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The index formula for the moduli of \(G\)-bundles on a curve. (English) Zbl 1193.14015

Let \(G\) be a reductive connected complex Lie group and let \(\mathfrak M\) be the moduli stack of algebraic \(G\)-bundles over a smooth projective algebraic curve. The authors prove the formulas for the index of \(K\)-theory classes over \(\mathfrak M\) conjectured by C. Teleman [London Math. Soc. Lect. Notes Ser. 308, 358–378 (2004; Zbl 1123.53048)]. For line bundles, the formulas generalize E. Verlinde’s which the authors extend to include the Atiyah–Bott classes. The formulas have Witten’s integrals over the moduli space of stable bundles as their large level limits. As an application, the Newstead–Ramanan conjecture on the vanishing of high Chern classes of certain moduli spaces of semi-stable \(G\)-bundles is proved.

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
19L10 Riemann-Roch theorems, Chern characters
19L47 Equivariant \(K\)-theory

Citations:

Zbl 1123.53048
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References:

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