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## Generalised Umbral Moonshine

## Abstract

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 study we introduce the notion of generalized umbral moonshine, which includes the generalized Mathieu moonshine 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 generalized umbral moonshine. We also discuss the possible origin of the generalized umbral moonshine.

- Authors:

- Korteweg-de Vries Institute for Mathematics, Amsterdam (The Netherlands); Univ. of Amsterdam, Amsterdam (The Netherlands)
- Univ. of Kentucky, Lexington, KY (United States)
- Stanford Univ., Stanford, CA (United States)

- Publication Date:

- Research Org.:
- SLAC National Accelerator Lab., Menlo Park, CA (United States)

- Sponsoring Org.:
- USDOE

- Contributing Org.:
- Korteweg-de Vries Institute for Mathematics, The Netherlands; University of Kentucky, USA; Stanford University, USA

- OSTI Identifier:
- 1507154

- Grant/Contract Number:
- AC02-76SF00515

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Symmetry, Integrability and Geometry: Methods and Applications

- Additional Journal Information:
- Journal Name: Symmetry, Integrability and Geometry: Methods and Applications; Journal ID: ISSN 1815-0659

- Publisher:
- Institute of Mathematics, National Academy of Sciences Ukraine

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; moonshine; mock modular form; finite group representations; group cohomology

### Citation Formats

```
Cheng, Miranda C. N., de Lange, Paul, and Whalen, Daniel P. Z.. Generalised Umbral Moonshine. United States: N. p., 2019.
Web. doi:10.3842/sigma.2019.014.
```

```
Cheng, Miranda C. N., de Lange, Paul, & Whalen, Daniel P. Z.. Generalised Umbral Moonshine. United States. doi:10.3842/sigma.2019.014.
```

```
Cheng, Miranda C. N., de Lange, Paul, and Whalen, Daniel P. Z.. Sat .
"Generalised Umbral Moonshine". United States. doi:10.3842/sigma.2019.014.
```

```
@article{osti_1507154,
```

title = {Generalised Umbral Moonshine},

author = {Cheng, Miranda C. N. and de Lange, Paul and Whalen, Daniel P. Z.},

abstractNote = {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 study we introduce the notion of generalized umbral moonshine, which includes the generalized Mathieu moonshine 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 H3(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 generalized umbral moonshine. We also discuss the possible origin of the generalized umbral moonshine.},

doi = {10.3842/sigma.2019.014},

journal = {Symmetry, Integrability and Geometry: Methods and Applications},

number = ,

volume = ,

place = {United States},

year = {2019},

month = {3}

}