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Analytic Conformal Blocks of $C_2$-cofinite Vertex Operator Algebras I: Propagation and Dual Fusion Products

Bin Gui, Hao Zhang

TL;DR

This work develops a theory of conformal blocks for $C_2$-cofinite VOAs beyond rationality, with the long-term aim of a sewing-factorization theorem for blocks over holomorphic families of compact Riemann surfaces. The authors advance propagation-based methods to define and study a dual fusion product $oxbackslash_{ rak X}(bW)$, proving its existence as a grading-restricted $bV^{ ensor M}$-module and constructing a canonical conformal block, all without requiring complete reducibility. A central insight is to relate self-sewing to disjoint sewing via auxiliary objects (e.g., a 3-point sphere ${ rak Q}$) and to derive a factorization formula equating dimensions of block spaces before and after sewing. The explicit construction of the dual fusion product via truncated dual blocks $ extscr T_{ rak X,a_ullet}^*(bW)$ and propagation theory provides a robust analytic-geometry framework that connects to HLZ-type tensor category ideas and Miyamoto’s pseudo-trace structures. Together, these results lay the groundwork for a general sewing-factorization theorem for $C_2$-cofinite VOAs in Part III and offer a geometric comprehension of pseudo-trace phenomena and permutation-twisted module phenomena in higher genus settings.

Abstract

This is the first paper of a three-part series in which we develop a theory of conformal blocks for $C_2$-cofinite vertex operator algebras (VOAs) that are not necessarily rational. The ultimate goal of this series is to prove a sewing-factorization theorem (and in particular, a factorization formula) for conformal blocks over holomorphic families of compact Riemann surfaces, associated to grading-restricted (generalized) modules of $C_2$-cofinite VOAs. In this paper, we prove that if $\mathbb V$ is a $C_2$-cofinite VOA, if $\mathfrak X$ is a compact Riemann surface with $N$ incoming marked points and $M$ outgoing ones, each equipped with a local coordinate, and if $\mathbb W$ is a grading-restricted $\mathbb V^{\otimes N}$-modules, then the ``dual fusion product" exists as a grading-restricted $\mathbb V^{\otimes M}$-module. Indeed, we prove a more general version of this result without assuming $\mathbb V$ to be $C_2$-cofinite. Our main method is a generalization of the propagation of conformal blocks.

Analytic Conformal Blocks of $C_2$-cofinite Vertex Operator Algebras I: Propagation and Dual Fusion Products

TL;DR

This work develops a theory of conformal blocks for -cofinite VOAs beyond rationality, with the long-term aim of a sewing-factorization theorem for blocks over holomorphic families of compact Riemann surfaces. The authors advance propagation-based methods to define and study a dual fusion product , proving its existence as a grading-restricted -module and constructing a canonical conformal block, all without requiring complete reducibility. A central insight is to relate self-sewing to disjoint sewing via auxiliary objects (e.g., a 3-point sphere ) and to derive a factorization formula equating dimensions of block spaces before and after sewing. The explicit construction of the dual fusion product via truncated dual blocks and propagation theory provides a robust analytic-geometry framework that connects to HLZ-type tensor category ideas and Miyamoto’s pseudo-trace structures. Together, these results lay the groundwork for a general sewing-factorization theorem for -cofinite VOAs in Part III and offer a geometric comprehension of pseudo-trace phenomena and permutation-twisted module phenomena in higher genus settings.

Abstract

This is the first paper of a three-part series in which we develop a theory of conformal blocks for -cofinite vertex operator algebras (VOAs) that are not necessarily rational. The ultimate goal of this series is to prove a sewing-factorization theorem (and in particular, a factorization formula) for conformal blocks over holomorphic families of compact Riemann surfaces, associated to grading-restricted (generalized) modules of -cofinite VOAs. In this paper, we prove that if is a -cofinite VOA, if is a compact Riemann surface with incoming marked points and outgoing ones, each equipped with a local coordinate, and if is a grading-restricted -modules, then the ``dual fusion product" exists as a grading-restricted -module. Indeed, we prove a more general version of this result without assuming to be -cofinite. Our main method is a generalization of the propagation of conformal blocks.
Paper Structure (52 sections, 65 theorems, 447 equations, 4 figures)

This paper contains 52 sections, 65 theorems, 447 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Sewing-factorization in the rational world
  3. Sewing-factorization beyond rationality
  4. The sewing-factorization theorem for $C_2$-cofinite VOAs
  5. Self-sewing regained; the factorization formula
  6. Construction of the dual fusion product $\boxbackslash_\mathfrak{X}(\mathbb W)$
  7. Notations
  8. Vertex operator algebras and conformal blocks
  9. Vertex operator algebras and their modules
  10. Admissible $\mathbb V_1\times \cdots \times \mathbb V_N$-modules
  11. Change of coordinate
  12. Sheaves of VOAs for compact Riemann surfaces
  13. Conformal blocks for compact Riemann surfaces
  14. Sheaves of VOAs for families of compact Riemann surfaces
  15. Conformal blocks for families of compact Riemann surfaces
  16. ...and 37 more sections

Key Result

Theorem 2

Choose $q\in\mathcal{D}_{r\rho}^\times=\{z\in\mathbb C:0<|z|<r\rho\}$. Then we have a well-defined linear map By "well-defined" we mean that $\mathcal{S}_q\uppsi_\mathbb M(w_\bullet)$ converges absolutely at $q$ for every $w_\bullet\in\mathbb W_\bullet$, and that the linear functional $\mathcal{S}_q\uppsi_\mathbb M:\mathbb W_\bullet\rightarrow\mathbb C$ is a conformal block (i.e., is an element o

Figures (4)

  • Figure 1: Self-sewing and disjoint sewing
  • Figure 2: An example of $\mathfrak{X}\#_{q_\bullet}\mathfrak Y$ where $N=2,K=3,M=3$. $\mathfrak{X}\#_{q_\bullet}\mathfrak Y$ has genus $5$
  • Figure 3: Transforming self-sewing to disjoint sewing
  • Figure 2.1:

Theorems & Definitions (200)

  • Remark 1
  • Theorem 2: Sewing-factorization theorem, cf. Gui-sewingconvergence
  • Remark 3
  • Theorem 5: Cf. Thm. \ref{['lb55']}
  • Theorem 6: Sewing-factorization
  • Corollary 7: Sewing-factorization
  • Corollary 8: Sewing-factorization
  • Remark 9
  • Example 10
  • Example 11
  • ...and 190 more