Born Geometry

Abstract

In this thesis, we summarize the work which we authored or co-authored during our PhD studies and also present additional details and ideas that are not found elsewhere. The main topic is the study of T-duality in theoretical physics through the lens of para-Hermitian geometry and Born geometry as well as the description of mathematical aspects of said geometries. In the summary portion, we introduce the D-bracket and a related notion of torsion on para-Hermitian manifolds, consequently using these geometric elements to define a unique connection with canonical properties analogous to the Levi-Civita connection in Riemannian geometry. We then discuss para-Hermitian geometry and Born geometry in the framework of generalized geometry, showing that both arise naturally in this context. We also show that the D-bracket can be recovered from the small and large Courant algebroids of the para-Hermitian manifold using the formalism of generalized geometry. Lastly, we discuss applications to theoretical physics beyond the immediate context of T-duality, showing that our generalized-geometric formulations of para-Hermitian geometry and Born geometry correspond to extended symmetries of two-dimensional non-linear sigma models. We also introduce the notion of para-Calabi-Yau manifolds and use this new geometry to study the semi-flat mirror symmetry. We show, in particular, that both the mirror manifolds carry Born structures and that the mirror map relates the symplectic moduli space of the Born geometry on one side to the complex and para-complex moduli on the other side. Additionally, we discuss the para-Hermitian geometry underlying the topological T-duality of Bouwknegt, Evslin and Mathai and present various new discussions and reformulations of known results.