Tropical super Gromov-Witten invariants

TL;DR

This paper introduces a tropical geometry approach to defining and computing super Gromov-Witten invariants for convex toric varieties, linking algebraic and tropical methods.

Contribution

It provides a new tropical framework for super Gromov-Witten invariants, including a procedure for computing the tropical Euler class of the SUSY normal bundle.

Findings

Super Gromov-Witten invariants can be computed tropically.

A procedure for computing the tropical Euler class of the SUSY normal bundle is described.

An example computation of a super Gromov-Witten invariant is provided.

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 $\overline{N}_{n, \beta}$ on $\overline{\mathcal{M}}_{0,n}(X, \beta)$, 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.