B. Jurčo, F. Moučka, J. Vysoký: Palatini Variation in Generalized Geometry and String Effective Actions (2023)

We develop the Palatini formalism within the framework of generalized Riemannian geometry of Courant algebroids. In this context, the Palatini variation of a generalized Einstein-Hilbert-Palatini action - formed using a generalized metric, a Courant algebroid connection (in contrary to the ordinary case, not necessarily a torsionless one) and a volume form - leads naturally to a proper notion of a generalized Levi-Civita connection and low-energy effective actions of string theory.

J. Vysoký: Graded Generalized Geometry (2022)

Generalized geometry finds many applications in the mathematical description of some aspects of string theory. In a nutshell, it explores various structures on a generalized tangent bundle associated to a given manifold. In particular, several integrability conditions can be formulated in terms of a canonical Dorfman bracket, an example of Courant algebroid. On the other hand, smooth manifolds can be generalized to involve functions of Z-graded variables which do not necessarily commute. This leads to a mathematical theory of graded manifolds. It is only natural to combine the two theories by exploring the structures on a generalized tangent bundle associated to a given graded manifold.

After recalling elementary graded geometry, graded Courant algebroids on graded vector bundles are introduced. We show that there is a canonical bracket on a generalized tangent bundle associated to a graded manifold. Graded analogues of Dirac structures and generalized complex structures are explored. We introduce differential graded Courant algebroids which can be viewed as a generalization of Q-manifolds. A definition and examples of graded Lie bialgebroids are given.

J. Vysoký: Global Theory of Graded Manifolds (2021)

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.

J. Vysoký: Hitchhiker's Guide to Courant Algebroid Relations (2020)

Courant algebroids provide a useful mathematical tool (not only) in string theory. It is thus important to define and examine their morphisms. To some extent, this was done before using an analogue of canonical relations known from symplectic geometry. However, it turns out that applications in physics require a more general notion. We aim to provide a self-contained and detailed treatment of Courant algebroid relations and morphisms. A particular emphasis is placed on providing enough motivating examples. In particular, we show how Poisson-Lie T-duality and Kaluza-Klein reduction of supergravity can be interpreted as Courant algebroid relations compatible with generalized metrics (generalized isometries).

B. Jurčo, J. Vysoký: Effective Actions for \(\sigma\)-Models of Poisson–Lie Type (2019)

(Quasi-)Poisson-Lie T-duality of string effective actions is described in the framework of generalized geometry of Courant algebroids. The approach is based on a generalization of Riemannian geometry in the context of Courant algebroids, including a proper version of a Levi-Civita connection. In our approach, the dilaton field is encoded in a Levi-Civita connection and its form is determined by the Courant algebroid geometry. Explicit examples of background solutions are provided using the approach developed in the paper.

B. Jurčo, J. Vysoký: Poisson-Lie T-duality of String Effective Actions:
A New Approach to the Dilaton Puzzle (2018)

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.

J. Vysoký: Kaluza-Klein Reduction of Low-Energy Effective Actions: Geometrical Approach (2017)

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.

B. Jurčo, J. Vysoký: Courant Algebroid Connections and String Effective Actions (2017)

Courant algebroids are a natural generalization of quadratic Lie algebras, appearing in various contexts in mathematical physics. A connection on a Courant algebroid gives an analogue of a covariant derivative compatible with a given fiber-wise metric. Imposing further conditions resembling standard Levi-Civita connections, one obtains a class of connections whose curvature tensor in certain cases gives a new geometrical description of equations of motion of low energy effective action of string theory. Two examples are given. One is the so called symplectic gravity, the second one is an application to the the so called heterotic reduction. All necessary definitions, propositions and theorems are given in a detailed and self-contained way.

B. Jurčo, J. Vysoký: Heterotic Reduction of Courant Algebroid Connections and Einstein-Hilbert Actions (2016)

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.

B. Jurčo, F.S. Khoo, P. Schupp, J. Vysoký: Generalized geometry and non-symmetric gravity (2015)

Generalized geometry provides the framework for a systematic approach to non-symmetric metric gravity theory and naturally leads to an Einstein-Kalb-Ramond gravity theory with totally anti-symmetric contortion. The approach is related to the study of the low-energy effective closed string gravity actions.

B. Jurčo, J. Vysoký: Leibniz algebroids, generalized Bismut connections and Einstein-Hilbert actions (2015)

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 \(H \).

B. Jurčo, P. Schupp, J. Vysoký: Extended generalized geometry and a DBI-type effective action
for branes ending on branes (2014)

Starting from the usual bosonic membrane action, we develop the geometry suitable for the description of p-brane backgrounds. Using the tools of generalized geometry we derive the generalization of string open-closed relations. Nambu-Poisson structures are used to generalize the concept of semiclassical noncommutativity of (\D\)-branes governed by Poisson tensor. We naturally describe the correspondence of recently proposed commutative and noncommutative versions of an effective action for (\p\)-branes ending on a (\p′\)-brane. We calculate the power series expansion of the action in background independent gauge. Leading terms in the double scaling limit are given by a generalization of a (semi-classical) matrix model.

B. Jurčo, P. Schupp, J. Vysoký: Nambu-Poisson Gauge Theory (2014)

We generalize noncommutative gauge theory using Nambu-Poisson structures to obtain a new type of gauge theory with higher brackets and gauge fields. The approach is based on covariant coordinates and higher versions of the Seiberg-Witten map. We construct a covariant Nambu-Poisson gauge theory action, give its first order expansion in the Nambu-Poisson tensor and relate it to a Nambu-Poisson matrix model.

B. Jurčo, P. Schupp, J. Vysoký: On the Generalized Geometry Origin of Noncommutative Gauge Theory (2013)

We discuss noncommutative gauge theory from the generalized geometry point of view. We argue that the equivalence between the commutative and semiclassically noncommutative DBI actions is naturally encoded in the generalized geometry of D-branes.

B. Jurčo, P. Schupp, J. Vysoký: p-Brane Actions and Higher Roytenberg Brackets (2013)

Motivated by the quest to understand the analog of non-geometric flux compactification in the context of M-theory, we study higher dimensional analogs of generalized Poisson sigma models and corresponding dual string and p-brane models. We find that higher generalizations of the algebraic structures due to Dorfman, Roytenberg and Courant play an important role and establish their relation to Nambu-Poisson structures.

J. Vysoký, L. Hlavatý: Poisson-Lie Sigma Models on Drinfel'd double (2012)

Poisson sigma models represent an interesting use of Poisson manifolds for the construction of a classical field theory. Their definition in the language of fibre bundles is shown and the corresponding field equations are derived using a coordinate independent variational principle. The elegant form of equations of motion for so called Poisson-Lie groups is derived. Construction of the Poisson-Lie group corresponding to a given Lie bialgebra is widely known only for coboundary Lie bialgebras. Using the adjoint representation of Lie group and Drinfel'd double we show that Poisson-Lie group can be constructed for general Lie bialgebra.

L. Hlavatý, V. Štěpán, J. Vysoký: Drinfel’d superdoubles and Poisson–Lie T-plurality in low dimensions (2010)

Defining the real Lie superalgebra as real \(\mathbb{Z}_{2}\)-graded vector space we classify real Manin supertriples and Drinfel’d superdoubles of superdimensions (2,2), (4,2), and (2,4). The Drinfel’d doubles of the superdimension (2,2) are then used for construction of the simplest σ-models related by Poisson–Lie T-plurality.