Modular linear differential equations of fourth order and minimal $\mathcal{W}$-algebras

TL;DR

This paper characterizes minimal $\\mathcal{W}$-algebras linked to the Deligne exceptional series at a specific level using a family of fourth-order modular linear differential equations, classifying those with solutions of conformal field theory type.

Contribution

It introduces a classification of fourth-order modular linear differential equations with solutions of CFT type, specifically characterizing minimal $\mathcal{W}$-algebras related to the Deligne series.

Findings

Solutions of the classified differential equations are explicitly described.

Characters of Ramond-twisted modules satisfy these differential equations.

The characterization applies to minimal $\mathcal{W}$-algebras at level $-h^\vee/6$.

Abstract

A characterization of the minimal $\mathcal{W}$-algebras associated with the Deligne exceptional series at level $-h^\vee/6$ is obtained by using one-parameter family of modular linear differential equations of order $4$. In particular, the characters of the Ramond-twisted modules of minimal $\mathcal{W}$-algebras related to the Deligne exceptional series satisfy one of these differential equations. In order to obtain the characterization, the differential equations in the one parameter family which have solutions of "CFT type" are classified, whose solutions are explicitly described.