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2 results

  • 5 - Moduli Stack of Quasi-parabolic G-Bundles and its Uniformization

  • Shrawan Kumar, University of North Carolina, Chapel Hill
  • Book:
    Conformal Blocks, Generalized Theta Functions and the Verlinde Formula
    Published online:
    19 November 2021
    Print publication:
    25 November 2021, pp 163-183

  • Summary

    We define the stack of G-bundles over smooth projective curves for a reductive algebraic group G as well as the stack of quasi-parabolic G-bundles over s-pointed smooth curves. Both of these stacks are smooth algebraic stacks. We define a tautological family of G-bundles over any smooth curve ? parameterized by the infinite Grassmannian. Then, we prove the Uniformization Theorem for both of the above two stacks. An important ingredient in the proof of the above two uniformization theorems is a result due to Drinfeld--Simpson asserting that for a family of G-bundles over ? parameterized by a scheme S, the pull-back of the family to some étale cover is trivial restricted to any affine open subset of ?. In particular, the uniformization theorems give a bijective parameterization of G-bundles as well as quasi-parabolic G-bundles over ?.

Conformal Blocks, Generalized Theta Functions and the Verlinde Formula
  • Shrawan Kumar
  • Published online:
    19 November 2021
    Print publication:
    25 November 2021
  • In 1988, E. Verlinde gave a remarkable conjectural formula for the dimension of conformal blocks over a smooth curve in terms of representations of affine Lie algebras. Verlinde's formula arose from physical considerations, but it attracted further attention from mathematicians when it was realized that the space of conformal blocks admits an interpretation as the space of generalized theta functions. A proof followed through the work of many mathematicians in the 1990s. This book gives an authoritative treatment of all aspects of this theory. It presents a complete proof of the Verlinde formula and full details of the connection with generalized theta functions, including the construction of the relevant moduli spaces and stacks of G-bundles. Featuring numerous exercises of varying difficulty, guides to the wider literature and short appendices on essential concepts, it will be of interest to senior graduate students and researchers in geometry, representation theory and theoretical physics.