Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 013, 61 pages      arXiv:2002.11125      https://doi.org/10.3842/SIGMA.2025.013

Modular Exercises for Four-Point Blocks - I

Miranda C.N. Cheng a, Terry Gannon b and Guglielmo Lockhart cd
a) Korteweg-de Vries Institute for Mathematics and Institute of Physics, University of Amsterdam, Amsterdam, The Netherlands
b) Department of Mathematics, University of Alberta, Canada
c) Institute of Physics, University of Amsterdam, The Netherlands
d) CERN, Theory Department, Geneva, Switzerland

Received May 08, 2024, in final form February 12, 2025; Published online February 28, 2025

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 $\Gamma(2)$, $\Gamma_0(2)$, 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.

Key words: conformal field theory; vertex operator algebras; modularity.

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