Extremal chiral $\mathcal N=4$ SCFT with $c=24$

TL;DR

The paper constructs an extremal chiral $ ext{N}=4$ SCFT with central charge $c=24$ by orbifolding the chiral boson theory on the Niemeier lattice $\,\Lambda_{A_2^{12}}$ with a $\, ext{Z}_2$ action. It identifies an $\, ext{su}(2)_4$ current algebra arising from invariant currents and, using twisted sectors, builds four dimension-$ frac{3}{2}$ supercurrents to realize the $ ext{N}=4$ superconformal algebra, and verifies that the Ramond sector reproduces the extremal $ ext{N}=4$ elliptic genus with index $m=4$. The construction breaks the full automorphism group $2.M_{12}$ down to $2 imes M_{11}$ by singling out one $A_2$ factor, and the resulting polar coefficients of the elliptic genus encode dimensions of $M_{11}$ representations. The work discusses broader implications for extremal CFTs, including potential twined (McKay–Thompson) extensions, connections to mock modular forms, and future exploration of other Niemeier lattice orbifolds.

Abstract

We construct an extremal chiral $\mathcal N=4$ superconformal field theory with central charge 24 from a $\mathbb Z_2$ orbifold of the chiral bosonic theory with target $\mathbb R^{24}/Λ$, where $Λ$ is the Niemeier lattice with root system $A_2^{12}$. This construction is analogous to constructions of extremal chiral $\mathcal N=1$ and $\mathcal N=2$ CFTs with $c=24$, where $Λ= Λ_{Leech}$ and the Niemeier lattice with root system $A_1^{24}$, respectively. The theory has a discrete symmetry group related to the sporadic group $M_{11}$.