Modular linear differential equations for four-point sphere conformal blocks

TL;DR

This paper develops modular linear differential equations (MLDEs) related to four-point sphere conformal blocks, connecting crossing symmetry, modular transformations, and Virasoro algebra, with explicit examples for various CFTs.

Contribution

It introduces a systematic construction of MLDEs for four-point conformal blocks, linking crossing symmetry with modular properties and providing explicit equations for different CFT cases.

Findings

Derived second order MLDEs for four-point blocks

Connected MLDE parameters with CFT data like central charge and operator dimensions

Presented explicit MLDE examples for BPZ and non-BPZ theories

Abstract

We construct modular linear differential equations (MLDEs) w.r.t. subgroups of the modular group whose solutions are Virasoro conformal blocks appearing in the expansion of a crossing symmetric 4-point correlator on the sphere. This uses a connection between crossing transformations and modular transformations. We focus specifically on second order MLDEs with the cases of all identical and pairwise identical operators in the correlator. The central charge, the dimensions of the above operators and those of the intermediate ones are expressed in terms of parameters that occur in such MLDEs. In doing so, the $q$-expansions of the solutions to the MLDEs are compared with those of Virasoro blocks; hence, Zamolodchikov's elliptic recursion formula provides an important input. Using the actions of respective subgroups, bootstrap equations involving the associated 3-point coefficients have been set up and solved as well in terms of the MLDE parameters. We present explicit examples of MLDEs corresponding to BPZ and novel non-BPZ equations, as well as unitary and non-unitary CFTs.