Double Field Theory: A Pedagogical Review

Double Field Theory (DFT) is a proposal to incorporate T-duality, a distinctive symmetry of string theory, as a symmetry of a field theory defined on a double configuration space. The aim of this review is to provide a pedagogical presentation of DFT and its applications. We first introduce some basic ideas on T-duality and supergravity in order to proceed to the construction of generalized diffeomorphisms and an invariant action on the double space. Steps towards the construction of a geometry on the double space are discussed. We then address generalized Scherk-Schwarz compactifications of DFT and their connection to gauged supergravity and flux compactifications. We also discuss U-duality extensions, and present a brief parcours on world-sheet approaches to DFT. Finally, we provide a summary of other developments and applications that are not discussed in detail in the review.Comment: 121 pages, invited review for Class. Quantum Grav; v2: Updated reference

On duality theory and pseudodifferential techniques for Colombeau algebras: generalized delta functionals, kernels and wave front sets

Summarizing basic facts from abstract topological modules over Colombeau generalized complex numbers we discuss duality of Colombeau algebras. In particular, we focus on generalized delta functionals and operator kernels as elements of dual spaces. A large class of examples is provided by pseudodifferential operators acting on Colombeau algebras. By a refinement of symbol calculus we review a new characterization of the wave front set for generalized functions with applications to microlocal analysis

A generalized logarithmic module and duality of Coxeter multiarrangements

We introduce a new definition of a generalized logarithmic module of multiarrangements by uniting those of the logarithmic derivation and the differential modules. This module is realized as a logarithmic derivation module of an arrangement of hyperplanes with a multiplicity consisting of both positive and negative integers. We consider several properties of this module including Saito's criterion and reflexivity. As applications, we prove a shift isomorphism and duality of some Coxeter multiarrangements by using the primitive derivation.Comment: 17 page

Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map

We introduce a general theory of parametrized objects in the setting of infinity categories. Although spaces and spectra parametrized over spaces are the most familiar examples, we establish our theory in the generality of objects of a presentable infinity category parametrized over objects of an infinity topos. We obtain a coherent functor formalism describing the relationship of the various adjoint functors associated to base-change and symmetric monoidal structures. Our main applications are to the study of generalized Thom spectra. We obtain fiberwise constructions of twisted Umkehr maps for twisted generalized cohomology theories using a geometric fiberwise construction of Atiyah duality. In order to characterize the algebraic structures on generalized Thom spectra and twisted (co)homology, we characterize the generalized Thom spectrum as a categorification of the well-known adjunction between units and group rings.Comment: Submission draft. Various changes, including rewrite in terms of infinity topoi and corrected discussion of functoriality of Atiyah dualit