Factorization of rank two theta functions. II: Proof of the Verlinde formula
type: Item
instance of: scholarly article
title: Factorization of rank two theta functions. II: Proof of the Verlinde formula
zbMATH Open document ID: 0847.32020
DOI: 10.1007/BF01446283
published in: Mathematische Annalen
publication date: 1996-02-25T00:00:00Z
full work available at URL: https://eudml.org/doc/165393
review text: The authors extend the arguments used previously in ibid. 297, No. 3, 417-466 (1993; Zbl 0788.32012) to get a general result concerning the factorization of holomorphic sections of line bundles on the moduli spaces of rank two parabolic bundles on Riemann surfaces. The main result is the proof of the Verlinde formula for \(SU(2)\): For all \(g \geq 2\) and all integers \(n_1, \dots, n_m\), one has the equality \(D^{g,k}_{n_1, \dots, n_m} = N^{g, k}_{n_1, \dots, n_m}\), where \[ D^{g,k}_{n_1, \dots, n_m} = \begin{cases} \dim \Hom_{\mathcal G} (V_{n_1} \otimes \cdots \otimes V_{n_m}, H^0 ({\mathcal A}, \Delta^{\oplus k}), \quad & \text{if } n_1, \dots, n_m \geq 0 \\ 0, \quad & \text{otherwise} \end{cases} \] \[ \text{and } N^{g,k}_{n_1, \dots, n_m} = \sum^k_{l = 0} (S_{l0})^{2 - 2g} {S_{ln_1} \over S_{l0}} \cdots {S_{ln_m} \over S_{l0}}. \]
Mathematics Subject Classification ID: 32G13
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 32G07
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 32L10
zbMATH DE Number: 840212
zbMATH Keywords: moduli space
zbMATH Keywords: factorization of holomorphic sections of line bundles
zbMATH Keywords: rank two parabolic bundles
zbMATH Keywords: Riemann surfaces
zbMATH Keywords: Verlinde formula
author: Georgios D. Daskalopoulos
author: Richard A. Wentworth
reviewed by: Rubén A. Hidalgo
MaRDI profile type: Publication
cites work: The Yang-Mills equations over Riemann surfaces
cites work: Dirac operators coupled to vector potentials
cites work: Conformal blocks and generalized theta functions
cites work: Generalized SU(2) theta functions
cites work: Hilbert polynomials of moduli spaces of rank 2 vector bundles. II
cites work: Variations of moduli of parabolic bundles
cites work: MODULI OF STABLE PAIRS FOR HOLOMORPHIC BUNDLES OVER RIEMANN SURFACES
cites work: Q4400687
cites work: The integrability criterion in SU(2) Chern-Simons gauge theory
cites work: Local degeneration of the moduli space of vector bundles and factorization of rank two theta functions. I
cites work: Infinite determinants, stable bundles and curvature
cites work: Groupe de Picard des variétés de modules de fibrés semi-stable sur les courbes algébriques. (Picard groups of moduli varieties of semi- stable bundles on algebraic curves)
cites work: Q4294360
cites work: SU(2) Chern-Simons theory at genus zero
cites work: Moduli of vector bundles on curves with parabolic structures
cites work: Classical and quantum conformal field theory
cites work: Q3863986
cites work: Stable bundles of rank 2 and odd degree over a curve of genus 2
cites work: Topological properties of some spaces of stable bundles
cites work: A vanishing theorem for semipositive line bundles over non-Kähler manifolds
cites work: Hilbert polynomials of moduli spaces of rank 2 vector bundles. I
cites work: Conformal field theory and the cohomology of the moduli space of stable bundles
cites work: Q3472274
cites work: Fusion rules and modular transformations in 2D conformal field theory
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description: scientific article; zbMATH DE number 840212