We introduce the category of holomorphic string algebroids, whose objects are Courant extensions of Atiyah Lie algebroids of holomorphic principal bundles, as considered by Bressler, and whose morphisms correspond to inner morphisms of the underlying holomorphic Courant algebroids in the sense of Severa. This category …
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Just like Atiyah Lie algebroids encode the infinitesimal symmetries of principal bundles, exact Courant algebroids are believed to encode the infinitesimal symmetries of -gerbes. At the same time, transitive Courant algebroids may be viewed as the higher analogue of Atiyah Lie algebroids, and the non-commutative a…
Introduces gauge theory for string algebroids, solving Calabi system.
problem Solving coupled equations for Calabi problem and Hull-Strominger system.
method Moment map picture with Hamiltonian gauge action, inner automorphisms of Courant algebroids.
result Moduli space carries pseudo-Kähler metric with Kähler potential given by dilaton functional.
In this paper, we use reduction by extended actions to give a construction of transitive Courant algebroids from string classes. We prove that T-duality commutes with the reductions and thereby determine global conditions for the existence of T-duals in heterotic string theory. In particular we find that T-duality exch…
Develops Palatini formalism in generalized geometry for string theory.
problem Formulating Palatini variation in generalized geometry.
method Palatini formalism within generalized Riemannian geometry of Courant algebroids.
result Natural emergence of generalized Levi-Civita connection and string effective actions.
Y-algebroids generalize symmetry structures for string/M-theory compactifications.
problem Capturing local symmetry structures for string/M-theory compactifications to various dimensions.
method Define Y-algebroids based on a tensor Y, classify M-exact E7-algebroids, and relate to E7(7)×R^+ generalised geometry.
result Y-algebroids match the generalised tangent space of E7(7)×R^+ generalised geometry, including possible fluxes.
We study the extension of a Lie algebroid by a representation up to homotopy, including semidirect products of a Lie algebroid with such representations. The extension results in a higher Lie algebroid. We give exact Courant algebroids and string Lie 2-algebras as examples of such extensions. We then apply this to obta…
Extends Lie bialgebroids for string and M theories with new calculus framework.
problem Formalize calculus on algebroids for string and M theories.
method Reinterpret matched pairs of Leibniz algebroids, examine algebroid axioms, construct double on direct sum.
result Construct Drinfel'd double of Lie bialgebroids for general algebroids.
We study the semidirect product of a Lie algebra with a representation up to homotopy and provide various examples coming from Courant algebroids, string Lie 2-algebras, and omni-Lie algebroids. In the end, we study the semidirect product of a Lie group with a representation up to homotopy and use it to give an integra…
New geometries defined for string models, filling gaps in the literature.
problem Developing mathematical structures for string models.
method Defining E-metric-connection geometries and locality structures.
result Unified framework for metric-affine and generalized geometries.
Defines T-duality and generalised Ricci flow relations using Courant algebroid relations.
problem Establishing compatibility between T-duality and generalised Ricci flow.
method Introducing Courant algebroid relations, invariant divergence operators, and generalised isometries.
result T-duality is compatible with generalised Ricci flow, and T-dual solutions are also solutions of generalised Ricci flow.
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 connect…
Study of exceptional algebroids in type IIA string theory.
problem Understanding the structure of algebroids in type IIA string theory.
method Warped compactifications and analysis of local moduli space.
result Local classification of algebroids reveals new scalar deformations and twists.
New algebroids allow studying various geometries simultaneously.
problem Constructing geometries on a broad class of algebroids.
method Introduced `anti-commutable` pre-Leibniz algebroids and admissible connections.
result Proved properties of admissible connections on various algebroids.
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 g…
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 …
Researchers link vertex algebras to non-Kähler solutions of the Hull-Strominger system.
problem Constructing representations of vertex algebras from non-Kähler solutions of the Hull-Strominger system.
method Embedding the N=2 superconformal vertex algebra in the chiral de Rham complex of a string Courant algebroid, with a condition on the Hermitian-Yang-Mills connection.
result Any solution of the Hull-Strominger system satisfying the Hermitian-Yang-Mills condition has an associated N=2 embedding.
