Generalized SU(2) theta functions
type: Item
instance of: scholarly article
title: Generalized SU(2) theta functions
zbMATH Open document ID: 0816.14013
DOI: 10.1007/BF01244310
author: Aaron Bertram
published in: Inventiones Mathematicae
publication date: 1995-07-10T00:00:00Z
full work available at URL: https://eudml.org/doc/144130
review text: In 1965, M. S. Narasimhan and C. S. Seshadri showed that the moduli space of stable rank-2 vector bundles with trivial determinant over a compact Riemann surface of genus \(g\) is diffeomorphic to the smooth real manifold parametrizing the irreducible SU(2)-representations of the fundamental group of the Riemann surface modulo conjugation. This result was generalized by V. Mehta and C. S. Seshadri in 1980, who proved that the manifold \(\Hom^{\text{irr}} (\Gamma, \text{SU(2))/SU(2)}\), associated with a Fuchsian group having only parabolic fixed points in the Siegel upper-half plane, can be regarded as a diffeomorphic model of the moduli space of certain special stable vector bundles of rank 2 with trivial determinant [cf. \textit{V. B. Mehta} and \textit{C. S. Seshadri}, Math. Ann. 248, 205-239 (1980; Zbl 0454.14006)]. The occuring vector bundles are called parabolic bundles, and their dependence on \(\Gamma\) is expressed by specifying data \((D,k)\), where \(D\) is an effective divisor of even degree on the curve and \(k\) a certain integer. The moduli space \(M(X,D,k)\) of semistable parabolic bundles carries a natural polarizing line bundle \({\mathcal L}\) which is called the generalized theta bundle on \((M,X,D,k)\). The space \(H^ 0 (M(X,D,k), {\mathcal L})\) of global sections of \({\mathcal L}\) is commonly called the space of generalized SU(2)-theta functions (with respect to the data \((D,k))\). The aim of the present paper is to compute the dimension of the \(\mathbb{C}\)-vector space \(H^ 0 (M(X, D, k), {\mathcal L})\) in terms of \(D,k\), and the genus \(g\) of \(X\). The result is the verification of Verlinde's formula, which has recently been established for a more general situation by \textit{G. Faltings} [J. Algebr. Geom. 3, No. 2, 347-374 (1994; Zbl 0809.14009)]. The author's approach, in this special case, is conceptually much simpler and yields, as a byproduct, some new information about certain sheaves on projective spaces. Moreover, the autho…
reviewed by: Werner Kleinert
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14H42
Mathematics Subject Classification ID: 14H55
Mathematics Subject Classification ID: 81T99
Mathematics Subject Classification ID: 14D22
Mathematics Subject Classification ID: 14K25
zbMATH DE Number: 549658
zbMATH Keywords: parabolic vector bundles
zbMATH Keywords: compact Riemann surface
zbMATH Keywords: generalized SU(2)- theta functions
zbMATH Keywords: Verlinde's formula
MaRDI profile type: Publication
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OpenAlex ID: W2085485921
description: scientific article; zbMATH DE number 549658