# Spinning the fuzzy sphere

## Abstract

Here, we construct various exact analytical solutions of the SO(3) BMN matrix model that correspond to rotating fuzzy spheres and rotating fuzzy tori. These are also solutions of Yang Mills theory compactified on a sphere times time and they are also translationally invariant solutions of the N = 1* field theory with a non-trivial chargedensity. The solutions we construct have a Ζ _{N} symmetry, where N is the rank of the matrices. After an appropriate ansatz, we reduce the problem to solving a set of polynomial equations in 2N real variables. These equations have a discrete set of solutions for each value of the angular momentum. We study the phase structure of the solutions for various values of N . Also the continuum limit where N → ∞, where the problem reduces to finding periodic solutions of a set of coupled differential equations. We also study the topology change transition from the sphere to the torus.

- Authors:

- Univ. of Cambridge (United Kingdom). Dept. of Applied Mathematics and Theoretical Physics; Univ. of California, Santa Barbara, CA (United States). Dept. of Physics
- Univ. of California, Santa Barbara, CA (United States). Dept. of Physics

- Publication Date:

- Research Org.:
- Univ. of California, Santa Barbara, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1455051

- Grant/Contract Number:
- SC0011702; AC05-06OR23100.

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2015; Journal Issue: 8; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Matrix Models; Non-Commutative Geometry; Supersymmetric gauge theory; AdS-CFT Correspondence

### Citation Formats

```
Berenstein, David, Dzienkowski, Eric, and Lashof-Regas, Robin. Spinning the fuzzy sphere. United States: N. p., 2015.
Web. doi:10.1007/JHEP08(2015)134.
```

```
Berenstein, David, Dzienkowski, Eric, & Lashof-Regas, Robin. Spinning the fuzzy sphere. United States. doi:10.1007/JHEP08(2015)134.
```

```
Berenstein, David, Dzienkowski, Eric, and Lashof-Regas, Robin. Thu .
"Spinning the fuzzy sphere". United States. doi:10.1007/JHEP08(2015)134. https://www.osti.gov/servlets/purl/1455051.
```

```
@article{osti_1455051,
```

title = {Spinning the fuzzy sphere},

author = {Berenstein, David and Dzienkowski, Eric and Lashof-Regas, Robin},

abstractNote = {Here, we construct various exact analytical solutions of the SO(3) BMN matrix model that correspond to rotating fuzzy spheres and rotating fuzzy tori. These are also solutions of Yang Mills theory compactified on a sphere times time and they are also translationally invariant solutions of the N = 1* field theory with a non-trivial chargedensity. The solutions we construct have a ΖN symmetry, where N is the rank of the matrices. After an appropriate ansatz, we reduce the problem to solving a set of polynomial equations in 2N real variables. These equations have a discrete set of solutions for each value of the angular momentum. We study the phase structure of the solutions for various values of N . Also the continuum limit where N → ∞, where the problem reduces to finding periodic solutions of a set of coupled differential equations. We also study the topology change transition from the sphere to the torus.},

doi = {10.1007/JHEP08(2015)134},

journal = {Journal of High Energy Physics (Online)},

number = 8,

volume = 2015,

place = {United States},

year = {2015},

month = {8}

}

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