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Bulletin of the London Mathematical Society Advance Access originally published online on May 7, 2008
Bulletin of the London Mathematical Society 2008 40(4):533-567; doi:10.1112/blms/bdn036
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© 2008 London Mathematical Society

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

Aravind Asok

Department of Mathematics
University of Washington
Box 354250
Seattle, WA 48195
USA
asok@math.washington.edu

Brent Doran

School of Mathematics
Institute for Advanced Study
Princeton, NJ 08540
USA
doranb@math.ias.edu

Frances Kirwan

The Mathematical Institute
24-29 St Giles
Oxford OX1 3LB
United Kingdom

Received 25 February 2007. Revision received 9 July 2007.

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.

2000 Mathematics Subject Classification 14H60 (14F42 14L24).


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