Exact Courant Algebroids Research Articles

Metric-Bourbaki algebroids: Cartan calculus for M-theory

String and M theories seem to require generalizations of usual notions of differential geometry on smooth manifolds. Such generalizations usually involve extending the tangent bundle to larger vector bundles equipped with various algebroid structures such as Courant algebroids, higher Courant algebroids, metric algebroids, or G-algebroids. The most general geometric scheme is not well understood yet, and a unifying framework for such algebroid structures is needed. Our aim in this paper is to propose such a general framework. Our strategy is to follow the hierarchy of defining axioms for a Courant algebroid: almost-Courant - metric - pre-Courant - Courant. In particular, we focus on the symmetric part of the bracket and the metric invariance property, and try to make sense of them in a manner as general as possible. These ideas lead us to define new algebroid structures which we dub Bourbaki and metric-Bourbaki algebroids, together with their almost- and pre-versions. For a special case of metric-Bourbaki algebroids that we call exact, we construct a collection of maps which generalize the Cartan calculus of exterior derivative, Lie derivative and interior product. This is done by a kind of reverse-mathematical analysis of the Ševera classification of exact Courant algebroids. By abstracting crucial properties of this collection of maps, we define the notion of Bourbaki calculus. Conversely, given an arbitrary Bourbaki calculus, we construct a metric-Bourbaki algebroid by building up a standard bracket that is analogous to the Dorfman bracket. Moreover, we prove that any exact metric-Bourbaki algebroid satisfying some further conditions has to have a bracket that is the twisted version of the standard bracket; a partly analogous result to Ševera classification. We prove that many physically and mathematically motivated algebroids from the literature are examples of these new algebroids, and when possible we construct a Bourbaki calculus on them. In particular, we show that the Cartan calculus can be seen as the Bourbaki calculus corresponding to an exact higher Courant algebroid. We also point out examples of Bourbaki calculi including the generalization of the Cartan calculus on vector bundle valued forms. One straightforward generalization of our constructions might be done by replacing the tangent bundle with an arbitrary Lie algebroid A. This step allows us to define an extension of our results, A-version, and extend our main results for them while proving many other algebroids from the literature fit into this framework.

Lie 2-algebras of symmetries of U(1)-bundle gerbes

This is a brief survey of some recent results on infinitesimal symmetries of bundle gerbes and their relation to three Lie 2-algebras associated to manifolds equipped with a closed 3-form, such as 2-plectic manifolds: the Poisson Lie 2-algebra of local observables, the Lie 2-algebra of sections of an associated exact Courant algebroid, and the Atiyah Lie 2-algebra.

Courant-Nijenhuis algebroids

We introduce Courant 1-derivations, which describe a compatibility between Courant algebroids and linear (1,1)-tensor fields and lead to the notion of Courant-Nijenhuis algebroids. We provide examples of Courant 1-derivations on exact Courant algebroids and show that holomorphic Courant algebroids can be viewed as special types of Courant-Nijenhuis algebroids. By considering Dirac structures, one recovers the Dirac-Nijenhuis structures of [5] (in the special case of the standard Courant algebroid) and obtains an equivalent description of Lie-Nijenhuis bialgebroids [9] via Manin triples.

B_{n}-Generalized Pseudo-Kähler Structures

We define the notions of B_{n}-generalized pseudo-Hermitian and B_{n}-generalized pseudo-Kähler structure on an odd exact Courant algebroid E. When E is in the standard form (or of type B_{n}) we express these notions in terms of classical tensor fields on the base of E. This is analogous to the bi-Hermitian viewpoint on generalized Kähler structures on exact Courant algebroids. We describe left-invariant B_{n}-generalized pseudo-Kähler structures on Courant algebroids of type B_{n} over Lie groups of dimension two, three and four.

Topological Dirac sigma models and the classical master equation

We present the construction of the classical Batalin–Vilkovisky (BV) action for topological Dirac sigma models. The latter are two-dimensional topological field theories that simultaneously generalise the completely gauged Wess–Zumino–Novikov–Witten model and the Poisson sigma model. Their underlying structure is that of Dirac manifolds associated to maximal isotropic and integrable subbundles of an exact Courant algebroid twisted by a 3-form. In contrast to the Poisson sigma model, the AKSZ construction is not applicable for the general Dirac sigma model. We therefore follow a direct approach for determining a suitable BV extension of the classical action functional with ghosts and antifields satisfying the classical master equation. Special attention is paid to target space covariance, which requires the introduction of two connections with torsion on the Dirac structure.

