Comparison of generalized theta functions
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 of semistable principal bundles with structure groups \(R\), \(G\) and \(Z_0\), together with canonically defined topological types. The main result of the paper generalizes a theorem (and its method of proof) by \textit{R. Donagi} and \textit{L. Tu} [cf. ``Theta functions for \(SL (n,C)\) versus \(GL (n,C)\)'', Math. Res. Lett. 1, No. 3, 345-357 (1994)] to arbitrary connected reductive algebraic groups. In the special case of a simply connected simple group \(R = G\), the author's result is covered by the famous Verlinde formula [cf. \textit{G. Faltings}, J. Algebr. Geom. 3, No. 2, 347-374 (1994; Zbl 0809.14009)]. In this regard, the comparison formula proved in the present paper may be understood as being an important contribution to further insights concerning the Verlinde formula and its significance in conformal field theories.
scientific article; zbMATH DE number 727834
scientific article