Building an Analogue of Majorana Theory for Mathieu Moonshine

2010年，Euchi，Ooguri和Tchikawa[4]观察到K3曲面的椭圆亏格在Mathieu群M24的不可约表示的维度方面具有自然分解，Mathieu群M24是一个零星的简单群，具有丰富的结构和应用。这一有趣的观察称为Mathieu Moonlight，非常令人想起通常被称为Monstous Moonlight的类似现象，即著名的J-函数在Virasoro代数的特征方面具有扩张，其系数是Monster群表示的维度，这是McKay和Thompson最先注意到的。在怪兽月光的背景下，这种观察最终被所谓的月光模[5]的构造所解释，月光模是由怪兽群体作用的一个顶点代数，Borcherds[1]用它来证明这一猜想。一个关键的观察结果为这种向量空间的存在提供了令人信服的证据，这来自于对所谓的McKay-Thompson级数的研究。它们是在用相应的字符替换展开系数后从J函数中获得的。由计算所谓的缠绕属的郑[2]，Gaberdiel，Hohenegger&Volpato[6，7]和Eguchi&Hikami[3]的工作进一步推进了Mathieu Moonlight[3]。这些是McKay-Thompson级数的类似，涉及到在椭圆亏格中插入一个M24群元，并且具有适当的模性质.从这些信息可以推导出所有傅里叶系数在M24表示下的分解，最近Gannon[8]抽象地证明了这一猜想.在其他方面，特别是在理论物理的背景下，人们已经做出了其他的观察，但对这一现象的完整的概念理解仍然是缺失的。在怪物月光的背景下，所谓的Majorana理论，其基本概念于2009年由A.A.Ivanov[9]引入，已被证明是研究Griess代数的子代数的原始工具，Griess代数是一种真正的交换非结合代数，其自同构群为Monster群。从那时起，各种有限群的Majorana表示的构造给出了关于Monstine Moonshine的非平凡信息，该项目的目的是从Mathieu Moonshine找到类似的理论。参考文献[1]R.E.Borcherds，Monstous Moonlight和Monstous Lie超代数，发明了。数学课。109,405-444(1992).[2]M.C.N.Cheng，K3表面，N=4 Dyons，和Mathieu群M24，Commun.数论物理。4(2010)，623[arxiv：1005.5415].[3]T.Eguchi，K.Hikami，关于K3曲面扭曲椭圆亏格的注记，Phys.让我们来吧。B694 446455(2011年)[arxiv：1008.4924][4]T.Eguchi，H.Ooguri和Y.Tchikawa，关于K3表面的注释和Mathieu群M24，Exper.数学课。20，91(2011年).[5]I.Frenkel，J.Lepowsky，A.Meurman Vertex算子代数和怪物，纯数学和应用数学，第134，学术出版社(1988年).[6]M.R.Gaberdiel，S.Hohenegger，R.Volpato Mathieu为K3，JHEP 09(2010)058；[arxiv：1006.0221].[7]M.R.Gaberdiel，S.Hohenegger，R.Volpato Mathieu Moonlight在K3的椭圆亏格，JHEP 10(2010)062；[8][8]T.Gannon，《关于Mathieu的大闹数学》，第301卷，322358页，2016年。[9]A.伊万诺夫，The Monster Group and Majorana Inolutions，剑桥大学数学剑桥丛书第176卷。出版社，剑桥，2009年。

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.