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Raviolo vertex algebras, cochains and conformal blocks

Luigi Alfonsi, Hyungrok Kim, Charles A. S. Young

Abstract

Raviolo vertex algebras were introduced recently by Garner and Williams in arXiv:2308.04414. Working at the level of cochain complexes, in the present paper we construct spaces of conformal blocks, or more precisely their duals, coinvariants, in the raviolo setting. We prove that the raviolo state-field map correctly captures the limiting behaviour of coinvariants as marked points collide.

Raviolo vertex algebras, cochains and conformal blocks

Abstract

Raviolo vertex algebras were introduced recently by Garner and Williams in arXiv:2308.04414. Working at the level of cochain complexes, in the present paper we construct spaces of conformal blocks, or more precisely their duals, coinvariants, in the raviolo setting. We prove that the raviolo state-field map correctly captures the limiting behaviour of coinvariants as marked points collide.
Paper Structure (34 sections, 10 theorems, 240 equations, 1 figure)

This paper contains 34 sections, 10 theorems, 240 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The raviolo vacuum module in cochain complexes
  3. The formal raviolo
  4. Functions on the formal raviolo
  5. Splitting
  6. Vacuum module
  7. State-field map
  8. Rational coinvariants and conformal blocks
  9. Fixed punctures
  10. Movable punctures and configuration space
  11. Taking coinvariants
  12. The usual vacuum Verma module and state-field map
  13. Ravioli configuration space
  14. Definition of $\mathrm{RavConf}_N$ by gluing Čech data
  15. Derived sections and the Thom-Sullivan functor
  16. ...and 19 more sections

Key Result

Lemma 1

Given a collection of homogeneous lowering operators $X^i \in {\mathfrak g} \mathbin\otimes {\mathbb C}\{\!\{ z\}\!\}_-$, $i=1,\dots,n$, we have The sum is over unshuffles, i.e. permutations $(\mu_1,\dots,\mu_m,\nu_1,\dots,\nu_{n-m})$ of $(1,\dots,n)$ such that $\mu_1<\dots<\mu_m$ and $\nu_1<\dots<\nu_{n-m}$, and $(-1)^{\chi(|X^1|,\dots,|X^n|,\mu,\nu)}$ is the Koszul sign of an unshuffle of the $

Figures (1)

  • Figure 1: Sketch of copies of the formal disc $D$, the formal punctured disc $D^\times$, and the formal raviolo $\mathrm{Rav}$ associated to a point $a$ in the complex plane.

Theorems & Definitions (24)

  • Lemma 1: Explicit formula for the state-field map
  • Lemma 2
  • proof
  • Theorem 3: Relation of the state-field map $Y$ to rational coinvariants
  • Remark 4
  • Theorem 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 14 more