Rationality and $C_2$-cofiniteness of certain diagonal coset vertex operator algebras

TL;DR

This paper proves the rationality and $C_2$-cofiniteness of certain diagonal coset vertex operator algebras for specific Lie algebras and admissible levels, and classifies modules in some cases.

Contribution

It establishes rationality and $C_2$-cofiniteness for diagonal coset VOAs associated with $so(2n)$ and $sl_2$ at admissible levels, and classifies modules for $sl_2$ cases.

Findings

Proves rationality and $C_2$-cofiniteness for $so(2n)$ diagonal cosets.

Proves rationality and $C_2$-cofiniteness for $sl_2$ diagonal cosets at admissible levels.

Classifies irreducible modules for $sl_2$ diagonal cosets when $k$ is a positive odd integer.

Abstract

In this paper, it is shown that the diagonal coset vertex operator algebra $C(L_{\mathfrak{g}}(k+2,0),L_{\mathfrak{g}}(k,0)\otimes L_{\mathfrak{g}}(2,0))$ is rational and $C_2$-cofinite in case $\mathfrak{g}=so(2n), n\geq 3$ and $k$ is an admissible number for $\hat{\mathfrak{g}}$. It is also shown that the diagonal coset vertex operator algebra $C(L_{sl_2}(k+4,0),L_{sl_2}(k,0)\otimes L_{sl_2}(4,0))$ is rational and $C_2$-cofinite in case $k$ is an admissible number for $\hat{sl_2}$. Furthermore, irreducible modules of $C(L_{sl_2}(k+4,0),L_{sl_2}(k,0)\otimes L_{sl_2}(4,0))$ are classified in case $k$ is a positive odd integer.