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

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
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