Algebroid Structure Research Articles

Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces

Let L=(L,[⋅,⋅],δ) be an algebraic Lie algebroid over a smooth projective curve X of genus g≥2 such that L is a line bundle whose degree is less than 2−2g. Let r and d be coprime numbers. We prove that the motivic class of the moduli space of L-connections of rank r and degree d over X does not depend on the Lie algebroid structure [⋅,⋅] and δ of L and neither on the line bundle L itself, but only on the degree of L (and of course on r, d and X). In particular it is equal to the motivic class of the moduli space of KX(D)-twisted Higgs bundles of rank r and degree d, for D any effective divisor with the appropriate degree. As a consequence, similar results (actually slightly stronger) are obtained for the corresponding E-polynomials. Some applications of these results are then deduced.

Compatible $E$-Differential Forms on Lie Algebroids over (Pre-)Multisymplectic Manifolds

We consider higher generalizations of both a (twisted) Poisson structure and the equivariant condition of a momentum map on a symplectic manifold. On a Lie algebroid over a (pre-)symplectic and (pre-)multisymplectic manifold, we introduce a Lie algebroid differential form called a compatible $E$-$n$-form. This differential form satisfies a compatibility condition, which is consistent with both the Lie algebroid structure and the (pre-)(multi)symplectic structure. There are many interesting examples such as a Poisson structure, a twisted Poisson structure and a twisted $R$-Poisson structure for a pre-$n$-plectic manifold. Moreover, momentum maps and momentum sections on symplectic manifolds, homotopy momentum maps and homotopy momentum sections on multisymplectic manifolds have this structure.

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.

The geometric constraints on Filippov algebroids

<abstract><p>Filippov $ n $-algebroids are introduced by Grabowski and Marmo as a natural generalization of Lie algebroids. On this note, we characterized Filippov $ n $-algebroid structures by considering certain multi-input connections, which we called Filippov connections, on the underlying vector bundle. Through this approach, we could express the $ n $-ary bracket of any Filippov $ n $-algebroid using a torsion-free type formula. Additionally, we transformed the generalized Jacobi identity of the Filippov $ n $-algebroid into the Bianchi-Filippov identity. Furthermore, in the case of rank $ n $ vector bundles, we provided a characterization of linear Nambu-Poisson structures using Filippov connections.</p></abstract>

Tripled Structures of Algebroids in Gauged Double Field Theory

Double field theory (DFT) is an effective theory of string theory. It has a manifest symmetry of T-duality. The gauge symmetry in DFT is related to some kind of algebroid structures, and they have a doubled structure. We focus on the gauge algebra of the O(D, D+n) gauged DFT and discuss an extension of the doubled structure. The gauge algebra of the O(D, D + n) gauged DFT has been described by the F-bracket. This bracket is related to some algebroid structures. We show that algebroids defined by the twisted C-bracket in the gauged DFT are built out of a direct sum of three Lie algebroids. They exhibit a “triple”, which we call the extended double, rather than a “double” structure. We also consider the geometrical realization of these structures in a (2D + n)-dimensional manifold.

Double field theory, twistors, and integrability in 4-manifolds

The search for a geometrical understanding of dualities in string theory, in particular T-duality, has led to the development of modern T-duality covariant frameworks such as Double Field Theory, whose mathematical structure can be understood in terms of generalized geometry and, more recently, para-Hermitian geometry. In this work we apply techniques associated to this doubled geometry to four-dimensional manifolds, and we show that they are particularly well-suited to the analysis of integrability in special spacetimes, especially in connection with Penrose's twistor theory and its applications to general relativity. This shows a close relationship between some of the geometrical structures in the para-Hermitian approach to double field theory and those in algebraically special solutions to the Einstein equations. Particular results include the classification of four-dimensional, possibly complex-valued, (para-)Hermitian structures in different signatures, the Lie and Courant algebroid structures of special spacetimes, and the analysis of deformations of (para-)complex structures. We also discuss a notion of “weighted algebroids” in relation to a natural gauge freedom in the framework. Finally, we analyze the connection with two- and three-dimensional (real and complex) twistor spaces, and how the former can be understood in terms of the latter, in particular in terms of twistor families.

A Lie Algebroid Structure for Vector Bundles of Finite Rank Isomorphic to Tangent Bundle of Their Base Space

We define a Lie algebroid structure for a class of vector bundles of rank k over a k-dimensional smooth manifold W, which are isomorphic to the tangent bundle TW. We construct an exciting example of these types of vector bundles. This example is constructed based on partial Caputo fractional derivatives. We call this vector bundle a fractional vector bundle and denote it by {mathscr {F}}^{nu } {mathscr {W}}.

