Matroids and amplitudes

TL;DR

This paper provides a combinatorial formula for intersection pairings on hyperplane arrangement complements, links them to matroid theory and Laplace transforms, and introduces a notion of scattering amplitudes for matroids with applications to physics.

Contribution

It introduces a closed combinatorial formula for intersection pairings, connects them to matroid theory and Laplace transforms, and defines matroid scattering amplitudes with physical properties.

Findings

Derived explicit formulas for intersection pairings.

Connected intersection forms to Laplace transforms of matroid fans.

Defined and analyzed scattering amplitudes for matroids.

Abstract

In the 1990s, Kita--Yoshida and Cho--Matsumoto introduced intersection forms on the twisted (co)homologies of hyperplane arrangement complements. We give a closed combinatorial formula for these intersection pairings. We show that these intersection pairings are obtained from (continuous and discrete) Laplace transforms of subfans of the Bergman fan of the associated matroid. We compute inverses of these intersection pairings, allowing us to identify (variants of) these intersection forms with the contravariant form of Schechtman--Varchenko, and the bilinear form of Varchenko. Building on parallel joint work with C. Eur, we define a notion of scattering amplitudes for matroids. We show that matroid amplitudes satisfy locality and unitarity, and recover biadjoint scalar amplitudes in the case of the complete graphic matroid. We apply our formulae for twisted intersection forms to deduce old and new formulae for scattering amplitudes.