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Tropicalization of super Gromov-Witten invariants

This paper establishes a tropical geometry framework for computing genus-0, Neveu-Schwarz marked super Gromov-Witten invariants of convex toric varieties by defining a procedure to calculate the tropical inverse Euler class of the SUSY normal bundle, thereby recovering known point invariants and explicitly computing an example for $\mathbb{P}^1$ .

Original authors: Artan Sheshmani, Shing-Tung Yau, Benjamin Zhou

Published 2026-09-24

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CC BY 4.0

Original authors: Artan Sheshmani, Shing-Tung Yau, Benjamin Zhou

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern mathematics, there is a field dedicated to counting the ways shapes can fit together, a discipline known as enumerative geometry. For decades, mathematicians have sought to understand the hidden patterns in these counts, often turning to a tool called tropical geometry to simplify the problem. Imagine taking a complex, curving shape and flattening it out until it looks like a skeleton made of straight lines and sharp corners; this is the essence of tropical geometry. It transforms difficult questions about curved spaces into more manageable puzzles about combinatorics and networks. This approach has already proven powerful in connecting the behavior of curved surfaces to the rigid logic of graphs, offering a new lens through which to view the fundamental structures of space.

Building on this foundation, a team of researchers has now extended this method to a more exotic frontier: the world of supergeometry. This branch of mathematics deals with spaces that include not just ordinary dimensions, but also "supersymmetric" ones, which are essential for describing certain physical theories about the universe. Specifically, the researchers focused on a type of count called a super Gromov-Witten invariant. These numbers act as a census, tallying the ways special maps can travel from a curved surface into a target space, while respecting the rules of supersymmetry. Until now, calculating these numbers for complex shapes was a formidable task, often requiring heavy machinery from algebraic geometry. The researchers asked a simple but profound question: could these super-census numbers also be translated into the language of tropical geometry, turning a problem of curved, supersymmetric maps into a problem of counting straight-line networks?

The answer, as detailed in their work, is a definitive yes. The team demonstrated that for a specific class of geometric spaces known as convex, toric varieties, these super Gromov-Witten invariants can indeed be defined and computed using tropical methods. They showed that the complex machinery required to count these supersymmetric maps could be replaced by a procedure that works entirely within the realm of tropical geometry. In the simplest case, where the target space is just a single point, they proved that the tropical count matches the known mathematical results exactly. This was a crucial first step, confirming that their new tropical definition was not just a guess, but a faithful translation of the existing theory.

To tackle the more general case of larger, multi-dimensional spaces, the researchers had to navigate a significant hurdle. In the standard theory, the calculation involves a specific mathematical object called a normal bundle, which describes how the space of possible maps sits inside a larger universe of all possible maps. In the supersymmetric world, this object is twisted by the extra dimensions of the theory, making it difficult to handle directly. The team introduced a procedure to compute the "inverse" of this twisted object using tropical tools. They did this by assuming that the families of maps they were studying could be broken down locally into simpler, product-like structures that behave well under tropicalization. Under this assumption, they constructed a map that takes a tropical cycle—a network of weighted lines—and produces the correct tropical count.

The power of this new method was put to the test with a concrete example involving a simple curved space known as the projective line. The researchers calculated a specific super Gromov-Witten invariant for this space, a number that describes the ways a genus-zero curve with two marked points can map into it. Using their tropical procedure, they derived a precise value for this count. The result was a specific fraction involving a variable that tracks the scaling of the supersymmetric dimensions. This calculation served as a proof of concept, showing that the abstract machinery they built could produce concrete, verifiable numbers.

Perhaps most importantly, the researchers showed that their general method for these larger spaces naturally collapses back to the simpler case of a single point. When they applied their tropical procedure to the point, the result perfectly matched the earlier, simpler calculation. This consistency is vital; it means the new, more complex definition is not an isolated invention but a coherent extension of the existing mathematical framework. By successfully translating these supersymmetric counts into the language of tropical geometry, the team has opened a new pathway for computation. They have shown that the intricate, curved world of super Gromov-Witten theory can be mapped onto the straight, combinatorial world of tropical curves, offering mathematicians a powerful new tool to explore the enumerative geometry of supersymmetric spaces.

