Exciton Scattering via Algebraic Topology

TL;DR

This paper develops an algebraic topology framework to analyze exciton scattering in molecules, establishing an index theorem that links global and local intersection indices, and provides bounds on the number of electronic excitations.

Contribution

It introduces a novel intersection theory approach for exciton scattering, connecting topology with chemical physics, and derives a sharp lower bound on exciton enumeration.

Findings

Established an index theorem relating global and local intersection indices.

Provided a sharp lower bound for the number of excitons in a molecule.

Applied topological methods to a chemical physics problem.

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 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. We provide a lower bound for this number. The bound is shown to be sharp in a limiting case.