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Elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula

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Zagier,  Don
Max Planck Institute for Mathematics, Max Planck Society;

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Zagier, D. (1996). Elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula. In Proceedings of the Hirzebruch 65 conference on algebraic geometry, Bar- Ilan University, Ramat Gan, Israel, May 2-7, 1993 (pp. 445-462). Ramat-Gan: Bar-Ilan University.

Cite as: http://hdl.handle.net/21.11116/0000-0004-38ED-3
Abstract
Let Ng, n, d denote the moduli space of semistable n-dimensional vector bundles over a fixed Riemann surface of genus g and having as determinant bundle a fixed line bundle of degree d. Its topology depends only on g, n and d\\pmod n. Over Ng, n, d there is a canonically defined line bundle \\Theta, generalizing the classical theta bundle over the Jacobian of a curve. A basic tool in the study of the moduli spaces is the determinant of the numbers \\dim H^0 (Ng, n, d, \\Theta^k) for variable k. An explicit formula for these numbers was conjectured by the physicist E. Verlinde. In the simplest case n=2, it says that \\dim H^0 (Ng, 2, 0, \\Theta^k)= D+ (g, k+2), \\dim H^0 (Ng, 2, 1, \\Theta^k)= D- (g, 2k+ 2), where D\\varepsilon (g, k)= \\biggl( k\\over 2 \\biggr)^g-1 \\sumj\\pmod k, j\\not\\equiv 0\\pmod k \\varepsilon^j-1 \\over \\sin^2g- 2 π j \\over k \\qquad (g, k\\in \\bbfN,\\ \\varepsilon= \\pm 1,\\ \\varepsilon^k= 1). The present paper discusses some of the many interesting number-theoretical and combinatorial aspects of the formula in the case n=2. Formulas for the rank 3 case and a duality formula for the general rank case are also discussed. Finally, a closed formula is given for the Betti number of the moduli space of arbitrary rank bundles over curves by solving explicitly the well-known Harder-Narasimhan-Atiyah-Bott recursion relation for these numbers.