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 t…
We reformulate the compatibility condition between a generalized metric and a small (non-maximal rank) Dirac structure in an exact Courant algebroid found in the context of the gauging of strings and formulated by means of two connections in purely Dirac-geometric terms. The resulting notion, a transverse generalized m…
In this paper, we construct a homotopy Poisson algebra of degree 3 associated to a split Lie 2-algebroid, by which we give a new approach to characterize a split Lie 2-bialgebroid. We develop the differential calculus associated to a split Lie 2-algebroid and establish the Manin triple theory for split Lie 2-algebroids…
The paper categorifies Lie and Courant algebroids, establishing correspondences and new constructions.
problem Categorification of Lie and Courant algebroids to better understand geometric structures.
method Introducing and studying new algebraic structures like VB-Lie 2-algebroids and VB-LWX 2-algebroids.
result Established correspondences and new constructions between Lie and Courant algebroids.
Introduces D-branes in para-Hermitian geometries using T-duality.
problem Defines D-branes in a new geometric framework.
method Uses para-Hermitian geometry and metric algebroids to define D-branes as conformal boundary conditions for open strings.
result Reveals D-branes as para-complex versions of topological A/B-branes.
A framework for constructing new kinds of gauge theories is suggested. Essentially it consists in replacing Lie algebras by Lie or Courant algebroids. Besides presenting novel topological theories defined in arbitrary spacetime dimensions, we show that equipping Lie algebroids E with a fiber metric having sufficiently …
Constructs brane current algebras from QP-manifolds, generalizing string currents.
problem Constructing brane current algebras from QP-manifolds.
method Using Poisson algebra and QP-manifolds (symplectic -algebroids), the paper derives a universal geometric form for Poisson brackets of brane currents.
result Derives a universal expression for 't Hooft anomaly in the presence of fluxes.
We begin by presenting a symmetric version of the circle equivariant T-duality result in a joint work of the second author with Siye Wu, thereby generalising the results there. We then initiate the study of twisted equivariant Courant algebroids and equivariant generalised geometry and apply it to our context. As befor…
We construct a new class of exact solutions describing spacetimes possessing Lie algebroid symmetry. They are described by generic off-diagaonal 5D metrics embedded in bosonic string gravity and possess nontrivial limits to the Einstein gravity. While we focus on nonholonomic vielbein transforms of the Schwarzschild me…
New flow defined to solve Hull-Strominger system, with estimates and convergence results.
problem Constructing solutions to the Hull-Strominger system of equations.
method Introducing a natural extension of pluriclosed flow and using string algebroids and higher gauge theory.
result Proves global existence and convergence of the flow on special backgrounds.
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.
Proposes a new geometric framework for M-theory algebroids.
problem Generalizations of differential geometry needed for M-theory.
method Hierarchy of axioms for Courant algebroids, focusing on symmetric part and metric invariance.
result Constructs new algebroid structures (Bourbaki and metric-Bourbaki) and defines Bourbaki pre-calculus.
We investigate geometric aspects of double field theory (DFT) and its formulation as a doubled membrane sigma-model. Starting from the standard Courant algebroid over the phase space of an open membrane, we determine a splitting and a projection to a subbundle that sends the Courant algebroid operations to the correspo…
Given a representation up to homotopy of a Lie algebroid on a 2-term complex of vector bundles, we define the corresponding holonomy as a strict 2-functor from a Weinstein path 2-groupoid to the gauge 2-groupoid of the underlying 2-term complex. We construct a corresponding transformation 2-groupoid and we prove that t…
We show that in analogy to the introduction of Poisson structures twisted by a closed 3-form by Park and Klimcik-Strobl, the study of three dimensional sigma models with Wess-Zumino term leads in a likewise way to twisting of Courant algebroid structures by closed 4-forms H. The presentation is kept pedagogical and acc…
We reexamine the notions of generalized Ricci tensor and scalar curvature on a general Courant algebroid, reformulate them using objects natural w.r.t. pull-backs and reductions, and obtain them from the variation of a natural action functional. This allows us to prove, in a very general setup, the compatibility of the…
Graded Generalized Geometrymath.DG
Combines generalized and graded geometry to explore new structures.
problem Exploring new structures on generalized tangent bundles of graded manifolds.
method Introduces canonical brackets, Dirac structures, and generalized complex structures.
result Canonical bracket on a generalized tangent bundle of a graded manifold.
