Comparison of generalized theta functions
type: Item
instance of: scholarly article
title: Comparison of generalized theta functions
zbMATH Open document ID: 0843.14013
DOI: 10.1215/S0012-7094-94-07618-7
published in: Duke Mathematical Journal
publication date: 1996-08-19T00:00:00Z
review text: Let \(C\) be a smooth complex projective curve of genus \(g\), and let \(R\) be a connected reductive complex Lie group. Then, due to the (meanwhile classical) results of \textit{A. Ramanathan} [Math. Ann. 213, 129-152 (1975; Zbl 0284.32019)], there is a suitable notion of (semi-)stability for principal \(R\)-bundles over \(C\) of given topological type \(\ell \in \pi_1 (R)\). Moreover, the equivalence classes of semi-stable principal \(R\)-bundles of topological type \(\ell \) are parametrized by the closed points of a normal, irreducible projective variety \({\mathfrak M}_C (R, \ell)\), which is a coarse moduli space for those objects. The goal of the present paper is to study certain line bundles \(L\) on the moduli spaces \({\mathfrak M}_C (R, \ell)\), their spaces of global sections \(H^0 ({\mathfrak M}_C (R, \ell)\), which are called the spaces of generalized theta functions on \({\mathfrak M}_C (R, \ell)\), and the behavior of their dimensions under the change of the structure group \(R\) by Lie group isogenies. For this purpose, the author defines a subgroup of the Picard group of the moduli stack of \({\mathfrak M}_C (R, \ell)\), whose elements are called the determinant line bundles on the moduli functor. Those determinant line bundles that actually live on the moduli space \({\mathfrak M}_C (R, \ell)\) itself are called the determinant line bundles on \({\mathfrak M}_C (R, \ell)\). Then a generalized theta function on \({\mathfrak M}_C( R, \ell)\) is defined to be a section of any ample determinant line bundle on \({\mathfrak M}_C (R, \ell )\). The author's main result is an equality that compares effectively the dimensions of spaces of generalized theta functions on different moduli spaces. More precisely, let \(G\) be the semisimple part of the structure group \(R\) and \(Z_0\) the connected component of the center of \(R \). Then the author's comparison theorem relates the dimensions of certain spaces of generalized theta functions on the moduli spaces…
reviewed by: Werner Kleinert
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14K25
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14M12
zbMATH DE Number: 727834
zbMATH Keywords: principal bundles on algebraic curves
zbMATH Keywords: semistability
zbMATH Keywords: moduli stack
zbMATH Keywords: determinant line bundles
zbMATH Keywords: generalized theta function
zbMATH Keywords: Verlinde formula
author: Tony Pantev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 727834