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arXiv · 2108.08806

Gromov-Witten theory and invariants of matroids

Dhruv Ranganathan·Jeremy Usatine
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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.

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Dhruv Ranganathan, Jeremy Usatine. 2022-05-02. Gromov-Witten theory and invariants of matroids. https://arxiv.org/abs/2108.08806

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