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Conformal blocks in algebraic geometry

Chiara Damiolini

Abstract

These notes survey the theory of (twisted) conformal blocks from an algebro-geometric perspective and have two main goals. The first one is to summarize the construction of conformal blocks from vertex operator algebras, and to describe their fundamental properties -- such as factorization and sewing -- which imply that conformal blocks define vector bundles on moduli spaces of curves whose ranks and Chern classes can be computed explicitly. The second aim is to describe how line bundles on moduli of principal bundles over a curve -- and their sections -- can be understood via (twisted) conformal blocks for (twisted) affine Lie algebras.

Conformal blocks in algebraic geometry

Abstract

These notes survey the theory of (twisted) conformal blocks from an algebro-geometric perspective and have two main goals. The first one is to summarize the construction of conformal blocks from vertex operator algebras, and to describe their fundamental properties -- such as factorization and sewing -- which imply that conformal blocks define vector bundles on moduli spaces of curves whose ranks and Chern classes can be computed explicitly. The second aim is to describe how line bundles on moduli of principal bundles over a curve -- and their sections -- can be understood via (twisted) conformal blocks for (twisted) affine Lie algebras.

Paper Structure

This paper contains 19 sections, 15 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Conformal blocks
  3. Conformal blocks from Lie Algebras
  4. Conformal blocks from VOAs
  5. Twisted conformal blocks
  6. Properties of conformal blocks
  7. Sheaves on moduli of curves
  8. Coherence
  9. Factorization and Sewing
  10. Connection
  11. Corollaries
  12. Vector bundles from coinvariants
  13. Chern Character
  14. Modular functor and tensor product
  15. Properties of twisted conformal blocks
  16. ...and 4 more sections

Key Result

Theorem 3.1.1

The map induces an isomorphism for every smooth point $Q \in C \setminus P_\bullet$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (33)

  • Definition 1
  • Example 2
  • Example 3
  • Definition 4
  • Remark 5
  • Example 6
  • Definition 7
  • Example 8
  • Remark 9
  • Theorem 3.1.1
  • ...and 23 more