Super AKSZ construction, integral forms, and the 2-dimensional mathcal{N} = (1, 1) sigma model

We discuss a natural extension of the AKSZ construction to the case where the source is given by a supermanifold with a chosen integral form. We then focus on the special case with the target given by a Courant algebroid. In the simplest case this leads to the BV version of the super Chern-Simons theory, as developed by Grassi-Maccaferri and Cremonini-Grassi. In the case of exact Courant algebroids we derive the 2-dimensional mathcal{N} = (1, 1) sigma model on the boundary, together with the Wess-Zumino term, paralleling the approach of Ševera in the bosonic case.

The standard cohomology of regular Courant algebroids

For any regular Courant algebroid E over a smooth manifold M with characteristic distribution F and ample Lie algebroid AE, we prove that there exists a canonical homological vector field on the graded manifold AE[1]⊕(TM/F)⁎[2] such that the resulting dg manifold ME, which we call the minimal model of the Courant algebroid E, encodes all cohomological information of E. Indeed, the standard cohomology of E can be identified with the cohomology of the function space on ME, which can be computed by a Hodge-to-de Rham type spectral sequence. We apply this result to generalized exact Courant algebroids and those arising from regular Lie algebroids.

A gravitational action with stringy Q and R fluxes via deformed differential graded Poisson algebras

We study a deformation of a 2-graded Poisson algebra where the functions of the phase space variables are complemented by linear functions of parity odd velocities. The deformation is carried by a 2-form B-field and a bivector Π, that we consider as gauge fields of the geometric and non-geometric fluxes H, f, Q and R arising in the context of string theory compactification. The technique used to deform the Poisson brackets is widely known for the point particle interacting with a U(1) gauge field, but not in the case of non-abelian or higher spin fields. The construction is closely related to Generalized Geometry: with an element of the algebra that squares to zero, the graded symplectic picture is equivalent to an exact Courant algebroid over the generalized tangent bundle E ≅ TM ⊕ T∗M, and to its higher gauge theory. A particular idempotent graded canonical transformation is equivalent to the generalized metric. Focusing on the generalized differential geometry side we construct an action functional with the Ricci tensor of a connection on covectors, encoding the dynamics of a gravitational theory for a contravariant metric tensor and Q and R fluxes. We also extract a connection on vector fields and determine a non-symmetric metric gravity theory involving a metric and H-flux.

Generalized Ricci flow on nilpotent Lie groups

Abstract We define solitons for the generalized Ricci flow on an exact Courant algebroid. We then define a family of flows for left-invariant Dorfman brackets on an exact Courant algebroid over a simply connected nilpotent Lie group, generalizing the bracket flows for nilpotent Lie brackets in a way that might make this new family of flows useful for the study of generalized geometric flows such as the generalized Ricci flow. We provide explicit examples of both constructions on the Heisenberg group. We also discuss solutions to the generalized Ricci flow on the Heisenberg group.

Metric-connection geometries on pre-Leibniz algebroids: A search for geometrical structure in string models

Metric-affine and generalized geometries are arguably the appropriate mathematical frameworks for Einstein’s theory of gravity and low-energy effective string field theory, respectively. In fact, mathematical structures in a metric-affine geometry are constructed on the tangent bundle, which is itself a Lie algebroid, whereas those in generalized geometries, which form the basis of double field theories, are constructed on Courant algebroids. Lie, Courant, and higher Courant algebroids, which are used in exceptional field theories, are all known to be special cases of pre-Leibniz algebroids. As mathematical structures on these algebroids are essential in string models, it is natural to work on a more unifying geometrical framework. Provided with some additional ingredients, the construction of such geometries can all be carried over to regular pre-Leibniz algebroids. We define below the notions of locality structures and locality projectors, which are some necessary ingredients. In terms of these structures, E-metric-connection geometries are constructed with (possibly) a minimum number of assumptions. Certain small gaps in the literature are also filled as we go along. E-Koszul connections, as a generalization of Levi–Cività connections, are defined and shown to be helpful for some results including a simple generalization of the fundamental theorem of Riemannian geometry. The existence and non-existence of E-Levi–Cività connections are discussed for certain cases. We also show that metric-affine geometries can be constructed in a unique way as special cases of E-metric-connection geometries. Some aspects of Lie and Lie-type algebroids are studied, where the latter are defined here as a generalization of Lie algebroids. Moreover, generalized geometries are shown to follow as special cases, and various properties of linear generalized-connections are proven in the present framework. Similarly, uniqueness of the locality projector in the case of exact Courant algebroids is proven, a result that explains why the curvature operator, defined with a projector in the double field theory literature, is a necessity.

A note on invariant generators for generalized subbundles

This note studies invariant generators for a certain class of invariant smooth generalized subbundles of exact Courant algebroids, which generalizes existing results for invariant vector subbundles of exact Courant algebroids. As a result, we provide a simple and geometric proof for Theorem 1 in [1].

