Weight one Jacobi forms and umbral moonshine

Cheng, Miranda C. N.; Duncan, John F. R.; Harvey, Jeffrey A.

doi:10.1088/1751-8121/aaa819

Cited by 16 publications

(17 citation statements)

References 45 publications

(124 reference statements)

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“…In particular, H ( ) e coincides with the Rademacher sum specified by its modular property and its pole at the cusps [18]. See [13,14,19,55] for a discussion on the cases where g = e.…”

Section: Umbral Moonshinementioning

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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“…In particular, H ( ) e coincides with the Rademacher sum specified by its modular property and its pole at the cusps [18]. See [13,14,19,55] for a discussion on the cases where g = e.…”

Section: Umbral Moonshinementioning

confidence: 99%

Generalised Umbral Moonshine

Cheng

Lange

Whalen³

2019

SIGMA

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“…Mathieu moonshine thus expanded the class of automorphic objects considered in moonshine to include mock modular forms and other weights. In a series of papers Cheng, Duncan and Harvey identified Mathieu moonshine as one of a family of correspondences between finite groups and mock modular forms [6,7,8]. They referred to this conjectured family of correspondences as umbral 2 moonshine.…”

Section: Umbral Moonshinementioning

confidence: 99%

Supersingular Elliptic Curves and Moonshine

Aricheta

2019

SIGMA

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We generalize a theorem of Ogg on supersingular j-invariants to supersingular elliptic curves with level. Ogg observed that the level one case yields a characterization of the primes dividing the order of the monster. We show that the corresponding analyses for higher levels give analogous characterizations of the primes dividing the orders of other sporadic simple groups (e.g., baby monster, Fischer's largest group). This situates Ogg's theorem in a broader setting. More generally, we characterize, in terms of supersingular elliptic curves with level, the primes arising as orders of Fricke elements in centralizer subgroups of the monster. We also present a connection between supersingular elliptic curves and umbral moonshine. Finally, we present a procedure for explicitly computing invariants of supersingular elliptic curves with level structure.

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“…§3). Although this is a simple manipulation it seems to be essential for umbral moonshine [CDH14a, CDH14b,CDH17], since in this more general setting weak Jacobi form formulations of the McKay-Thompson series are only known in some cases, whereas meromorphic Jacobi forms ψ (ℓ) g may be constructed in a uniform way (cf. §4 of [CDH14b], or §3 of this work).…”

Section: Thompson Series Hmentioning

confidence: 99%

Super vertex algebras, meromorphic Jacobi forms and umbral moonshine

Duncan

O'Desky

2018

Journal of Algebra

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The vector-valued mock modular forms of umbral moonshine may be repackaged into meromorphic Jacobi forms of weight one. In this work we constructively solve two cases of the meromorphic module problem for umbral moonshine. Specifically, for the type A Niemeier root systems with Coxeter numbers seven and thirteen, we construct corresponding bigraded super vertex operator algebras, equip them with actions of the corresponding umbral groups, and verify that the resulting trace functions on canonically twisted modules recover the meromorphic Jacobi forms that are specified by umbral moonshine. We also obtain partial solutions to the meromorphic module problem for the type A Niemeier root systems with Coxeter numbers four and five, by constructing super vertex operator algebras that recover the meromorphic Jacobi forms attached to maximal subgroups of the corresponding umbral groups.

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Weight one Jacobi forms and umbral moonshine

Cited by 16 publications

References 45 publications

Generalised Umbral Moonshine

Generalised Umbral Moonshine

Supersingular Elliptic Curves and Moonshine

Super vertex algebras, meromorphic Jacobi forms and umbral moonshine

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