Moonshine, Superconformal Symmetry, and Quantum Error Correction

Abstract

Special conformal field theories can have symmetry groups which are interesting sporadic finite simple groups. Famous examples include the Monster symmetry group of a $c=24$ two-dimensional conformal field theory (CFT) constructed by Frenkel, Lepowsky and Meurman, and the Conway symmetry group of a $c=12$ CFT explored in detail by Duncan and Mack-Crane. The Mathieu moonshine connection between the K3 elliptic genus and the Mathieu group $M_{24}$ has led to the study of K3 sigma models with large symmetry groups. A particular K3 CFT with a maximal symmetry group preserving $(4,4)$ superconformal symmetry was studied in beautiful work by Gaberdiel, Taormina, Volpato, and Wendland. The present paper shows that in both the GTVW and $c=12$ theories the construction of superconformal generators can be understood via the theory of quantum error correcting codes. The automorphism groups of these codes lift to symmetry groups in the CFT preserving the superconformal generators. In the case of the $N=1$ supercurrent of the GTVW model our result, combined with a result of T. Johnson-Freyd implies the symmetry group is the maximal subgroup of $M_{24}$ known as the sextet group. (The sextet group is also known as the holomorph of the hexacode.) Building on \cite{gtvw} the Ramond-Ramond sector of the GTVW model is related to the Miracle Octad Generator which in turn leads to a role for the Golay code as a group of symmetries of RR states. Moreover, $(4,1)$ superconformal symmetry suffices to define and decompose the elliptic genus of a K3 sigma model into characters of the $N=4$ superconformal algebra. The symmetry group preserving $(4,1)$ is larger than that preserving $(4,4)$.

only 53 pages

References in corpus (4)

• Flux Compactification
• Notes on the K3 Surface and the Mathieu group M_24
• Note on Twisted Elliptic Genus of K3 Surface
• Mathieu Moonshine in the elliptic genus of K3

Cited by in corpus (10)

• Narain CFTs and error-correcting codes on finite fields
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• Quantum stabilizer codes, lattices, and CFTs
• Boundary RG Flows for Fermions and the Mod 2 Anomaly
• Hecke Relations, Cosets and the Classification of 2d RCFTs
• Global Symmetries, Code Ensembles, and Sums Over Geometries
• Vertex operator superalgebra/sigma model correspondences: The four-torus case
• Non-invertible defects from the Conway SCFT to K3 sigma models I: general results
• Three Dimensional Pure Gravity and Generalized Hecke Operators
• Beauty And The Beast Part 2: Apprehending The Missing Supercurrent