Exciton scattering via algebraic topology

Abstract

This paper introduces an intersection theory problem for maps into a smooth manifold equipped with a stratification. We investigate the problem in the special case when the target is the unitary group [Formula: see text] and the domain is a circle. The first main result is an index theorem that equates a global intersection index with a finite sum of locally defined intersection indices. The local indices are integers arising from the geometry of the stratification. The result is used to study a well-known problem in chemical physics, namely, the problem of enumerating the electronic excitations (excitons) of a molecule equipped with scattering data.