Instances of higher geometry in field theory

Generalisations of geometry have emerged in various forms in the study of field theory and quantization. This mini-review focuses on the role of higher geometry in three selected physical applications. After motivating and describing some basic aspects of algebroid structures on bundles and (differential graded) Q-manifolds, we briefly discuss their relation to $$(\alpha )$$ the Batalin–Vilkovisky quantization of topological sigma models, $$(\beta )$$ higher gauge theories and generalized global symmetries and $$(\gamma )$$ tensor gauge theories, where the universality of their form and properties in terms of graded geometry is highlighted.

BFV extensions for mechanical systems with Lie-2 symmetry

We consider mechanical systems on ${T}^{*}M$ with possibly irregular and reducible first class constraints linear in the momenta, which thus correspond to singular foliations on $M$. According to a recent result, the latter ones have a Lie-infinity algebroid $(\mathcal{M},Q)$ covering them, where we restrict to the case of Lie-2 algebroids. We propose to consider ${T}^{*}\mathcal{M}$ as a potential Batalin-Fradkin-Vilkovisky (BFV) extended phase space of the constrained system, such that the canonical lift of the nilpotent vector field $Q$ yields automatically a solution to the BFV master equation. We show that in this case, the BFV extension of the Hamiltonian, providing a second corner stone of the BFV formalism, may be obstructed. We identify the corresponding complex governing this second extension problem explicitly (the first extension problem was circumvented by means of the lift of the Lie-2 algebroid structure). We repeatedly come back to the example of angular momenta on ${T}^{*}{\mathbb{R}}^{3}$: in this procedure, the standard free Hamiltonian does not have a BFV extension---while it does so on ${T}^{*}({\mathbb{R}}^{3}\{0})$, with a relatively involved ghost contribution singular at the origin.

‘Anti-commutable’ local pre-Leibniz algebroids and admissible connections

The concept of algebroid is convenient as a basis for constructions of geometrical frameworks. For example, metric-affine and generalized geometries can be written on Lie and Courant algebroids, respectively. Furthermore, string theories might make use of many other algebroids such as metric algebroids, higher Courant algebroids, or conformal Courant algebroids. Working on the possibly most general algebroid structure, which generalizes many of the algebroids used in the literature, is fruitful as it creates a chance to study all of them at once. Local pre-Leibniz algebroids are such general ones in which metric-connection geometries are possible to construct. On the other hand, the existence of the ‘locality operator’, which is present for the left-Leibniz rule for the bracket, necessitates the modification of torsion and curvature operators in order to achieve tensorial quantities. In this paper, this modification of torsion and curvature is explained from the point of view that the modification is applied to the bracket instead. This leads one to consider ‘anti-commutable’ local pre-Leibniz algebroids which satisfy an anti-commutativity-like property defined with respect to a choice of an equivalence class of connections. These ‘admissible’ connections are claimed to be the necessary ones while working on a geometry of algebroids. This claim is due to the fact that one can prove many desirable properties and relations if one uses only admissible connections. For instance, for admissible connections, we prove the first and second Bianchi identities, Cartan structure equations, Cartan magic formula, the construction of Levi-Civita connections, the decomposition of connection in terms of torsion and non-metricity. These all are possible because the modified bracket becomes anti-symmetric for an admissible connection so that one can apply the machinery of almost- or pre-Lie algebroids. We investigate various algebroid structures from the literature and show that they admit admissible connections which are metric-compatible in some generalized sense. Moreover, we prove that local pre-Leibniz algebroids that are not anti-commutable cannot be equipped with a torsion-free, and in particular Levi-Civita, connection.

Classification of generalized Einstein metrics on three-dimensional Lie groups

AbstractWe develop the theory of left-invariant generalized pseudo-Riemannian metrics on Lie groups. Such a metric accompanied by a choice of left-invariant divergence operator gives rise to a Ricci curvature tensor, and we study the corresponding Einstein equation. We compute the Ricci tensor in terms of the tensors (on the sum of the Lie algebra and its dual) encoding the Courant algebroid structure, the generalized metric, and the divergence operator. The resulting expression is polynomial and homogeneous of degree 2 in the coefficients of the Dorfman bracket and the divergence operator with respect to a left-invariant orthonormal basis for the generalized metric. We determine all generalized Einstein metrics on three-dimensional Lie groups.

Higher multi-Courant algebroids

The binary bracket of a Courant algebroid structure on (E,〈⋅,⋅〉) can be extended to a n-ary bracket on Γ(E), yielding a multi-Courant algebroid. These n-ary brackets form a Poisson algebra and were defined, in an algebraic setting, by Keller and Waldmann. We construct a higher geometric version of Keller-Waldmann Poisson algebra and define higher multi-Courant algebroids. As Courant algebroid structures can be seen as degree 3 functions on a graded symplectic manifold of degree 2, higher multi-Courant structures can be seen as functions of degree n≥3 on that graded symplectic manifold.

