Modular Exercises for Four-Point Blocks - I

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), Γ₀(2), or SL₂(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. 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 𝒯ₛ with the torus quantities of another family of theories 𝒯ₜ. A companion paper will delve into more examples and explore this sphere-torus duality.