ar X iv : 0 80 1 . 47 33 v 1 [ m at h . A G ] 3 0 Ja n 20 08 Yang - Mills theory and Tamagawa Numbers

The unexpected link which is the topic of this article was remarked on by Atiyah himself and his collaborator Raoul Bott in their fundamental 1983 paper [AB83] on the Yang-Mills equations over Riemann surfaces. In this paper Atiyah and Bott used ideas coming from Yang-Mills theory and equivariant Morse theory to derive inductive formulae for the Betti numbers of the moduli spaces M(n, d) of stable vector bundles of rank n and degree d over a fixed compact Riemann surface C of genus g ≥ 2, when n and d are coprime. (We will assume throughout this introduction that n and d are coprime integers with n > 0.) Equivalent formulae had been obtained earlier by Harder and Narasimhan [HN75] and Desale and Ramanan [DR75] using arithmetic techniques and the Weil conjectures. In the latter approach a crucial ingredient was the fact, proved by Weil, that the volume of a certain locally symmetric space attached to SLn with respect to a canonical measure – an invariant known as the Tamagawa number of SLn – is 1. Atiyah and Bott observed that although the two methods came from very different branches of mathematics, namely arithmetic and physics, there was a formal correspondence between them, with the Tamagawa number of SLn (or equivalently the function field analogue of the Siegel formula) playing, roughly speaking, the rôle of the cohomology of the classifying space of the gauge group in the Atiyah-Bott approach. They asked for a deeper understanding of this observation and in particular for a geometric explanation, exploiting the analogy with equivariant cohomology, of the fact that the Tamagawa number of SLn is 1. Contributions since then towards such understanding have included work by Bifet, Ghione and Letizia [Bif89, BGL94], providing another inductive procedure