Vector bundles on curves and generalized theta functions: recent results and open problems

Abstract

Riemann surface carries a natural line bundle, the determinant bundle. The space of sections of this line bundle (or its multiples) constitutes a natural non-abelian generalization of the spaces of theta functions on the Jacobian. There has been much progress in the last few years towards a better understanding of these spaces, including a rigorous proof of the celebrated Verlinde formula which gives their dimension. This survey paper tries to explain what is now known and what remains open.

15 pages, Plain TeX

References in corpus (3)

• Two Dimensional Gauge Theories Revisited
• Theta Functions for $\SL(n)$ versus $\GL(n)$
• The combinatorics of the Verlinde formulas

Cited by in corpus (23)

• The rank-level duality for non-abelian theta functions
• Stable maps and Quot schemes
• The action of outer automorphisms on bundles of chiral blocks
• Moduli of Abelian varieties, Vinberg theta-groups, and free resolutions
• Generalized theta linear series on moduli spaces of vector bundles on curves
• Equivariant Kaehler Geometry and Localization in the G/G Model
• Quantization via Mirror Symmetry
• Extensions of line bundles and Brill--Noether loci of rank-two vector bundles on a general curve
• Vector bundles on curves and theta functions
• Principal Schottky Bundles over Riemann surfaces
• Seiberg-Witten monopoles with multiple spinors on a surface times a circle
• The Chern classes of the Verlinde bundles
• Trace functionals on non-commutative deformations of moduli spaces of flat connections
• A representation theoretic approach to the WZW Verlinde formula
• Duality Construction of Moduli Spaces
• Moduli of rank 4 symplectic bundles over a curve of genus 2
• Factorization of generalized theta functions at reducible case
• Vector bundles and theta functions on curves of genus 2 and 3
• Elementary Transformations of Pfaffian Representations of Plane Curves
• Plane curves as Pfaffians
• Raynaud's vector bundles and base points of the generalized Theta divisor
• Moduli spaces of stable sheaves over quasi-polarized surfaces, and the relative Strange Duality morphism
• Rank four symplectic bundles without theta divisors over a curve of genus two