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

*(English)*Zbl 1166.14024This survey article, based on the third author’s presidential address to the LMS in 2005, studies the “unexpected links” between different approaches to computing the Betti numbers of the moduli space \(\mathcal{M}_C(n,d)\) of stable vector bundles of rank \(n\) and degree \(d\) over a fixed, compact Riemann surface \(C\) (and always \(\gcd(n,d)=1\)). These are Atiyah-Bott’s approach via Yang-Mills theory and equivariant Morse theory [M. F. Atiyah and R. Bott, Philos. Trans. R. Soc. Lond., A 308, 523–615 (1983; Zbl 0509.14014)], the arithmetic approach of G. Harder and M. S. Narasimhan [Math. Ann. 212, 215–248 (1975; Zbl 0324.14006)] and U. V. Desale and S. Ramanan [Math. Ann. 216, 233–244 (1975; Zbl 0317.14005)] using the Weil conjectures and Tamagawa measures, and a construction in the spirit of Bifet-Ghione-Letizia of \(\mathcal{M}_C(n,d)\) as a finite-dimensional GIT-quotient via maps into Grassmannians and matrix divisors.

These approaches are reviewed in §2, but the main aim of the paper is to translate the third approach into the setting of \(\mathbb{A}^1\)-homotopy theory and the motivic cohomology of Bloch and Voevodsky [C. Mazza, V. Voevodsky, C. Weibel, Lecture notes on motivic cohomology. Clay Mathematics Monographs 2. (Providence), RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute. (2006; Zbl 1115.14010)]. The authors ultimately present a study of the motivic cohomology of the moduli \(\mathcal{M}_C(n,d)\) of bundles on a smooth projective curve \(C\) over any algebraically closed field, and close the circle by relating it back to the Yang-Mills and the arithmetic approach.

In §3 the authors present concisely the theory of equivariant motivic cohomology and its basic properties; §4 reviews the cohomology of GIT quotients and its adaptation to the motivic setting. Finally, §5 studies the motivic cohomology of \(\mathcal{M}_C(n,d)\) over an arbitrary algebraically closed field, by treating \(\mathcal{M}_C(n,d)\) as a finite-dimensional GIT-quotient and applying the previously developed technology.

Throughout the paper, the authors wish to make evident the connections between the algebraic and the topological worlds, and in doing so they give a beautiful overview of a diverse range of subjects and how they all provide insight into the classical moduli problem.

These approaches are reviewed in §2, but the main aim of the paper is to translate the third approach into the setting of \(\mathbb{A}^1\)-homotopy theory and the motivic cohomology of Bloch and Voevodsky [C. Mazza, V. Voevodsky, C. Weibel, Lecture notes on motivic cohomology. Clay Mathematics Monographs 2. (Providence), RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute. (2006; Zbl 1115.14010)]. The authors ultimately present a study of the motivic cohomology of the moduli \(\mathcal{M}_C(n,d)\) of bundles on a smooth projective curve \(C\) over any algebraically closed field, and close the circle by relating it back to the Yang-Mills and the arithmetic approach.

In §3 the authors present concisely the theory of equivariant motivic cohomology and its basic properties; §4 reviews the cohomology of GIT quotients and its adaptation to the motivic setting. Finally, §5 studies the motivic cohomology of \(\mathcal{M}_C(n,d)\) over an arbitrary algebraically closed field, by treating \(\mathcal{M}_C(n,d)\) as a finite-dimensional GIT-quotient and applying the previously developed technology.

Throughout the paper, the authors wish to make evident the connections between the algebraic and the topological worlds, and in doing so they give a beautiful overview of a diverse range of subjects and how they all provide insight into the classical moduli problem.

Reviewer: Thomas Koeppe (Edinburgh)