ID: math/9809110

What is moonshine?

September 19, 1998

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R. E. Borcherds
Mathematics
Quantum Algebra
Group Theory

This is an informal write up of my talk in Berlin. It gives some background to Goddard's talk (math.QA/9808136) about the moonshine conjectures.

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Elementary introduction to Moonshine

May 2, 2016

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Shamit Kachru
High Energy Physics - Theory

These notes provide an elementary (and incomplete) sketch of the objects and ideas involved in monstrous and umbral moonshine. They were the basis for a plenary lecture at the 18th International Congress on Mathematical Physics, and for a lecture series at the Centre International de Recontres Mathematiques school on "Mathematics of String Theory."

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Monstrous Moonshine over Z?

April 11, 2018

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Scott Carnahan
Representation Theory
Number Theory

Monstrous Moonshine was extended in two complementary directions during the 1980s and 1990s, giving rise to Norton's Generalized Moonshine conjecture and Ryba's Modular Moonshine conjecture. Both conjectures have been unconditionally resolved in the last few years, so we describe some speculative conjectures that may extend and unify them.

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Monstrous moonshine and the classification of CFT

June 25, 1999

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Terry Gannon
Quantum Algebra

In these notes, based on lectures given in Istanbul, we give an introduction both to Monstrous Moonshine and to the classification of rational conformal field theories, using this as an excuse to explore several related structures and go on a little tour of modern math. We will discuss Lie algebras, modular functions, the finite simple group classification, vertex operator algebras, Fermat's Last Theorem, category theory, (generalised) Kac-Moody algebras, denominator identiti...

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Snowmass White Paper: Moonshine

January 31, 2022

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Sarah M. Harrison, Jeffrey A. Harvey, Natalie M. Paquette
Representation Theory

We present a brief overview of Moonshine with an emphasis on connections to physics. Moonshine collectively refers to a set of phenomena connecting group theory, analytic number theory, and vertex operator algebras or conformal field theories. Modern incarnations of Moonshine arise in various BPS observables in string theory and, via dualities, invariants in enumerative geometry. We survey old and new developments, and highlight some of the many open questions that remain.

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John McKay, Yang-Hui He
History and Overview
Algebraic Geometry
Number Theory
Representation Theory

These notes stem from lectures given by the first author (JM) at the 2008 "Moonshine Conference in Kashiwa" and contain a number of new perspectives and observations on Monstrous Moonshine. Because many new points have not appeared anywhere in print, it is thought expedient to update, annotate and clarify them (as footnotes), an editorial task which the second author (YHH) is more than delighted to undertake. We hope the various puzzles and correspondences, delivered in a per...

A short introduction to Monstrous Moonshine

February 7, 2019

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Valdo Tatitscheff
Number Theory
Group Theory
History and Overview
Mathematical Physics

This article is a short and elementary introduction to the monstrous moonshine aiming to be as accessible as possible. I first review the classification of finite simple groups out of which the monster naturally arises, and features of the latter that are needed in order to state the moonshine conjecture of Conway and Norton. Then I motivate modular functions and modular forms from the classification of complex tori, with the definitions of the J-invariant and its q-expansion...

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Postcards from the edge, or Snapshots of the theory of generalised Moonshine

September 10, 2001

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T. Gannon
Quantum Algebra
Group Theory
Number Theory

In 1978, John McKay made an intriguing observation: 196884=196883+1. Monstrous Moonshine is the collection of questions (and a few answers) inspired by this observation. Like moonlight itself, Moonshine is an indirect phenomenon. Just as in the theory of moonlight one must introduce the sun, so in the theory of Moonshine one should go well beyond the Monster. Much as a talk discussing moonlight may include a few words on sunsets or comet tails, so will we see snapshots of the...

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TASI Lectures on Moonshine

July 2, 2018

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Vassilis Anagiannis, Miranda C. N. Cheng
Representation Theory

The word moonshine refers to unexpected relations between the two distinct mathematical structures: finite group representations and modular objects. It is believed that the key to understanding moonshine is through physical theories with special symmetries. Recent years have seen a varieties of new ways in which finite group representations and modular objects can be connected to each other, and these developments have brought promises and also puzzles into the string theory...

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Comments on my papers

July 28, 2017

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G. Lusztig
Representation Theory

This document contains a description of several of my papers, including remarks on history and connection with subsequent work. It also contains some new results and conjectures.

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Monstrous Moonshine: The first twenty-five years

February 21, 2004

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T. Gannon
Quantum Algebra
Representation Theory

Twenty-five years ago, Conway and Norton published their remarkable paper `Monstrous Moonshine', proposing a completely unexpected relationship between finite simple groups and modular functions. This paper reviews the progress made in broadening and understanding that relationship.

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