Twisted (co)homology of Matroids

The study of the topology of hyperplane arrangement complements has long been a central part of combinatorial algebraic geometry. I will talk about intersection pairings on the twisted (co)homology for a hyperplane arrangement complement, first studied by Cho, Matsumoto, Kita, and Yoshida. There are two distinct pairings, one in deRham (co)homology, and one in Betti (co)homology. We show that these pairings are matroidal, give a tropical formula for them, and relate them to work of Schechtman and Varchenko.

This work is motivated by some questions in scattering amplitudes, which will be discussed in my (physics seminar) talk during the Combinatorics of Fundamental Physics workshop.