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 twodimensional 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 asmore »
- Authors:
-
- Univ. of Chicago, IL (United States). Enrico Fermi Inst.
- Rutgers Univ., Piscataway, NJ (United States). Dept. of Physics and Astronomy. NHETC
- Publication Date:
- Research Org.:
- Rutgers Univ., Piscataway, NJ (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC)
- OSTI Identifier:
- 1802014
- Grant/Contract Number:
- SC0010008
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of High Energy Physics (Online)
- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2020; Journal Issue: 5; Journal ID: ISSN 1029-8479
- Publisher:
- Springer Berlin
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Physics
Citation Formats
Harvey, Jeffrey A., and Moore, Gregory W. Moonshine, superconformal symmetry, and quantum error correction. United States: N. p., 2020.
Web. doi:10.1007/jhep05(2020)146.
Harvey, Jeffrey A., & Moore, Gregory W. Moonshine, superconformal symmetry, and quantum error correction. United States. https://doi.org/10.1007/jhep05(2020)146
Harvey, Jeffrey A., and Moore, Gregory W. Wed .
"Moonshine, superconformal symmetry, and quantum error correction". United States. https://doi.org/10.1007/jhep05(2020)146. https://www.osti.gov/servlets/purl/1802014.
@article{osti_1802014,
title = {Moonshine, superconformal symmetry, and quantum error correction},
author = {Harvey, Jeffrey A. and Moore, Gregory W.},
abstractNote = {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 twodimensional 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).},
doi = {10.1007/jhep05(2020)146},
journal = {Journal of High Energy Physics (Online)},
number = 5,
volume = 2020,
place = {United States},
year = {Wed May 27 00:00:00 EDT 2020},
month = {Wed May 27 00:00:00 EDT 2020}
}
Works referenced in this record:
Fermionic SPT phases in higher dimensions and bosonization
journal, October 2017
- Kapustin, Anton; Thorngren, Ryan
- Journal of High Energy Physics, Vol. 2017, Issue 10
Supersymmetry, supergravity theories and the dual spinor model
journal, April 1977
- Gliozzi, F.; Scherk, J.; Olive, D.
- Nuclear Physics B, Vol. 122, Issue 2
Noncommutative field theory
journal, November 2001
- Douglas, Michael; Nekrasov, Nikita
- Reviews of Modern Physics, Vol. 73, Issue 4
Algebras, BPS states, and strings
journal, March 1996
- Harvey, Jeffrey A.; Moore, Gregory
- Nuclear Physics B, Vol. 463, Issue 2-3
A Group of Order 8,315,553,613,086,720,000
journal, March 1969
- Conway, J. H.
- Bulletin of the London Mathematical Society, Vol. 1, Issue 1
Anomaly cancellations in supersymmetric D = 10 gauge theory and superstring theory
journal, December 1984
- Green, Michael B.; Schwarz, John H.
- Physics Letters B, Vol. 149, Issue 1-3
Factorizable dual model of pions
journal, August 1971
- Neveu, A.; Schwarz, J. H.
- Nuclear Physics B, Vol. 31, Issue 1
Quantum Computation and Quantum Information
book, January 2011
- Nielsen, Michael A.; Chuang, Isaac L.
- Cambridge University Press
Binary Codes and Vertex Operator (Super)Algebras
journal, April 1996
- Miyamoto, Masahiko
- Journal of Algebra, Vol. 181, Issue 1
Five-brane effective action in M-theory
journal, May 1997
- Witten, Edward
- Journal of Geometry and Physics, Vol. 22, Issue 2
The overarching finite symmetry group of Kummer surfaces in the Mathieu group M 24
journal, August 2013
- Taormina, A.; Wendland, K.
- Journal of High Energy Physics, Vol. 2013, Issue 8
A Hiker's Guide to K3.¶Aspects of N= (4,4) Superconformal Field Theory¶with Central Charge c= 6
journal, January 2001
- Nahm, Werner; Wendland, Katrin
- Communications in Mathematical Physics, Vol. 216, Issue 1
Heterotic string theory (I). The free heterotic string
journal, January 1985
- Gross, David J.; Harvey, Jeffrey A.; Martinec, Emil
- Nuclear Physics B, Vol. 256
The Generalised Complex Geometry of (p, q) Hermitian Geometries
journal, June 2019
- Hull, Christopher; Lindström, Ulf
- Communications in Mathematical Physics, Vol. 375, Issue 1
Mathieu Moonshine in the elliptic genus of K3
journal, October 2010
- Gaberdiel, Matthias R.; Hohenegger, Stefan; Volpato, Roberto
- Journal of High Energy Physics, Vol. 2010, Issue 10
An uplifting discussion of T-duality
journal, May 2018
- Harvey, Jeffrey A.; Moore, Gregory W.
- Journal of High Energy Physics, Vol. 2018, Issue 5
The 290 fixed-point sublattices of the Leech lattice
journal, February 2016
- Höhn, Gerald; Mason, Geoffrey
- Journal of Algebra, Vol. 448
Vacuum configurations for superstrings
journal, January 1985
- Candelas, P.; Horowitz, Gary T.; Strominger, Andrew
- Nuclear Physics B, Vol. 258
On the Algebras of BPS States
journal, October 1998
- Harvey, Jeffrey A.; Moore, Gregory
- Communications in Mathematical Physics, Vol. 197, Issue 3
Flux compactification
journal, May 2007
- Douglas, Michael R.; Kachru, Shamit
- Reviews of Modern Physics, Vol. 79, Issue 2
Bosonization on higher genus Riemann surfaces
journal, September 1987
- Alvarez-Gaumé, Luis; Bost, Jean-Benoît; Moore, Gregory
- Communications in Mathematical Physics, Vol. 112, Issue 3
Umbral moonshine
journal, January 2014
- Cheng, Miranda C. N.; Duncan, John F. R.; Harvey, Jeffrey A.
