Book Chapter10.1007/978-3-030-42400-8_1
A Short Introduction to the Algebra, Geometry, Number Theory and Physics of Moonshine
John F. R. Duncan
- 01 Jan 2020
- pp 1-85
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TL;DR: Moonshine as mentioned in this paper is a collection of coincidences connecting modular functions to the monster simple group, which was newly discovered at that time, and the effort to elucidate these connections led to new algebraic structures, and fruitful cross-fertilization between mathematics and physics.
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Abstract: Moonshine arose in the 1970s as a collection of coincidences connecting modular functions to the monster simple group, which was newly discovered at that time. The effort to elucidate these connections led to new algebraic structures, and fruitful cross-fertilization between mathematics and physics. In this century the field has been further enriched, with the discovery of new roles for finite groups in geometry, and new relations to number theory. We offer an introduction and invitation to this theory in these notes.
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Citations
Introduction to the Theory of Elliptic Hypergeometric Integrals
TL;DR: In this article, the key properties of elliptic hypergeometric integrals are described and a connection to the star-triangle relation and Coxeter relations for a permutation group is made.
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•Posted Content
An Overview of Penumbral Moonshine
TL;DR: In this article, the penumbral moonshine phenomenon was introduced and explained, which is a special case of a family of similar relationships between finite groups and vector-valued modular forms of a certain kind.
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Class numbers, cyclic simple groups, and arithmetic
TL;DR: In this article , a notion of optimal module for a finite group in the setting of holomorphic mock Jacobi forms was introduced, and the cyclic groups of prime order were classified in the special case of weight 2 and index 1.
An Overview of Penumbral Moonshine
John F. R. Duncan,Jeffrey A. Harvey,Brandon C. Rayhaun +2 more
TL;DR: This paper introduces "penumbral moonshine", a family of relationships between finite groups and vector-valued modular forms, generalizing Mathieu moonshine and Thompson moonshine, and explores its features, building on the concept of umbral moonshine.
References
•Posted Content
Module constructions for certain subgroups of the largest Mathieu group
TL;DR: In this article, the authors give vertex operator algebraic module constructions whose associated trace functions are meromorphic Jacobi forms, which are canonically associated to the mock modular forms of Mathieu moonshine.
Mathieu twining characters for K3
TL;DR: In this paper, the analogue of the McKay-Thompson series for the proposed Mathieu group action on the elliptic genus of K3 is analyzed. And the corresponding NS-sector twining char-acters have good modular properties and satisfy remarkable replication identities.
Snapshots of Conformal Field Theory
TL;DR: In this paper, the authors introduce conformal field theory, with a focus on those perspectives that are relevant for interpreting superconformal field theories by Calabi-Yau geometry.
Mathieu Moonshine and Orbifold K3s
Matthias R. Gaberdiel,Roberto Volpato +1 more
- 01 Jan 2014
TL;DR: In this article, it was shown that all cyclic torus orbifolds are exceptional in this sense, and that almost all of the exceptional cases are realized as cyclic Torus orbs.
Much ado about Mathieu
TL;DR: In this article, the integrality of multiplicities is proved using a small generalisation of Sturm's Theorem, while positivity involves a modification of a method of Hooley, for finding an effective bound on a family of Selberg-Kloosterman zeta functions at s = 3 / 4.