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Conformal blocks, fusion rules and the Verlinde formula. (English) Zbl 0848.17024

Teicher, Mina (ed.), Proceedings of the Hirzebruch 65 conference on algebraic geometry, Bar-Ilan University, Ramat Gan, Israel, May 2-7, 1993. Ramat-Gan: Bar-Ilan University, Isr. Math. Conf. Proc. 9, 75-96 (1996).
As the author writes in his abstract “a Rational Conformal Field Theory (RCFT) is a functor which associates to any Riemann surface with marked points a finite-dimensional vector space, so that certain axioms are satisfied; the Verlinde formula computes the dimension of these vector spaces. For some particular RCFTs associated to a compact Lie group \(G\) (the WZW-models), these spaces have a beautiful algebro-geometric interpretation as spaces of generalized theta functions, that is, sections of a determinant bundle (or its powers) over the moduli space of \(G\)-bundles on a Riemann surface.”
The author explains and develops further the proof of the Verlinde formula for the groups of type \(A,B,C,D\), and \(G\), following the work of Tsuchiya-Ueno-Yamada, Faltings, and Beauville-Laszlo. It is explained how the formula can be derived from the factorization rules. The Verlinde spaces are defined via coinvariants of certain tensor powers of highest weight modules of the Kac-Moody algebra associated to the group \(G\). The factorization rules are rules which should be fulfilled if one degenerates the curve with marked points to a union of curves with marked points of lower genera. The formalism of fusion rules and fusion rings is developed to encode them. This formalism is used to study the fusion ring of level \(l\) representations of the Kac-Moody algebra. By determining the character of these rings the Verlinde formula is proven.
This article can be highly recommended as an introduction to the original literature on the proof of the Verlinde formula. The correspondence with the generalized theta functions is not given in this article.
For the entire collection see [Zbl 0828.00035].

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
14H10 Families, moduli of curves (algebraic)
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
17B81 Applications of Lie (super)algebras to physics, etc.
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