Conformal blocks and generalized theta functions. (English) Zbl 0815.14015

Beauville, Arnaud; Laszlo, Yves

Commun. Math. Phys. 164, No. 2, 385-419 (1994).

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 \(p \in X\).

Reviewer: W.Kleinert (Berlin)

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

Keywords:

moduli space; compact Riemann surface; generalized theta bundle; Verlinde formula; conformal quantum field; representation theory of Kac-Moody algebras

Citations:

Zbl 0696.17010; Zbl 0752.17033

References:

[1]	[A-D-K] Arbarello, E., De Concini, C., Kac, V.: The infinite wedge representation and the reciprocity law for algebraic curves. Proc. of Symp. in Pure Math.49, 171–190 (1989) · Zbl 0699.22028
[2]	[B] Bourbaki, N.: Algèbre commutative, ch. 5 to 7. Paris: Masson 1985 · Zbl 0547.13002
[3]	[D-M] Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Publ. Math. IHES36, 75–110 (1969) · Zbl 0181.48803
[4]	[D-N] Drezet, J.M., Narasimhan, M.S.: Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques. Invent. Math.97, 53–94 (1989) · Zbl 0689.14012
[5]	[F] Faltings, G.: A proof for the Verlinde formula. J. Alg. Geom., to appear · Zbl 0809.14009
[6]	[G] Gepner, D.: Fusion rings and geometry. Commun. Math. Phys.141, 381–411 (1991) · Zbl 0752.17033
[7]	[K] Kac, V.: Infinite dimensional Lie algebras. Progress. in Math.44, Boston: Birkhäuser 1983 · Zbl 0537.17001
[8]	[Ku] Kumar, S.: Demazure character formula in arbitrary Kac-Moody setting. Invent. Math.89, 395–423 (1987) · Zbl 0635.14023
[9]	[K-N-R] Kumar, S., Narasimhan, M.S., Ramanathan, A.: Infinite Grassmannian and moduli space ofG-bundles. Preprint (1993)
[10]	[L] Laumon, G.: Un analogue global du cône nilpotent. Duke Math. J.57, 647–671 (1988) · Zbl 0688.14023
[11]	[L-MB] Laumon, G., Moret-Bailly, L.: Champs algébriques. Prépublication Université Paris-Sud (1992)
[12]	[M] Mathieu, O.: Formules de caractères pour les algèbres de Kac-Moody générales. Astérisque159–160 (1988)
[13]	[P-S] Peskine, C., Szpiro, L.: Dimension projective finie et cohomologie locale. Publ. Math. IHES42, 47–119 (1973) · Zbl 0268.13008
[14]	[SGA4 1/2] Cohomologie étale. Séminaire de Géométrie algébrique SGA4 1/2, par Deligne, P. Lecture Notes569. Berlin, Heidelberg, New York: Springer 1977
[15]	[SGA6] Théorie des intersections et théorème de Riemann-Roch. Séminaire de Géométrie algébrique SGA 6, dirigé par Berthelot, P., Grothendieck, A., Illusie, L. Lecture Notes225. Berlin, Heidelberg, New York: Springer 1971
[16]	[S1] Slodowy, P.: On the geometry of Schubert varieties attached to Kac-Moody Lie algebras. Can. Math. Soc. Conf. Proc.6, 405–442 (1984)
[17]	[S-W] Segal, G., Wilson, G.: Loop groups and equations of KdV type. Publ. Math. IHES61, 5–65 (1985) · Zbl 0592.35112
[18]	[T] Tate, J.: Residues of differentials on curves. Ann. Scient. Éc. Norm. Sup.1 (4ème série). 149–159 (1968) · Zbl 0159.22702
[19]	[Tu] Tu, L.: Semistable bundles over an elliptic curve. Adv. in Math.98, 1–26 (1993) · Zbl 0786.14021
[20]	[T-U-Y] Tsuchiya, A., Ueno, K., Yamada, Y.: Conformal field theory on universal family of stable curves with gauge symmetries. Adv. Studies in Pure Math.19, 459–566 (1989) · Zbl 0696.17010
[21]	[V] Verlinde, E.: Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys. B300, 360–376 (1988) · Zbl 1180.81120
[22]	[W] Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys.121, 351–399 (1986) · Zbl 0667.57005