Born Geometry in a Nutshell

Abstract

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.

Dynamical Dark Energy and Infinite Statistics

“…Here Σ is the worldsheet, the doubled target space variables X A = (x a /λ, xa /λ) combine the sum (x = x L + x R ) and the difference (x = x L − x R ) of the left-and right-movers on the string (a, A = 0, 1, • • • , d − 1 = 25, for the critical bosonic string), and λ = 1/ = √ α is the string length scale [39]. The mutually compatible dynamical fields ω AB (X), η AB (X), and H AB (X) are respectively: the antisymmetric symplectic structure ω AB , the symmetric polarization (doubly orthogonal) metric η AB , and the doubled symmetric metric H AB , which together define a Born geometry [26,40,41]. See also [42,43].…”

Modular polymer representations of the Weyl algebra

“…The companion paper [27] studies the formulation of a path integral in modular space. Interestingly, the duality between position and momentum highlighted by the modular representation is also the central theme of so-called Born reciprocity [28], whose implementation in the general relativistic setup is a fascinating open question studied in the field of Born geometry [29].…”