and tropical descendant calculus, offering a tropical path to compute and interpret SGW invariants for convex toric targets with potential extensions to log-geometry settings.

Abstract

We show that super Gromov-Witten invariants can be defined and computed by methods of tropical geometry. When the target is a point, the super invariants are descendant invariants on the moduli space of curves, which can be computed tropically. When the target is a convex, toric variety $X$, we describe a procedure to compute the tropical Euler class of the SUSY normal bundle $\overline{N}_{n, β}$ on $\overline{\mathcal{M}}_{0,n}(X, β)$, assuming it is locally tropicalizable in the sense of [CG], [CGM]. Then, we define the tropical, genus-0, $n$-marked, super Gromov-Witten invariant of $X$, and compute an example. This gives a tropical interpretation of super Gromov-Witten invariants of convex, toric varieties.

Tropical super Gromov-Witten invariants

TL;DR

The paper develops a tropical framework to define and compute super Gromov-Witten invariants, translating the problem into tropical geometry to handle both point targets and convex toric targets via a tropical inverse Euler class of the SUSY normal bundle. Central to the approach is an enlarged, decorated dual-graph moduli space that records powers of psi-classes, yielding a tropical formula for SGW(pt) as a sum over decorated graphs and enabling a tropical interpretation for toric targets through the extended cone complex . The SUSY normal bundle is shown to be a vector bundle for convex , and its tropical inverse equivariant Euler class is defined via tropical Chern classes of summands, allowing the definition of SGW. The work connects tropical psi-class machinery with super GW theory and provides computationally tractable tropical formulas, demonstrated through examples and consistency with known point-target results. Overall, it builds a bridge between super GW invariants and tropical descendant calculus, offering a tropical path to compute and interpret SGW invariants for convex toric targets with potential extensions to log-geometry settings.

Abstract

We show that super Gromov-Witten invariants can be defined and computed by methods of tropical geometry. When the target is a point, the super invariants are descendant invariants on the moduli space of curves, which can be computed tropically. When the target is a convex, toric variety , we describe a procedure to compute the tropical Euler class of the SUSY normal bundle on , assuming it is locally tropicalizable in the sense of [CG], [CGM]. Then, we define the tropical, genus-0, -marked, super Gromov-Witten invariant of , and compute an example. This gives a tropical interpretation of super Gromov-Witten invariants of convex, toric varieties.
Paper Structure (23 sections, 14 theorems, 37 equations, 5 figures)

This paper contains 23 sections, 14 theorems, 37 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Acknowledgments
  4. Preliminaries
  5. Tropical geometry
  6. Descendant invariants of $\overline{\mathcal{M}}_{0,n}$
  7. Tropicalization of $\overline{\mathcal{M}}_{0,n}$
  8. Tropical $\psi$-classes
  9. Tropical super Gromov-Witten invariants of a point
  10. Tropical super Gromov-Witten invariants of convex, toric varieties
  11. Super Gromov-Witten invariants
  12. SUSY normal bundles
  13. SUSY bundle $N_n$
  14. SUSY bundle $N_{(C, \phi)}$
  15. $ft_* \overline{N}_{n, \beta}$ is a vector bundle
  16. ...and 8 more sections

Key Result

Theorem 1

Suppose $Z \subset \mathcal{M} := \overline{\mathcal{M}}^{trop}_{0,n} \times \mathbb{Z}_{\geq 0}^n$ is the set of decorated dual graphs $\Gamma_{\vec{k}} = (\Gamma, k_1,\ldots, k_n)$, where $\Gamma$ is the dual graph of a smooth curve $C \in \overline{\mathcal{M}}^{trop}_{0,n}$ and $k_4+\ldots+k_n = where $W$ is defined in Definition def:W.

Figures (5)

  • Figure 2.1: For $f = x+y-1$, $V(f)$ (left) and $V(Trop(f))$ (right)
  • Figure 2.2: A stable curve $C \in \overline{\mathcal{M}}_{0,5}$ and its dual graph.
  • Figure 2.3: $\overline{\mathcal{M}}_{0,4}^{trop}$
  • Figure 3.1: Decorated dual graph for $C \in \overline{\mathcal{M}}_{0,n}$, where $k_i \in \mathbb{Z}_{\geq 0}$ remembers the power of $\psi_i$.
  • Figure 4.1: Cone complex $\Sigma_{L} = \Sigma_{\mathbb{P}^1} \times \mathbb{T}$.

Theorems & Definitions (59)

  • Theorem : Theorem \ref{['thm:SGW_point']}
  • Theorem : Theorem \ref{['thm:tropical_Euler_class']}
  • Corollary : Corollary \ref{['cor:connect']}
  • Remark 1.1
  • Remark 1.2
  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • Lemma 2.4: Koc, Lemma 25.2.3
  • Theorem 2.5: Witten's conjecture in genus-0, HM
  • ...and 49 more