Technical Summary: Tropicalization of Super Gromov-Witten Invariants

Problem Statement
The paper addresses the challenge of defining and computing genus-0, $n$ Neveu-Schwarz marked super Gromov-Witten (SGW) invariants for a convex, toric variety $X$ . While standard Gromov-Witten invariants count stable maps from Riemann surfaces, SGW invariants extend this theory to maps from super Riemann surfaces. Previous work by Keßler, Sheshmani, and Yau [KSY22, KSY23, KSY26] established the moduli space of super stable maps, $M^{super}_{0,n}(X,\beta)$ , and defined SGW invariants via $C^*$ -localization. This localization expresses the invariants as integrals over the classical moduli space $M_{0,n}(X,\beta)$ involving the inverse equivariant Euler class of the SUSY normal bundle $N_{n,\beta}$ . The central problem is to translate these complex geometric definitions into the combinatorial framework of tropical geometry, thereby enabling the computation of these invariants through tropical curves.

Methodology
The authors employ tropical geometry to construct a "tropicalization" of the super Gromov-Witten invariants. The methodology proceeds through the following steps:

Review of SUSY Normal Bundles: The paper recalls the definition of the SUSY normal bundle $N_{n,\beta} \to M_{0,n}(X,\beta)$ , which arises from the embedding of the classical moduli space into the super moduli space. The authors prove that for a convex variety $X$ , $N_{n,\beta}$ is a vector bundle of rank $c_1(\beta) + n - 2$ .
Tropicalization of the Point Case: For the target space $X = \text{pt}$ , the authors define tropical descendant invariants on the moduli space of tropical curves, $M^{trop}_{0,n}$ . They introduce a decorated moduli space $T_{0,n}$ of dual graphs equipped with powers of $\psi$ -classes. By utilizing results from [KM09], they establish an equivalence between the classical SGW invariants of a point and their tropical counterparts.
Tropicalization of the General Convex Toric Case: For a general convex, toric variety $X$ , the authors define a procedure to compute the tropical, inverse Euler class of the SUSY normal bundle. This relies on Assumption 4.5, which posits that families of stable maps can be locally trivialized by toroidal, spin varieties. Under this assumption, the authors construct the tropicalization of the spinor sheaf and the associated Chern classes.
Definition of Tropical SGW Invariants: Using the tropicalized inverse Euler class $e_{K,trop}(N_{n,\beta})^{-1}$ , the authors define the tropical SGW invariants for $X$ as the intersection of this class with the tropicalization of the virtual fundamental class and incidence conditions.

Key Contributions and Results

Theorem 3.11 (Point Case): The authors prove that the genus-0, $n$ -marked super Gromov-Witten invariants of a point, $SGW_{0,n}(\text{pt})$ , are equal to the tropical super Gromov-Witten invariants, $TSGW_{0,n}(\text{pt})$ . This is achieved by showing that the tropical intersection numbers on the moduli space of tropical curves recover the classical descendant invariants derived from the splitting of the SUSY bundle.
Theorem 4.10 (General Toric Case): The paper establishes the existence of a well-defined map that computes the tropical, inverse Euler class of the SUSY normal bundle $N_{n,\beta} \to M_{0,n}(X,\beta)$ . This map, $e_{K,trop}(N_{n,\beta})^{-1}$ , acts on the Chow group of tropical cycles of the extended cone complex of tropical stable maps, $TSM_{0,n}(\Sigma_X, \beta)$ .
Example 4.13 (Computation for $\mathbb{P}^1$ ): The authors explicitly compute a tropical, genus-0, 2-marked super Gromov-Witten invariant for $X = \mathbb{P}^1$ in the curve class $H$ . By analyzing the strata of the moduli space (smooth and singular domains) and applying the tropical inverse Euler class procedure, they derive the result:
$SGW^{trop}_{0,2}(\mathbb{P}^1, H) = \frac{1}{\kappa^3}$
where $\kappa$ is the identity character of the $C^*$ action.
Corollary 4.14 (Consistency): The authors demonstrate that their general definition for convex toric varieties recovers the result for the point case (Theorem 3.11) when evaluated over the smooth locus of the moduli space.

Significance and Claims
The paper claims to successfully extend the utility of tropical geometry from standard enumerative geometry to the realm of supergeometry. By defining tropical super Gromov-Witten invariants, the authors provide a combinatorial method to compute these invariants, which were previously accessible primarily through localization techniques in equivariant cohomology.

The significance lies in the formalization of the "tropicalization" of the SUSY normal bundle's inverse Euler class. The authors note that while the definition relies on Assumption 4.5 (local trivialization by toroidal, spin varieties), this assumption holds for specific cases like $\mathbb{P}^1$ , allowing for concrete computations. The work bridges the gap between the algebraic geometry of super Riemann surfaces and the combinatorial power of tropical curves, offering a new computational tool for super Gromov-Witten theory. The authors remain modest, noting that the general procedure is defined under specific structural assumptions regarding the families of stable maps.

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