The Lie group of automorphisms of a Courant algebroid and the moduli space of generalized metrics

We endow the group of automorphisms of an exact Courant algebroid over a compact manifold with an infinite dimensional Lie group structure modeled on the inverse limit of Hilbert spaces (ILH). We prove a slice theorem for the action of this Lie group on the space of generalized metrics. As an application, we show that the moduli space of generalized metrics is stratified by ILH submanifolds and relate it to the moduli space of usual metrics. Finally, we extend these results to odd exact Courant algebroids.

Transverse generalized metrics and 2d sigma models

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 metric, is also what is needed for the dynamics on the reduced phase space of a string theory.

Ricci flow, Killing spinors, and T-duality in generalized geometry

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 of the Hull-Strominger system. Our main result investigates a method to produce new solutions of the Ricci flow and the Killing spinor equations. For this, we consider T-duality between possibly topologically distinct torus bundles endowed with Courant structures, and demonstrate that solutions of the equations are exchanged under this symmetry. As applications, we give a mathematical explanation of the dilaton shift in string theory and prove that the Hull-Strominger system is preserved by T-duality.

String Principal Bundles and Courant Algebroids

Abstract Just like Atiyah Lie algebroids encode the infinitesimal symmetries of principal bundles, exact Courant algebroids encode the infinitesimal symmetries of $S^1$-gerbes. At the same time, transitive Courant algebroids may be viewed as the higher analog of Atiyah Lie algebroids and the noncommutative analog of exact Courant algebroids. In this article, we explore what the “principal bundles” behind transitive Courant algebroids are, and they turn out to be principal 2-bundles of string groups. First, we construct the stack of principal 2-bundles of string groups with connection data. We prove a lifting theorem for the stack of string principal bundles with connections and show the multiplicity of the lifts once they exist. This is a differential geometrical refinement of what is known for string structures by Redden, Waldorf, and Stolz–Teichner. We also extend the result of Bressler and Chen et al. on extension obstruction involving transitive Courant algebroids to the case of transitive Courant algebroids with connections, as a lifting theorem with the description of multiplicity once liftings exist. At the end, we build a morphism between these two stacks. The morphism turns out to be neither injective nor surjective in general, which shows that the process of associating the “higher Atiyah algebroid” loses some information and at the same time, only some special transitive Courant algebroids come from string bundles.

T-Duality in an H-Flux: Exchange of Momentum and Winding

Using our earlier proposal for Ramond-Ramond fields in an H-flux on loop space, we extend the Hori isomorphism of Bouwknegt-Evslin-Mathai from invariant differential forms, to invariant exotic differential forms such that the momentum and winding numbers are exchanged, filling in a gap in the literature. We also extend the compatibility of the action of invariant exact Courant algebroids on the T-duality isomorphism of Cavalcanti-Gualtieri, to the T-duality isomorphism on exotic invariant differential forms.

Symplectic structures on the integration of exact Courant algebroids

We construct an infinite-dimensional symplectic 2-groupoid as the integration of an exact Courant algebroid. We show that every integrable Dirac structure integrates to a "Lagrangian" sub-2-groupoid of this symplectic 2-groupoid. As a corollary, we recover a result of Bursztyn-Crainic-Weinstein-Zhu that every integrable Dirac structure integrates to a presymplectic groupoid.

Higher extensions of Lie algebroids

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 obtain a Lie 2-groupoid integrating an exact Courant algebroid.

On higher dimensional exact Courant algebroids

For a smooth manifold X we show an equivalence of categories between the category of O X ♯ [ n ] -extensions of T X ♯ and the category of higher-dimensional exact Courant algebroids on X . In addition, for any object in the category of R [ n ] dg-principal bundles over X ♯ we construct its Atiyah algebroid which gives rise to an example of O X ♯ [ n ] -extensions of T X ♯ .

Group Actions on DG-Manifolds and Exact Courant Algebroids

Let G be a Lie group acting by diffeomorphisms on a manifold M and consider the image of T[1]G and T[1]M, of G and M respectively, in the category of differential graded manifolds. We show that the obstruction to lift the action of T[1]G on T[1]M to an action on a \({\mathbb{R}[n]}\)-bundle over T[1]M is measured by the G equivariant cohomology of M. We explicitly calculate the differential graded Lie algebra of the symmetries of the \({\mathbb{R}[n]}\)-bundle over T[1]M and we use this differential graded Lie algebra to understand which actions are hamiltonian. We show how split Exact Courant algebroids could be obtained as the derived Leibniz algebra of the symmetries of \({\mathbb{R}[2]}\)-bundles over T[1]M, and we use this construction to propose that the infinitesimal symmetries of a split Exact Courant algebroid should be encoded in the differential graded Lie algebra of symmetries of a \({\mathbb{R}[2]}\)-bundle over T[1]M. With this setup at hand, we propose a definition for an action of a Lie group on an Exact Courant algebroid and we propose conditions for the action to be hamiltonian.