Geometry and topological photonics

Topological photonics provides a powerful framework to describe and understand many nontrivial wave phenomena in complex electromagnetic platforms. The topological index of a physical system is an abstract global property that depends on the family of operators that describes the propagation of Bloch waves. Here, we highlight that there is a profound geometrical connection between topological physics and the topological theory of mathematical surfaces. We show that topological band theory can be understood as a generalization of the topological theory of surfaces and that the genus of a surface can be regarded as a Chern number of a suitable operator defined over the surface. We point out some nontrivial implications of topology in the context of radiation problems and discuss why for physical problems the topological index is often associated with a bulk-edge correspondence.

Automated summary

Topological photonics offers a valuable framework to explain complex wave phenomena in electromagnetic systems. The topological index of a physical system is a global property dependent on the operators describing wave propagation. We establish a significant geometric connection between topological physics and the topological theory of mathematical surfaces. Our findings demonstrate that topological band theory extends the surface topological theory, wherein the surface genus can be considered as a Chern number of a suitable surface operator. We also explore the implications of topology in radiation problems and the bulk-edge correspondence in physical systems.