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One-Point Restricted Conformal Blocks and the Fusion Tensor Product

Jianqi Liu

TL;DR

The paper develops a one-point restricted conformal block framework on the three-pointed projective line to encode fusion data via a contracted tensor product $M^1\odot M^2$ that carries an $A(V)$-module structure. It proves an isomorphism $\mathscr{C}(\Sigma((M^3)',M^1,M^2)) \cong {\rm Hom}_{A(V)}(M^1\odot M^2,\Omega(M^3))$ under suitable hypotheses, providing a purely algebraic bridge between intertwining operators and Zhu-algebra representations. For strongly rational VOAs, this contracted product realizes the fusion tensor product on $\mathsf{Mod}(A(V))$ via the Huang–Lepowsky $P(z)$-tensor framework and its adjoint equivalence with $\mathsf{Adm}(V)$; the approach yields the correct fusion rules in key examples, including affine, Virasoro, and Ising theories. The work advances a rigorous algebraic route to fusion in VOAs, clarifying the relationship between $A(V)$-modules and VOA intertwiners and offering a versatile tool for computing and understanding fusion in nontrivial cases.

Abstract

We investigate a one-point restriction of conformal blocks on $(\mathbb{P}^1,\infty,1,0)$ associated with modules over a vertex operator algebra. By restricting the module attached to the point $\infty$ to its bottom degree, we obtain a new formula for computing fusion rules in terms of a left $A(V)$-module $M^1\odot M^2$ over the Zhu algebra $A(V)$. As a consequence, for strongly rational VOAs, the construction of $M^1\odot M^2$ induces the fusion tensor product on the module category $\mathsf{Mod}(A(V))$.

One-Point Restricted Conformal Blocks and the Fusion Tensor Product

TL;DR

The paper develops a one-point restricted conformal block framework on the three-pointed projective line to encode fusion data via a contracted tensor product that carries an -module structure. It proves an isomorphism under suitable hypotheses, providing a purely algebraic bridge between intertwining operators and Zhu-algebra representations. For strongly rational VOAs, this contracted product realizes the fusion tensor product on via the Huang–Lepowsky -tensor framework and its adjoint equivalence with ; the approach yields the correct fusion rules in key examples, including affine, Virasoro, and Ising theories. The work advances a rigorous algebraic route to fusion in VOAs, clarifying the relationship between -modules and VOA intertwiners and offering a versatile tool for computing and understanding fusion in nontrivial cases.

Abstract

We investigate a one-point restriction of conformal blocks on associated with modules over a vertex operator algebra. By restricting the module attached to the point to its bottom degree, we obtain a new formula for computing fusion rules in terms of a left -module over the Zhu algebra . As a consequence, for strongly rational VOAs, the construction of induces the fusion tensor product on the module category .

Paper Structure

This paper contains 34 sections, 29 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. Space of VOA-conformal blocks on three-pointed ${\mathbb P}^1$
  3. Preliminaries of VOAs
  4. Vertex operator algebras and modules
  5. Intertwining operators and fusion rules
  6. Contragredient modules
  7. Borcherds' Lie algebra and Zhu's algebra associated to a VOA
  8. Generalized Verma module associated to an $A(V)$-module
  9. The chiral Lie algebra ancillary to $({\mathbb P}^1,\infty,1,0)$
  10. Space of three-pointed conformal blocks on ${\mathbb P}^1$
  11. Restriction of the chiral Lie algebra at $\infty$
  12. Spanning elements of $\infty$-restricted chiral Lie algebra
  13. The augmented ideal of the $\infty$-restricted chiral Lie algebra
  14. Space of $\infty$-restricted three-pointed conformal blocks
  15. $\infty$-restricted three-pointed conformal blocks on ${\mathbb P}^1$
  16. ...and 19 more sections

Key Result

Lemma 2.7

DLM98. A VOA $V$ is rational if and only if $A(V)$ is semisimple and $(\bar{M}(-)\dashv \Omega(-)):\mathsf{Mod}(A(V))\rightleftarrows \mathsf{Adm}(V)$ is an adjoint equivalence.

Theorems & Definitions (71)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • Definition 2.8
  • Remark 2.9
  • Proposition 2.10
  • ...and 61 more