We give a covariant realization of the doubled sigma-model formulation of duality-symmetric string theory within the general framework of para-Hermitian geometry. We define a notion of generalized metric on a para-Hermitian manifold and discuss its relation to Born geometry. We show that a Born geometry uniquely define…
Extends Drinfel'd doubles to include twists and generalizes Lie bialgebroids.
problem Tackles the extension of Drinfel'd doubles and Lie bialgebroids to include twists and generalizes them.
method Introduces a framework of calculus on algebroids and examines compatibility conditions for various algebroid properties.
result Introduces the notion of proto bialgebroids and their Drinfel'd doubles, generalizing both bialgebroids and proto Lie bialgebroids.
Constructs BPS complexes and Chern--Simons theories from G-structures.
problem Infinitesimal moduli space computation and supersymmetric systems.
method Universal algebraic construction of BPS complexes and associated linearised BV Chern--Simons theories.
result Reproduces classic examples in gauge theory and constructs heterotic superpotential functionals.
We geometrize six-dimensional pure Supergravity by means of an exact Courant algebroid, whose Severa class is defined through the Supergravity three-form , equipped with a generalized metric and a compatible, torsion-free, generalized connection. The Supergravity equations of motion follow from the v…
It has been shown recently that the geometry of D-branes in general topologically twisted (2,2) sigma-models can be described in the language of generalized complex structures. On general grounds such D-branes (called generalized complex (GC) branes) must form a category. We compute the BRST cohomology of open strings …
We introduce a notion of Ricci flow in generalized geometry, extending a previous definition by Gualtieri on exact Courant algebroids. Special stationary points of the flow are given by solutions to first-order differential equations, the Killing spinor equations, which encompass special holonomy metrics with solutions…
Defines Dirac pairs on Jacobi algebroids, generalizing Lie algebroids.
problem Generalizing Dirac pairs to Jacobi algebroids.
method Introducing Dirac pairs on Jacobi algebroids and showing their relationship to Lie algebroids.
result Dirac pairs on Jacobi algebroids characterize compatible structures.
Paper constructs various algebroids using n-systems and metric n-systems.
problem Creating algebroids from n-systems and metric n-systems.
method Using n-systems and metric n-systems to construct algebroids.
result Explicit computations for all resulting structure maps.
Constructing -Lie algebroids via connections
problem Constructing Lie algebroids and -Lie algebroids
method Using connections generated by finite families of differential operators and dual sections
result Providing sufficient conditions for generating families to determine Lie algebroid and -Lie algebroid structures
In this paper, we introduce the notion of a pre-symplectic algebroid, and show that there is a one-to-one correspondence between pre-symplectic algebroids and symplectic Lie algebroids. This result is the geometric generalization of the relation between left-symmetric algebras and symplectic (Frobenius) Lie algebras. A…
Introduces new construction for Courant algebroids and curved structures.
problem Understanding and classifying Courant algebroids and their lifts.
method Introduces Courant algebroid lift and curved Courant algebroids, establishing connections to various geometric structures.
result Established a classification of exact curved Courant algebroids and related connections to various geometric structures.
This paper provides an alternative, much simpler, definition for Li-Bland's LA-Courant algebroids, or Poisson Lie 2-algebroids, in terms of split Lie 2-algebroids and self-dual 2-representations. This definition generalises in a precise sense the characterisation of (decomposed) double Lie algebroids via matched pairs …
E-Courant algebroidsmath.DG
In this paper, we introduce the notion of -Courant algebroids, where is a vector bundle. It is a kind of generalized Courant algebroid and contains Courant algebroids, Courant-Jacobi algebroids and omni-Lie algebroids as its special cases. We explore novel phenomena exhibited by -Courant algebroids and provid…
The Free Courant Algebroidmath.DG
We introduce the category of generalized Courant algebroids and show that it admits a free object on any anchored vector bundle. The free Courant algebroid is built from two components: the generalized Courant algebroid associated to a symmetric Leibniz algebroid and the free symmetric Leibniz algebroid on an anchored …