Yang-Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics

Abstract

Atiyah and Bott used equivariant Morse theory applied to the Yang–Mills functional to calculate the Betti numbers of moduli spaces of vector bundles over a Riemann surface, rederiving inductive formulae obtained from an arithmetic approach which involved the Tamagawa number of SLn. This article attempts to survey and extend our understanding of this link between Yang–Mills theory and Tamagawa numbers, and to explain how methods used over the last three decades to study the singular cohomology of moduli spaces of bundles on a smooth projective curve over ℂ can be adapted to the setting of 𝔸1-homotopy theory to study the motivic cohomology of these moduli spaces over an algebraically closed field.