Serre-Swan Theorem for Graded Vector Bundles

Abstract: In ordinary geometry, Serre-Swan theorem relates a geometrical definition of vector bundles to finitely generated projective modules. This fundamental result allows one to work with vector bundles in an entirely algebraic way.

Graded vector bundles over graded manifolds can be introduced wither as a particular graded manifolds, or as sheaves of modules (of their sections). It is expected that they correspond to finitely generated projective modules in a similar fashion. However, since graded vector bundles cannot be described by their fibers, one cannot use the standard arguments to prove the theorem.

Basic definitions and a sketch of the proof are presented.

Graded Jet Geometry

Abstract: We start by a general graded vector bundle over a general graded manifold and show how one can define differential operators on its sheaf of sections. This can be used to define graded jet bundles. One must avoid the traditional fiber-wise construction to do so. Instead, we show how everything can be done on the level of sheaves of its sections. All necessary notions from graded geometry are recalled.

Palatini Variation in Supergravity

Abstract: Generalized geometry proved to be a right mathematical tool for the description of an action and equations of motion of the bosonic sector of supergravity. In particular, the action can be written in the Einstein-Hilbert fashion using the particular class of Levi-Civita Courant algebroid connections on a generalized tangent bundle. To justify these choices, we resort to a simple idea. One can consider a general Courant algebroid and write an action for three independent dynamical fields - a volume form on the base manifold, a generalized metric and a general Courant algebroid connection. Amazingly, the corresponding equations of motion tie those fields together in a way resembling the well-known Palatini variation in general relativity. Necessary mathematical notions are recalled in this talk.

Graded Generalized Geometry

Abstract: One can consider graded vector bundles over \(\mathbb{Z}\)-graded manifolds equipped with a non-degenerate bilinear form of an arbitrary degree. By considering a graded version of the Dorfman bracket, we arrive to a suitable set of axioms for graded Courant algebroids. We discuss graded Dirac structures and generalized complex structures and show how to use them to obtain (differential) graded Poisson and symplectic manifolds.

Graded Manifolds: Some Issues

Abstract: A need for a geometrical theory with integer graded coordinates arose both in geometry (Courant algebroids, Poisson geometry) and physics (AKSZ and BV formalism). Based on the approach of Berezin-Leites and Kostant to supermanifolds, \(\mathbb{Z}\)-graded manifolds are usually defined as (graded) locally ringed spaces, that is certain sheaves of graded commutative algebras over (second countable Hausdorff) topological spaces, locally isomorphic to a suitable "local model".

This approach works with no major issues for non-negatively (or non-positively) graded manifolds, which is sufficient for most of the applications. However; if one tries to include coordinates of both positive and negative degrees, issues appear on several levels. This was addressed recently by M. Fairon by extending the local model sheaf. Interestingly, this modification creates a new subtle issue on the level of \(\mathbb{Z}\)-graded linear algebra.

This talk intends to point out the aforementioned issues and to offer the modifications required to obtain a consistent theory of \(\mathbb{Z}\)-graded manifolds with coordinates of an arbitrary degree.

Courant algebroid morphisms revisited

Abstract: In recent years, Courant algebroids had become a geometrical useful tool in string theory. As for any mathematical structure, one naturally attempts to establish the notion of Courant algebroid morphism. Although this was done twenty years ago, the most general definition remains relatively unknown. Similarly to the category of symplectic manifolds, the space of morphisms is not large enough. Based on the Weinstein's idea of symplectic "category" and its Lagrangian relations, one has to allow for a more general notion. This has its cost - not all morphisms can be composed. A generalization of this approach is presented. One can show that some relevant physical problems naturally fit into this framework, e.g. Poisson-Lie T-duality, Kaluza-Klein reduction of supergravity, or generalized geometry inclusion into para-Hermitian geometry (possible geometrical framework for DFT).

Plurality of effective actions: Quasi-Poisson case

Abstract: In 1995 Klimčík and Ševera proposed a new kind of duality for two-dimensional sigma models targeted in two mutually dual Poisson-Lie groups. On the level of corresponding low-energy effective actions, one has to find the correct formulas for dilaton fields. This can be done either by careful analysis of the associated path integral densities or (rather surprisingly) using the Levi-Civita connections on Courant algebroids.

A Manin pair is formed by a quadratic Lie algebra and its Lagrangian subalgebra. Whenever it integrates to a Lie group D and its subgroup G, one obtains a quasi-Poisson structure on G and a quasi-Poisson action of G on the coset space S = D/G. One can build an effective theory targeted on S. However, there can be several such subgroups G. Using the language of Courant algebroids, we prove the following: starting with a certain data on the quadratic Lie algebra, one can construct fields on all possible target spaces S satisfying the respective equations of motion.

Courant Algebroid Connections: Applications in String Theory

Abstract: Courant algebroids generalize quadratic Lie algebras in a natural way and they find their application throughout the mathematical physics. In particular, an analogue of Levi-Civita connections can be used to find a geometrical description for equations of motion of string low-energy effective actions. This observation allows one to employ the tools of geometry to derive some intriguing relations of the effective theories. As an example, the supergravity analogue of Kaluza-Klein reduction and (quasi)-Poisson-Lie T-duality are shown.

Kaluza-Klein reduction of Supergravity: Geometric approach

Abstract: An idea to obtain the gauge theory of electromagnetism in the presence of a gravitational field from the simpler theory in higher dimensions dates back to 1919. Various generalizations of this concept allow one to obtain a non-Abelian Yang-Mills model in this way. In particular, we seek for such procedure for so called heterotic supergravity. It turns out that equations of motion of such thery can be obtained as geometrical constraints imposed onto a generalized Ricci tensor of certain Levi-Civita connection on Courant algebroids. This allows one to use a known procedure of Courant algebroid reductions to find the suitable reduction of supergravity theories.

Generalized geometry and effective actions for strings and branes

Abstract: Lost in the tides of time.

Nambu sigma models and their algebraic structure

Abstract: Lost in the tides of time.