The Moonshine Anomaly

Abstract

The anomaly for the Monster group \({\mathbb{M}}\) acting on its natural (aka moonshine) representation \({V^\natural}\) is a particular cohomology class \({\omega^\natural \in {\rm H}^3(\mathbb{M},{\rm U}(1))}\) that arises as a conformal field theoretic generalization of the second Chern class of a representation. This paper shows that \({\omega^\natural}\) has order exactly 24 and is not a Chern class. In order to perform this computation, this paper introduces a finite-group version of T-duality, which is used to relate \({\omega^\natural}\) to the anomaly for the Leech lattice CFT.

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  • 28 November 2019

    Unfortunately, the handling editor was wrongly published in the original online publication of this article.

  • 28 November 2019

    Unfortunately, the handling editor was wrongly published in the original online publication of this article.

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Acknowledgements

I discussed substantial portions of this work with David Treumann, who pointed out that \({\omega^\natural}\) might not be a Chern class, and with Marcel Bischoff, who told me a separate conformal-net-theoretic proof of Example 2.1.1 using the main results of [GL96,LX04]. I thank both of them for their time and attention. Leonard Soicher kindly shared with me matrices satisfying his presentation for Co1; these matrices were first computed by Richard Parker and are reproduced in Appendix A. I would also like to thank Lakshya Bhardwaj, Richard Borcherds, Kevin Costello, Davide Gaiotto, Nora Ganter, Andre Henriques, David Jordan, and Alex Weekes for helpful suggestions and discussions, and the referees for their valuable comments. 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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Johnson-Freyd, T. The Moonshine Anomaly. Commun. Math. Phys. 365, 943–970 (2019). https://doi.org/10.1007/s00220-019-03300-2

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