Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- 1 An Introduction to Affine Lie Algebras and the Associated Groups
- 2 Space of Vacua and its Propagation
- 3 Factorization Theorem for Space of Vacua
- 4 Fusion Ring and Explicit Verlinde Formula
- 5 Moduli Stack of Quasi-parabolic G-Bundles and its Uniformization
- 6 Parabolic G-Bundles and Equivariant G-Bundles
- 7 Moduli Space of Semistable G-Bundles Over a Smooth Curve
- 8 Identification of the Space of Conformal Blocks with the Space of Generalized Theta Functions
- 9 Picard Group of Moduli Space of G-Bundles
- Appendix A Dynkin Index
- Appendix B C-Space and C-Group Functors
- Appendix C Algebraic Stacks
- Appendix D Rank-Level Duality (A Brief Survey) (by Swarnava Mukhopadhyay)
- Bibliography
- Index
9 - Picard Group of Moduli Space of G-Bundles
Published online by Cambridge University Press: 19 November 2021
- Frontmatter
- Contents
- Preface
- Introduction
- 1 An Introduction to Affine Lie Algebras and the Associated Groups
- 2 Space of Vacua and its Propagation
- 3 Factorization Theorem for Space of Vacua
- 4 Fusion Ring and Explicit Verlinde Formula
- 5 Moduli Stack of Quasi-parabolic G-Bundles and its Uniformization
- 6 Parabolic G-Bundles and Equivariant G-Bundles
- 7 Moduli Space of Semistable G-Bundles Over a Smooth Curve
- 8 Identification of the Space of Conformal Blocks with the Space of Generalized Theta Functions
- 9 Picard Group of Moduli Space of G-Bundles
- Appendix A Dynkin Index
- Appendix B C-Space and C-Group Functors
- Appendix C Algebraic Stacks
- Appendix D Rank-Level Duality (A Brief Survey) (by Swarnava Mukhopadhyay)
- Bibliography
- Index
Summary
The main aim of this chapter is to determine the Picard group of the moduli space M(G) of semistable G-bundles over a smooth projective curve ? explicitly and show that it is generated by the theta bundles. In fact, it is shown that the theta bundle corresponding to the fundamental representation with the minimal Dynkin index freely generates the Picard group. In particular, it is isomorphic with the group of integers. We further prove that M(G) is Gorenstein and we identify its dualizing line bundle. Moreover, we prove the vanishing of the higher cohomology of the theta bundles over M(G). The moduli space M(G) is identified as a weighted projective space for ? an elliptic curve. This identification allows us to directly prove the above results in the case of genus-1 curves.
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- Publisher: Cambridge University PressPrint publication year: 2021