$C_2$-Cofiniteness and Rationality of the Icosahedral Orbifold $V_{L_2}^{A_5}$

Abstract

Let $L_2=\mathbb{Z}\alpha$ be the rank-one root lattice with $(\alpha ,\alpha )=2$, and let $A_5$ act on the lattice vertex operator algebra $V_{L_2}$ through an icosahedral subgroup of $\mathrm{Aut}(V_{L_2})\cong PS{L}_2(\mathbb{C})$. We prove that the fixed-point vertex operator algebra $V_{L_2}^{A_5}$ is strongly rational.

Substantially revised version. We have added many details to the proofs and exposition, corrected several inaccuracies in the previous version, and substantially improved the organization, clarity, and readability of the manuscript