The McKay-Thompson series of Mathieu Moonshine modulo two

Creutzig, Thomas; Hoehn, Gerald; Miezaki, Tsuyoshi

doi:10.48550/arxiv.1211.3703

Cited by 6 publications

(20 citation statements)

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“…In Section 3 we also obtain an evenness result conjectured by several authors: Theorems A and B are both used in [12] to prove a conjecture in Umbral Moonshine [7]. Indeed, their Theorem 1.2 is far stronger than Conjecture 5.11 in [7] (specialised to M 24 ), and a little weaker than Conjecture 5.12 in [7].…”

Section: Introductionmentioning

confidence: 71%

“…An immediate corollary of Theorem 4 is the validity of Conjecture 5.11 in Umbral Moonshine [7]. However it should be remarked that the proof of this in [12] actually established a much stronger statement. Oddness of McKay-Thompson coefficients is far less trivial than strict positivity of certain multiplicities, since as we see almost every multiplicity will be strictly positive.…”

Section: Qed To Lemmamentioning

confidence: 87%

“…Incidentally, it is curious that imaginary quadratic fields play a role in Umbral Moonshine (see Section 5.4 of [7]) while the equivalent theory of positive-definite quadratic forms plays a crucial role in our Section 4 proof. Perhaps this can supply a deeper explanation for Conjecture 5.12 than would arise from arguments as in [12].…”

Section: Introductionmentioning

confidence: 90%

“…The pair (3, 1A) has S = 1A∪3A∪3B and M (3,1A) = R 3 -span{(1, 1, 1), (0, 9, 3), (0, 0, 9)}, so we need to establish (44,12,28,4,4, 0, 0), (−24, 8, 24, 8, 0, 0, 0), (−208, −16, 16, 0, 0, 0, 0), (448, 64, 0, 0, 0, 0, 0), using obvious no-tation. Therefore we need to verify…”

Section: Weak K3-mathieu Moonshine I: Integralitymentioning

confidence: 99%

“…The analysis of [30] and our Theorem C hints that the 'ultimate' symmetry lies somewhere between M 24 and Conway. The split extension Z 12 2 ×M 24 , a maximal subgroup of Co 0 , is the only such group and seems worth a look. (The evenness property was used in [12] to prove one of the Umbral Moonshine conjectures, and together with Theorem A proves that the elliptic genus of Enriques surfaces decomposes into a sum of M 12 characters.…”

Section: Speculationsmentioning

confidence: 99%

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Much ado about Mathieu

Gannon

2016

Advances in Mathematics

152

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Eguchi, Ooguri and Tachikawa observed that the coefficients of the elliptic genus of type II string theory on K3 surfaces appear to be dimensions of representations of the largest Mathieu group. Subsequent work by several people established a candidate for the elliptic genus twisted by each element of M 24 . In this paper we prove that the resulting sequence of class functions are true characters of M 24 , proving the Eguchi-Ooguri-Tachikawa 'Mathieu Moonshine' conjecture. 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. We also prove the evenness property of the multiplicities, as conjectured by several authors, and use that to investigate the proposal of Gaberdiel-Hohenegger-Volpato that Mathieu Moonshine lifts to the Conway groups Co 0 and Co 1 . We identify the role group cohomology plays in both Mathieu Moonshine and Monstrous Moonshine; in particular this gives a cohomological interpretation for the non-Fricke elements in Norton's Generalised Monstrous Moonshine conjecture, and gives a lower bound for H 3 (B Co1 ; C × ).

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Section: Introductionmentioning

confidence: 71%

Section: Qed To Lemmamentioning

confidence: 87%

Section: Introductionmentioning

confidence: 90%

Section: Weak K3-mathieu Moonshine I: Integralitymentioning

confidence: 99%

Section: Speculationsmentioning

confidence: 99%

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Much ado about Mathieu

Gannon

2016

Advances in Mathematics

152

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Landau-Ginzburg orbifolds and symmetries of K3 CFTs

Cheng

Ferrari

Harrison

et al. 2017

J. High Energ. Phys.

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Recent developments in the study of the moonshine phenomenon, including umbral and Conway moonshine, suggest that it may play an important role in encoding the action of finite symmetry groups on the BPS spectrum of K3 string theory. To test and clarify these proposed K3-moonshine connections, we study Landau-Ginzburg orbifolds that flow to conformal field theories in the moduli space of K3 sigma models. We compute K3 elliptic genera twined by discrete symmetries that are manifest in the UV description, though often inaccessible in the IR. We obtain various twining functions coinciding with moonshine predictions that have not been observed in physical theories before. These include twining functions arising from Mathieu moonshine, other cases of umbral moonshine, and Conway moonshine. For instance, all functions arising from M 11 ⊂ 2.M 12 moonshine appear as explicit twining genera in the LG models, which moreover admit a uniform description in terms of its natural 12-dimensional representation. Our results provide strong evidence for the relevance of umbral moonshine for K3 symmetries, as well as new hints for its eventual explanation.

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Umbral moonshine and the Niemeier lattices

2014

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In this paper we relate umbral moonshine to the Niemeier lattices: the 23 even unimodular positive-definite lattices of rank 24 with non-trivial root systems. To each Niemeier lattice we attach a finite group by considering a naturally defined quotient of the lattice automorphism group, and for each conjugacy class of each of these groups we identify a vector-valued mock modular form whose components coincide with mock theta functions of Ramanujan in many cases. This leads to the umbral moonshine conjecture, stating that an infinite-dimensional module is assigned to each of the Niemeier lattices in such a way that the associated graded trace functions are mock modular forms of a distinguished nature. These constructions and conjectures extend those of our earlier paper, and in particular include the Mathieu moonshine observed by Eguchi-Ooguri-Tachikawa as a special case. Our analysis also highlights a correspondence between genus zero groups and Niemeier lattices. As a part of this relation we recognise the Coxeter numbers of Niemeier root systems with a type A component as exactly those levels for which the corresponding classical modular curve has genus zero.

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The McKay-Thompson series of Mathieu Moonshine modulo two

Cited by 6 publications

References 0 publications

Much ado about Mathieu

Much ado about Mathieu

Landau-Ginzburg orbifolds and symmetries of K3 CFTs

Umbral moonshine and the Niemeier lattices

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