Modular Invariance in Two-Dimensional Conformal Field Theories

Abstract

Modular invariance is the symmetry under the action of the modular group which two-dimensional conformal field theories enjoy. In this work, we will explore two disparate applications of the constraints implied by this symmetry by utilising the methods of modular bootstrap. We will first investigate the modular linear differential equation approach to the classification of rational conformal field theories. After a brief discussion of the known results, we will quickly review the original work in arXiv:1912.04298, which uses a contour-integral representation of RCFT characters to develop an algorithm to compute the modular S-matrix. We will then present a detailed review of the original work in arXiv:2002.01949 on the classification of three-character RCFT. In this work, we conjectured several infinite families of quasi-characters in order 3 with Wronskian index, l = 0, studied their modular properties and used them to explicitly construct physical three-character rational conformal field theories with higher l, in some sense mirroring the progress in the two-character case. We will then move on to the other application of modular bootstrap motivated from the AdS3/CFT2 correspondence and the search for pure gravity. After a brief review of the well-known Cardy formula, we will use a modular bootstrap equation implied by a generalisation of the modular S transformation, to derive a universal density of large spin Virasoro primaries, in the lightcone limit. On the way, we will also derive an upper bound on the twist gap for a generic CFT, as well as an upper bound on the mass of the lowest massive excitation of a theory of AdS3 quantum gravity.