An Extremal N=2 Superconformal Field Theory

TL;DR

This work constructs an explicit extremal ${N}=2$ superconformal field theory at central charge $c=24$ by forming a ${Z}_2$ orbifold of the ${A_1^{24}}$ Niemeier lattice theory and then forming a double-cover that supports ${N}=2$ superconformal symmetry. The resulting chiral spectrum realizes the extremal elliptic genus for $m=4$, with the Ramond-sector expansion matching $Z_{EG}^{m=4,RR}=y^{-4}+46+y^{4}+O(q)$ and exhibiting an $M_{23}$ symmetry in the spectrum. The construction uses a judicious choice of a $U(1)$ current and a supercurrent built from twisted sector ground states to satisfy the ${N}=2$ OPEs, and demonstrates spectral-flow invariance consistent with a weak Jacobi form of weight $0$ and index $4$. This provides a concrete example supporting the idea of chiral gravity duals and highlights the roles of sporadic groups and error-correcting codes in quantum gravity contexts.

Abstract

We provide an example of an extremal chiral ${\cal N}=2$ superconformal field theory at $c=24$. The construction is based on a ${\mathbb Z}_2$ orbifold of the theory associated to the $A_{1}^{24}$ Niemeier lattice. The statespace is governed by representations of the sporadic group $M_{23}$.