Kaluza-Klein Reduction of Low-Energy Effective Actions: Geometrical Approach

Equations of motion of low-energy string effective actions can be conveniently described in terms of generalized geometry and Levi-Civita connections on Courant algebroids. This approach is used to propose and prove a suitable version of the Kaluza-Klein-like reduction. Necessary geometrical tools are recalled.Comment: Some comments added based on the journal review. A few of minor typos correcte

Heterotic Reduction of Courant Algebroid Connections and Einstein-Hilbert Actions

We discuss Levi-Civita connections on Courant algebroids. We define an appropriate generalization of the curvature tensor and compute the corresponding scalar curvatures in the exact and heterotic case, leading to generalized (bosonic) Einstein-Hilbert type of actions known from supergravity. In particular, we carefully analyze the process of the reduction for the generalized metric, connection, curvature tensor and the scalar curvature.Comment: New section and several references added based on the journal revie

Leibniz algebroids, generalized Bismut connections and Einstein-Hilbert actions

Connection, torsion and curvature are introduced for general (local) Leibniz algebroids. Generalized Bismut connection on $TM \oplus \Lambda^{p} T^{\ast}M$ is an example leading to a scalar curvature of the form $R + H^2$ for a closed $(p+2)$-form $H$

Poisson-Lie T-duality of String Effective Actions: A New Approach to the Dilaton Puzzle

For a particular class of backgrounds, equations of motion for string sigma models targeted in mutually dual Poisson-Lie groups are equivalent. This phenomenon is called the Poisson-Lie T-duality. On the level of the corresponding string effective actions, the situation becomes more complicated due to the presence of the dilaton field. A novel approach to this problem using Levi-Civita connections on Courant algebroids is presented. After the introduction of necessary geometrical tools, formulas for the Poisson-Lie T-dual dilaton fields are derived. This provides a version of Poisson-Lie T-duality for string effective actions.Comment: One subsection added, several typos and minor mistakes correcte

Global Theory of Graded Manifolds

A theory of graded manifolds can be viewed as a generalization of differential geometry of smooth manifolds. It allows one to work with functions which locally depend not only on ordinary real variables, but also on $\mathbb{Z}$-graded variables which can either commute or anticommute, according to their degree. To obtain a consistent global description of graded manifolds, one resorts to sheaves of graded commutative associative algebras on second countable Hausdorff topological spaces, locally isomorphic to a suitable "model space". This paper aims to build robust mathematical foundations of geometry of graded manifolds. Some known issues in their definition are resolved, especially the case where positively and negatively graded coordinates appear together. The focus is on a detailed exposition of standard geometrical constructions rather then on applications. Necessary excerpts from graded algebra and graded sheaf theory are included

Graded Jet Geometry

Jet manifolds and vector bundles allow one to employ tools of differential geometry to study differential equations, for example those arising as equations of motions in physics. They are necessary for a geometrical formulation of Lagrangian mechanics and the calculus of variations. It is thus only natural to require their generalization in geometry of $\mathbb{Z}$-graded manifolds and vector bundles. Our aim is to construct the $k$-th order jet bundle $\mathfrak{J}^{k}_{\mathcal{E}}$ of an arbitrary $\mathbb{Z}$-graded vector bundle $\mathcal{E}$ over an arbitrary $\mathbb{Z}$-graded manifold $\mathcal{M}$. We do so by directly constructing its sheaf of sections, which allows one to quickly prove all its usual properties. It turns out that it is convenient to start with the construction of the graded vector bundle of $k$-th order (linear) differential operators $\mathfrak{D}^{k}_{\mathcal{E}}$ on $\mathcal{E}$. In the process, we discuss (principal) symbol maps and a subclass of differential operators whose symbols correspond to completely symmetric $k$-vector fields, thus finding a graded version of Atiyah Lie algebroid. Necessary rudiments of geometry of $\mathbb{Z}$-graded vector bundles over $\mathbb{Z}$-graded manifolds are recalled