Interplay of Mathematics and Physics

INTERPLAY OF MATHEMATICS AND PHYSICS: ATIYAH—SINGER INDEX THEORY AND GAUGE THEORIES OF PARTICLE PHYSICS: Atiyah—Singer Index Theory, which relates the solution space of certain types of partial differential equations (PDEs) to topological data of the PDE, is a highlight of modern mathematics. It also has major physical consequences in gauge theories of particle physics where it implies a connection between fermionic zeromodes and gauge field topology. This explains, e.g., the large mass of the eta’ meson in the theory of the strong nuclear interactions. Furthermore, a families version of index theory elucidates the anomalies whose presence spoils the consistency of certain types of quantum field theories known as chiral gauge theories (which includes the unified Standard Model of particle physics). This leads to anomaly cancellation conditions which restrict the allowable particle contents of such theories.