Theta functions, geometric quantization and unitary Schottky bundles
نویسندگان
چکیده
We study geometric quantization of moduli spaces of vector bundles on an algebraic curve X, and its relation to theta functions. In limits when the complex structure of X degenerates, we describe vector spaces of distributions with Verlinde dimension, associated to the quantization of some of these moduli spaces in real polarizations. We show that special bases on these spaces, defined using a trinion decomposition of X, are related by modular transformation matrices for characters of affine Lie algebras, which appear naturally in the Blattner-Kostant-Sternberg (BKS) pairing for different quantizations of the moduli space.
منابع مشابه
Quantization, Classical and Quantum Field Theory and Theta-Functions.
Arnaud Beauville’s survey ”Vector bundles on Curves and Generalized Theta functions: Recent Results and Open Problems” [Be] appeared 10 years ago. This elegant survey is short (16 pages) but provides a complete introduction to a specific part of algebraic geometry. To repeat his succes now we need more pages, even though we assume that the reader is already acquainted with the material presente...
متن کامل1 1 A pr 1 99 9 Quantization and “ theta functions ”
Geometric Quantization links holomorphic geometry with real geometry, a relation that is a prototype for the modern development of mirror symmetry. We show how this treatment can be used to construct a special basis in every space of conformal blocks. This is a direct generalization of the basis of theta functions with characteristics in every complete linear system on an Abelian variety (see [...
متن کاملGeometric quantization of Hamiltonian actions of Lie algebroids and Lie groupoids
We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line bundle. Next, we discuss a version of Kähler quantization suitable for this s...
متن کاملSchottky Uniformization and Vector Bundles over Riemann Surfaces
We study a natural map from representations of a free group of rank g in GL(n,C), to holomorphic vector bundles of degree 0 over a compact Riemann surface X of genus g, associated with a Schottky uniformization of X . Maximally unstable flat bundles are shown to arise in this way. We give a necessary and sufficient condition for this map to be a submersion, when restricted to representations pr...
متن کاملThe Geometry of Unitary 2-Representations of Finite Groups and their 2-Characters
Motivated by topological quantum field theory, we investigate the geometric aspects of unitary 2-representations of finite groups on 2-Hilbert spaces, and their 2-characters. We show how the basic ideas of geometric quantization are ‘categorified’ in this context: just as representations of groups correspond to equivariant line bundles, 2-representations of groups correspond to equivariant gerb...
متن کامل