### Abstract

We investigate a relation of the contravariant geometry to the emergent gravity from noncommutative gauge theories. We give a refined formulation of the contravariant gravity, and provide solutions to the contravariant Einstein equation. We linearize the equation around background solutions, including curved ones. A noncommutative gauge theory on the Moyal plane can be rewritten as an ordinary gauge theory on a curved background via the Seiberg-Witten map; this is known as emergent gravity. We show that this phenomenon also occurs for a gauge theory on a noncommutative homogeneous Kähler background. We argue that the resulting geometry can be naturally described by the contravariant geometry under an identification of the fluctuation of the Poisson tensor with the field strength obtained by the Seiberg-Witten map. These results indicate that the contravariant gravity is a suitable framework for noncommutative spacetime physics.

Original language | English |
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Article number | 055009 |

Journal | Classical and Quantum Gravity |

Volume | 35 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2018 Feb 1 |

### Keywords

- Noncommutative gauge theory
- Poisson geometry
- Seiberg-Witten map

### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)

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## Cite this

*Classical and Quantum Gravity*,

*35*(5), [055009]. https://doi.org/10.1088/1361-6382/aaa533