English
 
Help Privacy Policy Disclaimer
  Advanced SearchBrowse

Item

ITEM ACTIONSEXPORT

Released

Book Chapter

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

MPS-Authors
/persons/resource/persons236497

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

External Resource
No external resources are shared
Fulltext (public)
There are no public fulltexts stored in PuRe
Supplementary Material (public)
There is no public supplementary material available
Citation

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.