Abstract
AbstractMonstrous moonshinerelates distinguished modular functions to the representation theory of the Monster "Image missing". The celebrated observations that$$ 1=1,\ \ \ 196884=1+196883,\ \ \ 21493760=1+196883+21296876,\ \ \ \dots\dots (*)$$1=1,196884=1+196883,21493760=1+196883+21296876,……(*)illustrate the case ofJ(τ)=j(τ)−744, whose coefficients turn out to be sums of the dimensions of the 194 irreducible representations of "Image missing". Such formulas are dictated by the structure of the graded monstrous moonshine modules. Recent works in moonshine suggest deep relations between number theory and physics. Number theoretic Kloosterman sums have reappeared in quantum gravity, and mock modular forms have emerged as candidates for the computation of black hole degeneracies. This paper is a survey of past and present research on moonshine. We also compute the quantum dimensions of the monster orbifold and obtain exact formulas for the multiplicities of the irreducible components of the moonshine modules. These formulas imply that such multiplicities are asymptotically proportional to dimensions. For example, the proportion of 1’s in (*) tends to$$\frac{\dim(\chi_{1})}{\sum_{i=1}^{194}\dim(\chi_{i})}=\frac{1}{5844076785304502808013602136}=1.711\ldots \times 10^{-28}. $$dim(χ1)∑i=1194dim(χi)=15844076785304502808013602136=1.711…×10−28.2010 Mathematics Subject Classification:11F11; 11 F22; 11F37; 11F50; 20C34; 20C35
Highlights
This story begins innocently with peculiar numerics, and in its present form exhibits connections to conformal field theory, string theory, quantum gravity, and the arithmetic of mock modular forms
Zhu explained the modularity of the graded dimension of V in [237], by proving that this is typical for vertex operator algebras satisfying certain natural hypotheses, and Dong-Li-Mason extended Zhu’s work in [79], obtaining modular invariance results for graded trace functions arising from the action of a finite group of automorphisms
We will give a brief review of quantum gravity, since it is an important area of physical inquiry which has played a role in the development of moonshine, but we must first warn the reader: problems have been identified with the existing conjectures that relate the monster to gravity, and the current status of this connection is uncertain
Summary
This story begins innocently with peculiar numerics, and in its present form exhibits connections to conformal field theory, string theory, quantum gravity, and the arithmetic of mock modular forms. One of the most powerful such applications occurred in moonshine, when Borcherds introduced a particular example - the monster Lie algebra m - and used it to prove [18] the moonshine conjectures of Conway-Norton His method entailed using monsterequivariant versions of the denominator identity for m to verify that the coefficients of the McKay-Thompson series Tg, defined by (3.2) according to the Frenkel-LepowskyMeurman construction of V , satisfy the replication formulas conjectured by ConwayNorton in [58]. Zhu explained the modularity of the graded dimension of V in [237], by proving that this is typical for vertex operator algebras satisfying certain natural (but restrictive) hypotheses, and Dong-Li-Mason extended Zhu’s work in [79], obtaining modular invariance results for graded trace functions arising from the action of a finite group of automorphisms. We will give a brief review of quantum gravity, since it is an important area of physical inquiry which has played a role in the development of moonshine, but we must first warn the reader: problems have been identified with the existing conjectures that relate the monster to gravity, and the current status of this connection is uncertain
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
