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Conformal blocks and generalized theta functions. (English) Zbl 0815.14015
Let \(SU_ X (r)\) be the moduli space of semistable rank-\(r\) vector bundles over a compact Riemann surface \(X\) of genus \(g\), whose determinant bundle is trivial. There is a natural polarizing line bundle \(L\) over \(SU_ X (r)\), the so-called generalized theta bundle. The \(\mathbb{C}\)-vector space of sections of \(L^{\otimes k}\), \(k \in \mathbb{N}\), is called the space of generalized theta functions of order \(k\) on \(SU_ X (r)\), and its dimension may be computed by the famous Verlinde formula.
The Verlinde formula was first discovered by physicists, in the context of conformal quantum field theory [cf. E. Verlinde, “Fusion rules and modular transformations in \(2d\) conformal field theory”, Nucl. Phys. B 300, No. 3, 360-376 (1988)], served then as a conjectural problem (even for more general semistable vector bundles on curves) in algebraic geometry for some years, and was recently affirmatively established, partly in special cases, by several authors [A. Bertram – A. Szenes (1991), M. Thaddeus (1992), G. Faltings (1993)].
The aim of the present paper is to give a proof of the Verlinde formula, in the case mentioned above, by explicitly relating the constructions and arguments of the physicists to computing \(H^ 0 (SU_ X(r), L^{\otimes k})\). This is done by establishing a canonical isomorphism between this space and the so-called space of conformal blocks of level \(k\), denoted by \(B_ k (r)\), which naturally arises in the representation theory of Kac-Moody algebras and their applications in conformal quantum field theory. Having constructed this canonical isomorphism in a mathematically rigorous way, the authors derive the Verlinde formula from the dimension formula for \(B_ k (r)\), which has been computed, in a purely combinatorial manner, by A. Tsuchiya, K. Ueno and Y. Yamada [in Integrable systems in quantum field theory and statistical mechanics, Proc. Sympos. 1988, Adv. Stud. Pure Math. 19, 459-566 (1989; Zbl 0696.17010)], and also by D. Gepner [Commun. Math. Phys. 141, 381-411 (1991; Zbl 0752.17033)].
At the end of the paper, the results are generalized to the case of semistable vector bundles of arbitrary degree and determinant \({\mathcal O}_ X (dp)\), \(d\) being a fixed integer and \(p \in X\).

MSC:
14H42 Theta functions and curves; Schottky problem
14H60 Vector bundles on curves and their moduli
81T20 Quantum field theory on curved space or space-time backgrounds
14K25 Theta functions and abelian varieties
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
14H55 Riemann surfaces; Weierstrass points; gap sequences
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