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Gromov–Witten theory and invariants of matroids

Published version
Peer-reviewed

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Authors

Ranganathan, D 
Usatine, J 

Abstract

jats:titleAbstract</jats:title>jats:pWe 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.</jats:p>

Description

Keywords

Article, 14N35, 14T20, 14N20

Journal Title

Selecta Mathematica, New Series

Conference Name

Journal ISSN

1022-1824
1420-9020

Volume Title

28

Publisher

Springer Science and Business Media LLC