Modular Exercises for Four-Point Blocks -- I

TL;DR

This work proves that sphere four-point chiral blocks of rational VOAs are vector-valued modular forms for groups $Γ(2)$, $Γ_0(2)$, or ${\rm SL}_2(\mathbb{Z})$, and that full four-point correlators achieved by pairing holomorphic and anti-holomorphic blocks are modular invariant. By isolating the Dehn-twist–independent data into core blocks, the authors show these core blocks form weight-zero vector-valued modular forms with modular data determined by braiding and fusing matrices, enabling systematic construction via Rademacher sums, RH theory, MLDEs, or hybrid methods. The paper then develops multiple examples across Virasoro minimal models, RCFTs at $c=1$, WZW models, and Liouville theory to illustrate how modular techniques determine conformal blocks and correlators, and it discusses a sphere–torus correspondence linking sphere quantities to torus data in related theories. This modular framework provides powerful constraints and computational tools for chiral blocks and fusion coefficients across a broad class of VOAs and CFTs, with a companion paper promised to extend the catalogue of examples and systematize sphere–torus dualities.

Abstract

The well-known modular property of the torus characters and torus partition functions of (rational) vertex operator algebras (VOAs) and 2d conformal field theories (CFTs) has been an invaluable tool for studying this class of theories. In this work we prove that sphere four-point chiral blocks of rational VOAs are vector-valued modular forms for the groups $Γ(2)$, $Γ_0(2)$, or $\mathrm{SL}_2(\mathbb{Z})$. Moreover, we prove that the four-point correlators, combining the holomorphic and anti-holomorphic chiral blocks, are modular invariant. In particular, in this language the crossing symmetries are simply modular symmetries. This gives the possibility of exploiting the available techniques and knowledge about modular forms to determine or constrain the physically interesting quantities such as chiral blocks and fusion coefficients, which we illustrate with a few examples. We also highlight the existence of a sphere-torus correspondence equating the sphere quantities of certain theories ${\mathcal T}_s$ with the torus quantities of another family of theories ${\mathcal T}_t$. A companion paper will delve into more examples and explore more systematically this sphere-torus duality.

Figures (4)

• Figure 1: The four-point chiral block.
• Figure 2: A braid $\beta\in {\rm Br}_3$ giving rise to the permutation $(13)\in {\rm Sym}(3)$.
• Figure 3: Diagram corresponding to the intertwiner $\Phi_{M_2,M_3}^{M_1}$.
• Figure 4: Schematic depiction of the torus-sphere correspondence between ${\mathcal{T}}_t$ and ${\rm Sym}^2{\mathcal{T}}_t$.

Theorems & Definitions (6)

• Theorem 5.1
• Corollary 5.2
• Theorem 5.3
• Theorem 6.1
• Theorem 7.1
• Corollary 7.2