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Nodal degeneration of chiral algebras

Elchanan Nafcha

Abstract

Given a family of stable curves, we define a sheaf of factorization algebras associated to any universal factorization algebra, and prove a gluing formula for the corresponding sheaf of chiral homology, generalizing the sheaves of vertex algebras and the associated Verlinde formula for gluing of conformal blocks.

Nodal degeneration of chiral algebras

Abstract

Given a family of stable curves, we define a sheaf of factorization algebras associated to any universal factorization algebra, and prove a gluing formula for the corresponding sheaf of chiral homology, generalizing the sheaves of vertex algebras and the associated Verlinde formula for gluing of conformal blocks.

Paper Structure

This paper contains 22 sections, 47 theorems, 263 equations.

Table of Contents

  1. Introduction
  2. Motivation and main results
  3. Overview
  4. Notations and conventions
  5. Ind-coherent sheaves over infinite type stacks
  6. Relation to previous work
  7. Acknowledgements
  8. Chiral and factorization algebras
  9. Chiral monoidal structure
  10. Chiral and factorization modules
  11. Chiral Homology
  12. Universal factorization algebras
  13. Moduli spaces of nodal curves
  14. Stable weighted pointed curves
  15. Moduli of semistable modifications for (g,n) = (0,2)
  16. ...and 7 more sections

Key Result

Theorem 1.2

(Theorem thm:loc-fact) For any family of smooth curves $X/S$, there exists a classifying map such that pullback map $\pi_{\mathop{\mathrm{dR}}\nolimits}^{X/S,!}$ sends a universal factorization algebra to an $S$-family of factorization algebras $\mathcal{A}_{X/S}$ over $X/S$.

Theorems & Definitions (142)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Lemma 1.7
  • Lemma 1.8
  • Definition 2.1
  • Lemma 2.2
  • ...and 132 more