Moduli spaces of parabolic fibres and conformal blocks
type: Item
instance of: scholarly article
title: Moduli spaces of parabolic fibres and conformal blocks
zbMATH Open document ID: 0877.14031
DOI: 10.1215/S0012-7094-96-08408-2
author: Christian Pauly
published in: Duke Mathematical Journal
publication date: 1996-10-30T00:00:00Z
review text: Let \(C\) be a smooth, irreducible, projective complex curve. Using a double coset construction, \textit{A. Beauville} and \textit{Y. Laszlo} [Commun. Math. Phys. 164, No. 2, 385-419 (1994; Zbl 0815.14015)] built up a canonical isomorphism between the vector space of conformal blocs studied in conformal field theory and the space of sections of an ample line bundle on the space of semistable vector bundles of fixed rank with trivial determinant. The present article extends this isomorphism setting into correspondence the space of sections of an ample line bundle on the space \(M(r,k, \vec{a}, \vec{n})\) of semistable parabolic bundles on \(C\) associated to a fixed finite set \(I\) of points of \(C\), and the dual of the vector space \(V_C(I, \vec{\lambda})\) of conformal blocs associated to \(I\) and a set \(\vec{\lambda} = \{\lambda_x\}_{x\in I}\) of dominant weights of the simple Lie algebra \(sl_r(\mathbb{C})\). The weights \(\vec{\lambda}\) depend on the parabolic weights \((k, \vec{a}, \vec{n})\). Using the factorizations rules for conformal blocs [see \textit{A. Tsuchiya}, \textit{K. Ueno} and \textit{Y. Yamada} in: Integrable systems in quantum field theory and statistical mechanics, Proc. Symp., Kyoto and Kyuzeso 1988, Adv. Stud. Pure Math. 19, 459-566 (1989; Zbl 0696.17010)], this allows the author to derive a Verlinde-type formula for the moduli spaces of parabolic vector bundles, i.e. to calculate the dimension of the space of sections cited above.
reviewed by: I.Quandt
Mathematics Subject Classification ID: 14K25
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 81T99
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 14H60
zbMATH DE Number: 922159
zbMATH Keywords: parabolic bundles
zbMATH Keywords: conformal blocs
zbMATH Keywords: generalized theta functions
zbMATH Keywords: Verlinde formula
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 922159