A proof of the Thompson moonshine conjecture

Griffin, Michael; Mertens, Michael H.

doi:10.1186/s40687-016-0084-7

Cited by 18 publications

(22 citation statements)

References 40 publications

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“…More explicitly, we proposed in Section 3 six conditions of generalised umbral moonshine conjecture and provided evidence in the form of proposing all the twistedtwined functions for all cases of generalised umbral moonshine. A proof of this conjecture should be attaianable following the methods of [32,45].…”

Section: Summary and Discussionmentioning

confidence: 99%

Generalised Umbral Moonshine

Cheng

Lange

Whalen³

2019

SIGMA

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Umbral moonshine describes an unexpected relation between 23 finite groups arising from lattice symmetries and special mock modular forms. It includes the Mathieu moonshine as a special case and can itself be viewed as an example of the more general moonshine phenomenon which connects finite groups and distinguished modular objects. In this paper we introduce the notion of generalised umbral moonshine, which includes the generalised Mathieu moonshine [Gaberdiel M.R., Persson D., Ronellenfitsch H., Volpato R., Commun. Number Theory Phys. 7 (2013), 145-223] as a special case, and provide supporting data for it. A central role is played by the deformed Drinfel'd (or quantum) double of each umbral finite group G, specified by a cohomology class in H 3 (G, U (1)). We conjecture that in each of the 23 cases there exists a rule to assign an infinite-dimensional module for the deformed Drinfel'd double of the umbral finite group underlying the mock modular forms of umbral moonshine and generalised umbral moonshine. We also discuss the possible origin of the generalised umbral moonshine.

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Section: Summary and Discussionmentioning

confidence: 99%

Generalised Umbral Moonshine

Cheng

Lange

Whalen³

2019

SIGMA

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“…Duncan, Griffin, and the second author gave a survey [7] of the advancements in the theory of moonshine and its applications to physics as of 2015, including their proof [8] of the Umbral Moonshine Conjecture of Cheng, Duncan and Harvey [2]. Along these lines, there has been a flurry of recent work (for example, see [9,13]).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Higher width moonshine

Dawsey

Ono

2020

Advances in Mathematics

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Weak moonshine for a finite group G is the phenomenon where an infinite dimensional graded G-modulehas the property that its trace functions, known as McKay-Thompson series, are modular functions. Recent work by Dehority, Gonzalez, Vafa, and Van Peski established that weak moonshine holds for every finite group. Since weak moonshine only relies on character tables, which are not isomorphism class invariants, non-isomorphic groups can have the same McKay-Thompson series. We address this problem by extending weak moonshine to arbitrary width s ∈ Z + . For each 1 ≤ r ≤ s and each irreducible character χ i , we employ Frobenius' r-character extension χ (r) i : G (r) → C to define McKay-Thompson series of V (r)G := V G × · · · × V G (r copies) for each r-tuple in G (r) := G × · · · × G (r copies). These series are modular functions. We find that complete width 3 weak moonshine always determines a group up to isomorphism. Furthermore, we establish orthogonality relations for the Frobenius r-characters, which dictate the compatibility of the extension of weak moonshine for V G to width s weak moonshine.

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“…Schematically, they can be most naturally viewed as a sum of two contributions F 3,g = Projected Rademacher Sum + θ -corrections , (5.25) where the first term is the Rademacher sum of the unique polar term 2q −3 for the suitable group Γ g and multiplier system, projected onto the Kohnen plus space, and the second term again involves linear combinations of θ (d 2 τ), where d are certain integers compatible with the group Γ g . We refer to [99], and in particular [104] for more details.…”

Section: Thompson Moonshinementioning

confidence: 99%

TASI Lectures on Moonshine

Cheng¹,

Anagiannis²

2018

Proceedings of Theoretical Advanced Study Institute Summer School 2017 "Physics at the Fundamental Frontier" — PoS(TASI2017)

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The word moonshine refers to unexpected relations between 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 variety of new ways in which finite group representations and modular objects can be connected to each other, and these developments have brought promises but also puzzles into the string theory community. These lecture notes aim to bring graduate students in theoretical physics and mathematical physics to the forefront of this active research area. In Part II of this note, we review the various cases of moonshine connections, ranging from the classical monstrous moonshine established in the last century to the most recent findings. In Part III, we discuss the relation between the moonshine connections and physics, especially string theory. After briefly reviewing a recent physical realisation of monstrous moonshine, we will describe in some details the mystery of the physical aspects of umbral moonshine, and also mention some other setups where string theory black holes can be connected to moonshine. To make the exposition self-contained, we also provide the relevant background knowledge in Part I, including sections on finite groups, modular objects, and two-dimensional conformal field theories. This part occupies half of the pages of this set of notes and can be skipped by readers who are already familiar with the relevant concepts and techniques.

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A proof of the Thompson moonshine conjecture

Abstract: In this paper, we prove the existence of an infinite-dimensional graded supermodule for the finite sporadic Thompson group Th whose McKay-Thompson series are weakly holomorphic modular forms of weight 1 2 satisfying properties conjectured by Harvey and Rayhaun.

Cited by 18 publications

References 40 publications

Generalised Umbral Moonshine

Generalised Umbral Moonshine

Higher width moonshine

TASI Lectures on Moonshine

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