Topological aspects of zero modes in cavity resonators

TL;DR

This paper explores the connection between zero modes in electromagnetic cavity resonators and the cavity's topological properties, revealing how topological invariants determine zero-mode characteristics.

Contribution

It establishes a mathematical relationship linking electromagnetic zero modes to topological invariants like homology groups and Euler characteristic.

Findings

Zero modes' dimension equals the cavity's homology group dimension.

The alternating sum of zero-mode dimensions relates to the Euler characteristic.

Topological invariants influence electromagnetic mode structure.

Abstract

We discuss the relationship between the zero modes of electromagnetic fields in a cavity resonator and the cavity's topological characteristics. We show that the dimension of the electromagnetic zero-mode space coincides with the dimension of the corresponding homology group of the cavity. Moreover, we prove that the alternating sum of the dimensions of the electromagnetic zero-mode spaces is closely related to the Euler characteristic of the cavity boundary, and hence to the integral of the curvature.