Building an Analogue of Majorana Theory for Mathieu Moonshine

In 2010 Eguchi, Ooguri and Tachikawa [4] observed that the elliptic genus of a K3 surface has a natural decomposition in terms of dimensions of irreducible representations of the Mathieu group M24, a sporadic simple group with a great wealth of structure and applications.

This intriguing observation, called Mathieu Moonshine, is very reminiscent of a similar phenomenon usually referred to as Monstrous Moonshine, namely that the famous J-function has an expansion in terms of characters of the Virasoro algebra whose coefficients are dimensions of Monster group representations, as was first noted by McKay and Thompson. In the context of Monstrous Moonshine, this observation was eventually explained by the construction of the so-called Moonshine module [5], a vertex algebra acted on by the Monster group and used by Borcherds [1] to prove the conjecture.

One key observation that provided convincing evidence for the existence of such a vector space came from considering the so-called McKay-Thompson series. These are obtained from the J-function upon replacing the expansion coefficients by their corresponding characters. The Mathieu Moonshine was pushed further - indeed, made well defined - by the work of Cheng [2], Gaberdiel, Hohenegger & Volpato [6, 7] and Eguchi & Hikami [3], who calculated the so-called twining genera. These are the analogues of the McKay-Thompson series, involving the insertion of an M24 group element into the elliptic genus and possess the appropriate modular properties.

From this information one can deduce the decomposition of all Fourier coefficients in terms of M24 representations and the conjecture was recently proved abstractly by Gannon [8]. On the way other observations have been made, especially in the context of theoretical physics, but a full conceptual understanding of the phenomenon is still missing.

In the context of Monstrous Moonshine, the so-called Majorana Theory, whose basic concepts were introduced in 2009 by A. A. Ivanov [9], has proved to be an original tool to examine the subalgebras of the Griess algebra, a real commutative non-associative algebra that has the Monster group as its auto-morphism group. Since then, the construction of Majorana representations of various finite groups has given non-trivial information about the Monstrous Moonshine and the project is to find a similar theory coming from the Mathieu Moonshine.

References
[1] R. E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109, 405-444 (1992).
[2] M.C.N. Cheng, K3 Surfaces, N=4 Dyons, and the Mathieu group M24, Commun. Number Theory Phys. 4 (2010), 623 [arXiv:1005.5415].
[3] T. Eguchi, K. Hikami, Note on twisted elliptic genus of K3 surface, Phys. Lett. B694 446455 (2011) [arXiv:1008.4924]
[4] T. Eguchi, H. Ooguri and Y. Tachikawa, Notes on the K3 Surface and the Mathieu group M24, Exper. Math. 20, 91 (2011).
[5] I. Frenkel, J. Lepowsky, A. Meurman Vertex operator algebras and the Monster, Pure and Applied Mathematics, 134, Academic Press (1988).
[6] M. R. Gaberdiel, S. Hohenegger, R. Volpato Mathieu twining characters for K3, JHEP 09 (2010) 058; [arXiv:1006.0221].
[7] M. R. Gaberdiel, S. Hohenegger, R. Volpato Mathieu Moonshine in the elliptic genus of K3, JHEP 10 (2010) 062; [arXiv:1008.3778v3].
[8] T. Gannon, Much ado about Mathieu, Adv. Math., vol. 301, pp. 322358, 2016.
[9] A. A. Ivanov, The Monster Group and Majorana Involutions, volume 176 of Cambridge Tracts in Mathematics, Cambridge Univ. Press, Cambridge, 2009.