Moonshine, superconformal symmetry, and quantum error correction
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Received: April 2, 2020 Accepted: April 29, 2020 Published: May 27, 2020
Moonshine, superconformal symmetry, and quantum error correction
a
Enrico Fermi Institute and Department of Physics, University of Chicago, 5620 Ellis Ave., Chicago IL 60637 b NHETC and Department of Physics and Astronomy, Rutgers University, 126 Frelinghuysen Rd., Piscataway NJ 08855, U.S.A.
E-mail: [email protected], [email protected] 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 M24 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 [41]. 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 M24 known as the sextet group. (The sextet group is also known as the holomorph of the hexacode.) Building on [41] 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). Keywords: Conformal Field Models in String Theory, Conformal Field Theory, Discrete Symmetries ArXiv ePrint: 2003.13700
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)146
JHEP05(2020)146
Jeffrey A. Harveya and Gregory W. Mooreb
Contents 1 Introduction
1 3 4 5 9 11
3 Supercurrents, states, and codes 3.1 The GTVW model 3.2 (4, 4) superconformal algebra in the GTVW model 3.3 The hexacode representation of the N = 1 supercurrent 3.4 Why the image of P defines a superconformal current 3.5 The relation of the N = 4 supercurrents to codes
13 13 14 16 18 20
4 Relation to Mathieu moonshine 4.1 Statement of Mathieu moonshine 4.2 The Mukai theorem 4.3 Quantum Mukai theorem 4.4 Evading a no-go theorem
21 21 22 22 23
5 Symmetries of supercurrents 5.1 The stabilizer group within Q6 5.2 Further symmetries of Ψ: lifting the hexacode automorphisms 5.3 The stabilizer group of Im(P ) within SU
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