  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.

Algebraic Geometry;Combinatorics