We formulate a kinematical extension of Double Field Theory on a 2d-dimensional para-Hermitian manifold (P, η, ω) where the O(d, d) metric η is supplemented by an almost symplectic two-form ω. Together η and ω define an almost bi-Lagrangian structure K which provides a splitting of the tangent bundle T P = L ⊕L into two Lagrangian subspaces. In this paper a canonical connection and a corresponding generalised Lie derivative for the Leibniz algebroid on T P are constructed. We find integrability conditions under which the symmetry algebra closes for general η and ω, even if they are not flat and constant. This formalism thus provides a generalisation of the kinematical structure of Double Field Theory. We also show that this formalism allows one to reconcile and unify Double Field Theory with Generalised Geometry which is thoroughly discussed.

We present a global construction of a so-called D-bracket appearing in the physics literature of Double Field Theory (DFT) and show that if certain integrability criteria are satisfied, it can be seen as a sum of two Courant algebroid brackets. In particular, we show that the local picture of the extended space-time used in DFT fits naturally in the geometrical framework of para-Hermitian manifolds and that the data of an (almost) para-Hermitian manifold is sufficient to construct the D-bracket. Moreover, the twists of the bracket appearing in DFT can be interpreted in this framework geometrically as a consequence of certain deformations of the underlying para-Hermitian structure. *

It has been known for a while that the effective geometrical description of compactified strings on d-dimensional target spaces implies a generalization of geometry with a doubling of the sets of tangent space directions. This generalized geometry involves an O(d, d) pairing η and an O(2d) generalized metric H. More recently it has been shown that in order to include T-duality as an effective symmetry, the generalized geometry also needs to carry a phase space structure or more generally a para-Hermitian structure encoded into a skew-symmetric pairing ω. The consistency of string dynamics requires this geometry to satisfy a set of compatibility relations that form what we call a Born geometry. In this work we prove an analogue of the fundamental theorem of Riemannian geometry for Born geometry. We show that there exists a unique connection which preserves the Born structure (η, ω, H) and which is torsionless in a generalized sense. This resolves a fundamental ambiguity that is present in the double field theory formulation of effective string dynamics.

We give a concise summary of the para-Hermitian geometry that describes a doubled target space fit for a covariant description of T-duality in string theory. This provides a generalized differentiable structure on the doubled space and leads to a kinematical setup which allows for the recovery of the physical spacetime. The picture can be enhanced to a Born geometry by including dynamical structures such as a generalized metric and fluxes which are related to the physical background fields in string theory. We then discuss a generalization of the Levi-Civita connection in this setting-the Born connection-and a twisting of the kinematical structure in the presence of fluxes.

According to formalism, a mathematician is not concerned with mysterious metaphysical entities but with mathematical symbols. As a result, mathematical entities become simply sensible signs. However, the price that has to be paid for this move seems to be too high, for mathematics, at present considered to be the queen of sciences, turns out to be a to a contentless game. That is why it seems absurd to regard numbers and all mathematical entities as mere symbols. The aim of our paper is to show the reasons that have led some philosophers and mathematicians to adopt the view that mathematical terms in the proper sense refer to nothing and mathematical propositions have no real content. At the same time we want to explain how formalism helped to overcome the traditional concept of science.