On intermediate Lie algebra $E_{7+1/2}$

TL;DR

This paper introduces new vertex operator algebras related to the intermediate Lie algebra $E_{7+1/2}$, explores their properties, and conjectures rationality conditions and a Weyl dimension formula for their representations.

Contribution

It proposes novel VOAs associated with $E_{7+1/2}$, conjectures their rationality at certain levels, and introduces a Weyl dimension formula for their irreducible representations.

Findings

Conjecture that $(E_{7+1/2})_k$ is rational for levels $k \,\leq\, 5$

Construction of VOA characters and coset models for $E_{7+1/2}$

Prediction of conformal weights at levels 3, 4, 5

Abstract

$E_{7+1/2}$ is an intermediate Lie algebra filling a hole between $E_7$ and $E_8$ in the Deligne-Cvitanovi\'c exceptional series. It was found independently by Mathur, Muhki, Sen in the classification of 2d RCFTs via modular linear differential equations (MLDE) and by Deligne, Cohen, de Man in representation theory. In this paper we propose some new vertex operator algebras (VOA) associated with $E_{7+1/2}$ and give some useful information at small levels. We conjecture that the affine VOA $(E_{7+1/2})_k$ is rational if and only if the level $k$ is at most $5$, and provide some evidence from the viewpoint of MLDE. We propose a conjectural Weyl dimension formula for infinitely many irreducible representations of $E_{7+1/2}$, which generates almost all irreducible representations of $E_{7+1/2}$ with level $k\leq 4$. More concretely, we propose the affine VOA $E_{7+1/2}$ at level 2 and the rank-two instanton VOA associated with $E_{7+1/2}$. We compute the VOA characters and provide some coset constructions. These generalize the previous works of Kawasetsu for affine VOA $E_{7+1/2}$ at level 1 and of Arakawa--Kawasetsu at level $-5$. We then predict the conformal weights of affine VOA $E_{7+1/2}$ at level $3,4,5$.