Gromov-Witten theory and invariants of matroids

TL;DR

This paper develops new Gromov-Witten invariants for matroids, linking algebraic geometry and combinatorics, and shows these invariants depend only on the matroid structure in certain cases.

Contribution

It introduces a novel Gromov-Witten theoretic approach to matroids, connecting invariants with the Chow groups of rational curves in toric varieties.

Findings

Invariants coincide with virtual classes for realizable matroids.

Quantum cohomology of wonderful models depends only on the matroid.

Toric intersection theory encodes Gromov-Witten invariants as weighted fans.

Abstract

We use techniques from Gromov-Witten theory to construct new invariants of matroids taking value in the Chow groups of spaces of rational curves in the permutohedral toric variety. When the matroid is realizable by a complex hyperplane arrangement, our invariants coincide with virtual fundamental classes used to define the logarithmic Gromov-Witten theory of wonderful models of arrangement complements, for any logarithmic structure supported on the wonderful boundary. When the boundary is empty, this implies that the quantum cohomology ring of a hyperplane arrangement's wonderful model is a combinatorial invariant, i.e., it depends only on the matroid. When the boundary divisor is maximal, we use toric intersection theory to convert the virtual fundamental class into a balanced weighted fan in a vector space, having the expected dimension. We explain how the associated Gromov-Witten theory is completely encoded by intersections with this weighted fan. We include a number of questions whose positive answers would lead to a well-defined Gromov-Witten theory of non-realizable matroids.