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Holomorphic SCFTs with small index

Part of: Lie algebras and Lie superalgebras Homology and cohomology theories Theory of error-correcting codes and error-detecting codes General mathematical topics and methods in quantum theory

Published online by Cambridge University Press:  18 January 2021

Davide Gaiotto  and
Theo Johnson-Freyd
Davide Gaiotto*
Affiliation:
Perimeter Institute for Theoretical Physics, Waterloo, Ontario
Theo Johnson-Freyd
Affiliation:
Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia e-mail: theojf@dal.ca
*
e-mail: dgaiotto@perimeterinstitute.ca
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Abstract

We observe that every self-dual ternary code determines a holomorphic $\mathcal N=1$ superconformal field theory. This provides ternary constructions of some well-known holomorphic $\mathcal N=1$ superconformal field theories (SCFTs), including Duncan’s “supermoonshine” model and the fermionic “beauty and the beast” model of Dixon, Ginsparg, and Harvey. Along the way, we clarify some issues related to orbifolds of fermionic holomorphic CFTs. We give a simple coding-theoretic description of the supersymmetric index and conjecture that for every self-dual ternary code this index is divisible by $24$ ; we are able to prove this conjecture except in the case when the code has length $12$ mod $24$ . Lastly, we discuss a conjecture of Stolz and Teichner relating $\mathcal N=1$ SCFTs with Topological Modular Forms. This conjecture implies constraints on the supersymmetric indexes of arbitrary holomorphic SCFTs, and suggests (but does not require) that there should be, for each k, a holomorphic $\mathcal N=1$ SCFT of central charge $12k$ and index $24/\gcd (k,24)$ . We give ternary code constructions of SCFTs realizing this suggestion for $k\leq 5$ .

Keywords

Supersymmetryvertex operator algebraslinear codesmoonshineTMF

MSC classification

Primary: 17B69: Vertex operators; vertex operator algebras and related structures
Secondary: 94B05: Linear codes, general 81Q60: Supersymmetry and quantum mechanics 55N20: Generalized (extraordinary) homology and cohomology theories
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 2 , April 2022 , pp. 573 - 601
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

We thank Noam D. Elkies for many valuable conversations, some of which were hosted by mathoverflow.net, and for helping with a number of calculations. We also thank Greg Moore, Jeff Harvey, and an anonymous referee for comments on a draft of this paper. Research at the Perimeter Institute for Theoretical Physics is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.

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