On E. Verlinde’s formula in the context of stable bundles. (English) Zbl 0820.14024

Bott, Raoul

Int. J. Mod. Phys. A 6, No. 16, 2847-2858 (1991).

This paper reports on some ideas around Verlinde’s formula without giving details or proofs. Thus it could be considered as a leisure introduction to the subject. The following topics are discussed:
1. Classical theta functions and Jacobians of Riemann surfaces. The theta functions are interpreted as holomorphic sections of powers of the line bundle, associated to the theta-divisor on the Jacobian, and the dimension formula of the space of this sections in terms of the Riemann- Roch formula is explained.
2. The nonabelian analogue is briefly discussed, i.e. the Jacobian is replaced by the moduli space of semistable vector bundles of rank \(n\) and with prescribed determinant, such that the holomorphic Euler characteristic of the bundles vanishes. Then the bundles which have nonzero holomorphic sections form a divisor, in perfect analogy to the rank 1 case and the classical theta-divisor. Verlinde’s formula is then interpreted as a formula for the dimension of tensor powers of the associated line bundle.
3. A group theoretical approach and A. Szenes’ proof of Verlinde’s formula (for the case \(n = 2)\) are briefly discussed.
4. The fusion rules are very briefly discussed.

Reviewer: H.Kurke (Berlin)

MSC:

14H60	Vector bundles on curves and their moduli
14K25	Theta functions and abelian varieties
14C20	Divisors, linear systems, invertible sheaves
32L81	Applications of holomorphic fiber spaces to the sciences

Keywords:

theta functions; Jacobians of Riemann surfaces; Verlinde formula