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Jan Vysoky

Publications and source records attributed to Jan Vysoky.

At least 19 recordsLinked to original sources
arXiv2026

Flows on Graded Manifolds

Flows of vector fields are an essential tool in differential geometry, with countless applications in both theory and practice. While they have been extensively studied for ordinary manifolds and supermanifolds, a treatment of flows in $\mathbb{Z}$-graded geometry is currently missing. $\mathbb{Z}$-graded manifolds constitute a natural generalization of supermanifolds, allowing functions in coordinates of arbitrary integer degree. In this paper, flows of vector fields on $\mathbb{Z}$-graded manifolds are defined, and it is proved that every vector field admits a unique maximal flow (in the odd case, under the assumption that the vector field is homological). Vector fields invariant under flows are examined. Conditions under which the flows of two vector fields commute are investigated, and the interaction between flows corresponding to related vector fields is studied. The main body of the paper is intentionally kept short, and we hope it is accessible to readers with only minimal prior knowledge of $\mathbb{Z}$-graded geometry.

math.DG↗
arXiv2025

Three Examples of Graded Lie Groups

Lie theory is, beyond any doubt, an absolutely essential part of differential geometry. It is therefore necessary to seek its generalization to $\mathbb{Z}$-graded geometry. In particular, it is vital to construct non-trivial and explicit examples of graded Lie groups and their corresponding graded Lie algebras. Three fundamental families of graded Lie groups are developed in this paper: the general linear group associated with any graded vector space, the graded orthogonal group associated with a graded vector space equipped with a metric, and the graded symplectic group associated with a graded vector space equipped with a symplectic form. We provide both a direct geometric construction and a functor-of-points perspective. It is shown that their corresponding Lie algebras are isomorphic to the anticipated subalgebras of the graded Lie algebra of linear endomorphisms. Isomorphisms of graded Lie groups induced by linear isomorphisms, as well as possible applications, are also discussed.

math.DG↗
arXiv2025

Threefold Nature of Graded Vector Bundles

Graded vector bundles over a given $\mathbb{Z}$-graded manifold can be defined in three different ways: certain sheaves of graded modules over the structure sheaf of the base graded manifold, finitely generated projective graded modules over the algebra of global functions on the base graded manifold, or locally trivial graded manifolds with a suitable linear structure. We argue that all three approaches are the same. More precisely, the respective categories are proved to be equivalent.

math.DG↗
arXiv2023

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.

math.DG↗
arXiv2023

Graded Generalized Geometry

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 $\mathbb{Z}$-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.

math.DG↗
arXiv2022

Palatini Variation in Generalized Geometry and String Effective Actions

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.

hep-th↗
arXiv2021

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.

math.DG↗
arXiv2020

Hitchhiker's Guide to Courant Algebroid Relations

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).

math.DG↗
arXiv2019

Effective Actions for $σ$-Models of Poisson-Lie Type

(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.

hep-th↗
arXiv2018

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.

hep-th↗
arXiv2017

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.

hep-th↗
arXiv2016

Courant Algebroid Connections and String Effective Actions

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.

math-ph↗
arXiv2016

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.

hep-th↗
arXiv2015

Geometry of Membrane Sigma Models

String theory still remains one of the promising candidates for a unification of the theory of gravity and quantum field theory. One of its essential parts is relativistic description of moving multi-dimensional objects called membranes (or p-branes) in a curved spacetime. On the classical field theory level, they are described by an action functional extremalising the volume of a manifold swept by a propagating membrane. This and related field theories are collectively called membrane sigma models. Differential geometry is an important mathematical tool in the study of string theory. It turns out that string and membrane backgrounds can be conveniently described using objects defined on a direct sum of tangent and cotangent bundles of the spacetime manifold. Mathematical field studying such object is called generalized geometry. Its integral part is the theory of Leibniz algebroids, vector bundles with a Leibniz algebra bracket on its module of smooth sections. Special cases of Leibniz algebroids are better known Lie and Courant algebroids. This thesis is divided into two main parts (the second one is ommited in this preprint). In the first one, we review the foundations of the theory of Leibniz algebroids, generalized geometry, extended generalized geometry, and NambuPoisson structures. The main aim is to provide the reader with a consistent introduction to the mathematics used in the published papers. The text is a combination both of well known results and new ones. We emphasize the notion of a generalized metric and of corresponding orthogonal transformations, which laid the groundwork of our research.

math.DG↗
arXiv2015

Generalized geometry and non-symmetric gravity

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.

hep-th↗
arXiv2014

Extended generalized geometry and a DBI-type effective action for branes ending on branes

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.

hep-th↗
arXiv2014

Nambu-Poisson Gauge Theory

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.

hep-th↗