# 3D string theory and Umbral moonshine

## Abstract

Here, the simplest string theory compactifications to 3D with 16 supercharges—the heterotic string on T 7, and type II strings on $$K3 \times T^3$$ —are related by U-duality, and share a moduli space of vacua parametrized by $$O(8, 24;{{\mathbb Z}}) ~\backslash ~O(8, 24)~ /~ (O(8) \times O(24))$$ . One can think of this as the moduli space of even, self-dual 32-dimensional lattices with signature (8,24). At 24 special points in moduli space, the lattice splits as $$\Gamma^{8, 0} \oplus \Gamma^{0, 24}$$ . $$\Gamma^{0, 24}$$ can be the Leech lattice or any of 23 Niemeier lattices, while $$\Gamma^{8, 0}$$ is the E _{8} root lattice. We show that starting from this observation, one can find a precise connection between the Umbral groups and type IIA string theory on K3. This may provide a natural physical starting point for understanding Mathieu and Umbral moonshine. The maximal unbroken subgroups of Umbral groups in 6D (or any other limit) are those obtained by starting at the associated Niemeier point and moving in moduli space while preserving the largest possible subgroup of the Umbral group. To illustrate the action of these symmetries on BPS states, we discuss the computation of certain protected four-derivative terms in the effective field theory, and recover facts about the spectrum and symmetry representations of 1/2-BPS states.

- Authors:

- Stanford Univ., Palo Alto, CA (United States)
- Stanford Univ., Palo Alto, CA (United States); SLAC National Accelerator Lab., Menlo Park, CA (United States)

- Publication Date:

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

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1408232

- Grant/Contract Number:
- [AC02-76SF00515]

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Physics. A, Mathematical and Theoretical

- Additional Journal Information:
- [ Journal Volume: 50; Journal Issue: 40]; Journal ID: ISSN 1751-8113

- Publisher:
- IOP Publishing

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; umbral moonshine; K3; string compactifications

### Citation Formats

```
Kachru, Shamit, Paquette, Natalie M., and Volpato, Roberto. 3D string theory and Umbral moonshine. United States: N. p., 2017.
Web. doi:10.1088/1751-8121/aa6e07.
```

```
Kachru, Shamit, Paquette, Natalie M., & Volpato, Roberto. 3D string theory and Umbral moonshine. United States. doi:10.1088/1751-8121/aa6e07.
```

```
Kachru, Shamit, Paquette, Natalie M., and Volpato, Roberto. Tue .
"3D string theory and Umbral moonshine". United States. doi:10.1088/1751-8121/aa6e07. https://www.osti.gov/servlets/purl/1408232.
```

```
@article{osti_1408232,
```

title = {3D string theory and Umbral moonshine},

author = {Kachru, Shamit and Paquette, Natalie M. and Volpato, Roberto},

abstractNote = {Here, the simplest string theory compactifications to 3D with 16 supercharges—the heterotic string on T 7, and type II strings on $K3 \times T^3$ —are related by U-duality, and share a moduli space of vacua parametrized by $O(8, 24;{{\mathbb Z}}) ~\backslash ~O(8, 24)~ /~ (O(8) \times O(24))$ . One can think of this as the moduli space of even, self-dual 32-dimensional lattices with signature (8,24). At 24 special points in moduli space, the lattice splits as $\Gamma^{8, 0} \oplus \Gamma^{0, 24}$ . $\Gamma^{0, 24}$ can be the Leech lattice or any of 23 Niemeier lattices, while $\Gamma^{8, 0}$ is the E 8 root lattice. We show that starting from this observation, one can find a precise connection between the Umbral groups and type IIA string theory on K3. This may provide a natural physical starting point for understanding Mathieu and Umbral moonshine. The maximal unbroken subgroups of Umbral groups in 6D (or any other limit) are those obtained by starting at the associated Niemeier point and moving in moduli space while preserving the largest possible subgroup of the Umbral group. To illustrate the action of these symmetries on BPS states, we discuss the computation of certain protected four-derivative terms in the effective field theory, and recover facts about the spectrum and symmetry representations of 1/2-BPS states.},

doi = {10.1088/1751-8121/aa6e07},

journal = {Journal of Physics. A, Mathematical and Theoretical},

number = [40],

volume = [50],

place = {United States},

year = {2017},

month = {9}

}

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