- Communications in Number Theory and Physics, Vol. 8, Issue 2
superconformal field theory
journal, August 1988
- Dixon, L.; Ginsparg, P.; Harvey, J.
- Nuclear Physics B, Vol. 306, Issue 3
K3 surfaces, $\mathcal{N}=4$ dyons and the Mathieu group $M_{24}$
journal, January 2010
- Cheng, Miranda C. N.
- Communications in Number Theory and Physics, Vol. 4, Issue 4
Heterotic string theory
journal, March 1986
- Gross, David J.; Harvey, Jeffrey A.; Martinec, Emil
- Nuclear Physics B, Vol. 267, Issue 1
Exact results for N = 2 compactifications of heterotic strings
journal, September 1995
- Kachru, Shamit; Vafa, Cumrun
- Nuclear Physics B, Vol. 450, Issue 1-2
Note on twisted elliptic genus of K3 surface
journal, January 2011
- Eguchi, Tohru; Hikami, Kazuhiro
- Physics Letters B, Vol. 694, Issue 4-5
A K3 sigma model with $ \mathbb{Z}_2^8 $ : $ {{\mathbb{M}}_{20 }} $ symmetry
journal, February 2014
- Gaberdiel, Matthias R.; Taormina, Anne; Volpato, Roberto
- Journal of High Energy Physics, Vol. 2014, Issue 2
Second-quantized mirror symmetry
journal, November 1995
- Ferrara, Sergio; Harvey, Jeffrey A.; Strominger, Andrew
- Physics Letters B, Vol. 361, Issue 1-4
Conformal field theory, triality and the monster group
journal, February 1990
- Dolan, L.; Goddard, P.; Montague, P.
- Physics Letters B, Vol. 236, Issue 2
The Quantum Hamming and Hexacodes
journal, June 1998
- Beth, Th.; Grassl, M.
- Fortschritte der Physik, Vol. 46, Issue 4-5
Framed Vertex Operator Algebras, Codes and the Moonshine Module
journal, April 1998
- Dong, Chongying; Griess Jr., Robert L.; H�hn, Gerald
- Communications in Mathematical Physics, Vol. 193, Issue 2
Beauty and the beast: Superconformal symmetry in a monster module
journal, June 1988
- Dixon, L.; Ginsparg, P.; Harvey, J.
- Communications in Mathematical Physics, Vol. 119, Issue 2
Spin TQFTs and fermionic phases of matter
journal, October 2016
- Gaiotto, Davide; Kapustin, Anton
- International Journal of Modern Physics A, Vol. 31, Issue 28n29
A new combinatorial approach to M 24
journal, January 1976
- Curtis, R. T.
- Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 79, Issue 1
Heterotic String
journal, February 1985
- Gross, David J.; Harvey, Jeffrey A.; Martinec, Emil
- Physical Review Letters, Vol. 54, Issue 6
Introduction to M(atrix) theory and noncommutative geometry
journal, April 2002
- Konechny, Anatoly; Schwarz, Albert
- Physics Reports, Vol. 360, Issue 5-6
Bosonization in arbitrary genus
journal, September 1986
- Alvarez-Gaumé, Luis; Moore, Gregory; Nelson, Philip
- Physics Letters B, Vol. 178, Issue 1
Notes on the K3 Surface and the Mathieu Group M 24
journal, March 2011
- Eguchi, Tohru; Ooguri, Hirosi; Tachikawa, Yuji
- Experimental Mathematics, Vol. 20, Issue 1
Sphere Packings, Lattices and Groups
book, January 1999
- Conway, J. H.; Sloane, N. J. A.
- Grundlehren der mathematischen Wissenschaften
On the unitary representations of N=2 and N=4 superconformal algebras
journal, August 1988
- Eguchi, Tohru; Taormina, Anne
- Physics Letters B, Vol. 210, Issue 1-2
All (4,1): Sigma models with (4, q) off-shell supersymmetry
journal, March 2017
- Hull, Chris; Lindström, Ulf
- Journal of High Energy Physics, Vol. 2017, Issue 3
Superconformal algebras and string compactification on manifolds with SU(n) holonomy
journal, March 1989
- Eguchi, Tohru; Ooguri, Hirosi; Taormina, Anne
- Nuclear Physics B, Vol. 315, Issue 1
A natural representation of the Fischer-Griess Monster with the modular function J as character
journal, May 1984
- Frenkel, I. B.; Lepowsky, J.; Meurman, A.
- Proceedings of the National Academy of Sciences, Vol. 81, Issue 10
Monstrous Moonshine
journal, October 1979
- Conway, J. H.; Norton, S. P.
- Bulletin of the London Mathematical Society, Vol. 11, Issue 3
Dirichlet Branes and Ramond-Ramond Charges
journal, December 1995
- Polchinski, Joseph
- Physical Review Letters, Vol. 75, Issue 26
Ultra-violet behaviour of two-dimensional supersymmetric non-linear σ-models
journal, January 1987
- Howe, P. S.; Papadopoulos, G.
- Nuclear Physics B, Vol. 289
A web of 2d dualities: ${\bf Z}_2$ gauge fields and Arf invariants
journal, January 2019
- Karch, Andreas; Tong, David; Turner, Carl
- SciPost Physics, Vol. 7, Issue 1
The quaternionic lattice for 2G2(4) and its maximal subgroups
journal, August 1982
- Wilson, Robert A.
- Journal of Algebra, Vol. 77, Issue 2
Monstrous moonshine and monstrous Lie superalgebras
journal, December 1992
- Borcherds, Richard E.
- Inventiones Mathematicae, Vol. 109, Issue 1