## Abstract

In this paper we begin the development of a relative version of T-duality in generalized complex geometry which we propose as a manifestation of mirror symmetry. Let M be an n-dimensional smooth real manifold, V a rank n real vector bundle on M, and ∇ a flat connection on V. We define the notion of a ∇-semi-flat generalized almost complex structure on the total space of V. We show that there is an explicit bijective correspondence between ∇-semi-flat generalized almost complex structures on the total space of V and ∇^{v}-semi-flat generalized almost complex structures on the total space of ∇^{v}. We show that semi-flat generalized complex structures give rise to a pair of transverse Dirac structures on the base manifold. We also study the ways in which our results generalize some aspects of T-duality such as the Buscher rules. We show explicitly how spinors are transformed and discuss the induces correspondence on branes under certain conditions.

Original language | English |
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Pages (from-to) | 533-558 |

Number of pages | 26 |

Journal | Journal of Geometry and Physics |

Volume | 56 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2006 |

Externally published | Yes |

## Keywords

- 14J32
- Dirac manifolds
- Generalized complex manifolds
- Mirror symmetry
- T-duality

## ASJC Scopus subject areas

- Mathematical Physics
- General Physics and Astronomy
- Geometry and Topology