Momentum section on Courant algebroid and constrained Hamiltonian mechanics

We propose a generalization of the momentum map on a symplectic manifold with a Lie algebra action to a Courant algebroid structure. The theory of a momentum section on a Lie algebroid is generalized to the theory compatible with a Courant algebroid. As an example, we identify the momentum section in a constrained Hamiltonian mechanics with Courant algebroid symmetry. Moreover, we construct cohomological formulations by considering the BFV and BV formalism of this Hamiltonian system. The Weil algebra for this structure is constructed.

Weyl quantization of degree 2 symplectic graded manifolds

Let S be a spinor bundle of a pseudo-Euclidean vector bundle (E,g) of even rank. We introduce a new filtration on the algebra D(M,S) of differential operators on S. As a main property, the associated graded algebra grD(M,S) is proved to be isomorphic to the algebra O(M) of smooth functions on M, where M is the degree 2 symplectic graded manifold canonically associated to (E,g). Accordingly, we establish the Weyl quantization of M as a map WQħ:O(M)→D(M,S), and prove that WQħ satisfies all the desired properties of quantizations. As an application, we obtain a bijection between Courant algebroid structures (E,g,ρ,〚⋅,⋅〛) equivalently characterized by Hamiltonian generating functions on M, and skew-symmetric Dirac generating operators D∈D(M,S). The operator D2 gives a new invariant of (E,g,ρ,〚⋅,⋅〛) which generalizes the square norm of the Cartan 3-form of a quadratic Lie algebra. We study this invariant in detail in the particular case of E being the double of a Lie bialgebroid (A,A⁎).

Generalization of the concept of classical [formula omitted]-matrix to Lie algebroids

We present some new constructions of Lie algebroids starting from vector fields on manifold M. The tangent bundle TM possess a natural structure of Lie algebroid, but we use these fields to construct a collection of interesting new algebroid structures. Next, we show that these constructions can be used in a more general situation, starting from an arbitrary Lie algebroid over M. In the final step, we show that after limiting ourselves to Lie algebras these formulas as a special case contain brackets well known in theory of classical r-matrices. We can think of our constructions as extending the concept of classical r-matrices to Lie algebroids. Several examples illustrate the importance of these constructions.

Global aspects of doubled geometry and pre-rackoid

The integration problem of a C-bracket and a Vaisman (metric, pre-DFT) algebroid that are geometric structures of double field theory (DFT) is analyzed. We introduce a notion of a pre-rackoid as a global group-like object for an infinitesimal algebroid structure. We propose several realizations of pre-rackoid structures: One realization is that elements of a pre-rackoid are defined by cotangent paths along doubled foliations in a para-Hermitian manifold. Another realization is proposed as a formal exponential map of the algebroid of DFT. We show that the pre-rackoid reduces to a rackoid that is the integration of the Courant algebroid when the strong constraint of DFT is imposed. Finally, for a physical application, we show an implementation of the (pre-)rackoid in a three-dimensional topological sigma model.

The Hopf algebroid structure of differentially recursive sequences

A differentially recursive sequence over a differential field is a sequence of elements satisfying a homogeneous differential equation with non-constant coefficients (namely, Taylor expansions of elements of the field) in the differential algebra of Hurwitz series. The main aim of this paper is to explore the space of all differentially recursive sequences over a given field with a non-zero differential. We show that these sequences form a two-sided vector space that admits, in a canonical way, a structure of Hopf algebroid over the subfield of constant elements. We prove that it is the direct limit, as a left comodule, of all spaces of formal solutions of linear differential equations and that it satisfies, as Hopf algebroid, an additional universal property. When the differential on the base field is zero, we recover the Hopf algebra structure of linearly recursive sequences.

BV and BFV for the H-Twisted Poisson Sigma Model

We present the BFV and the BV extension of the Poisson sigma model (PSM) twisted by a closed 3-form H. There exist superfield versions of these functionals such as for the PSM and, more generally, for the AKSZ sigma models. However, in contrast to those theories, they depend on the Euler vector field of the source manifold and contain terms mixing data from the source and the target manifold. Using an auxiliary connection $$\nabla $$ on the target manifold M, we obtain alternative, purely geometrical expressions without the use of superfields, which are new also for the ordinary PSM and promise adaptations to other Lie algebroid-based gauge theories: The BV functional, in particular, is the sum of the classical action, the Hamiltonian lift of the (only on-shell nilpotent) BRST differential, and a term quadratic in the antifields which is essentially the basic curvature and measures the compatibility of $$\nabla $$ with the Lie algebroid structure on $$T^*M$$ . We finally construct a $$\hbox {Diff}(M)$$ -equivariant isomorphism between the two BV formulations.

The Lie groupoid analogue of a symplectic Lie group

A symplectic Lie group is a Lie group with a left-invariant symplectic form. Its Lie algebra structure is that of a quasi-Frobenius Lie algebra. In this note, we identify the groupoid analogue of a symplectic Lie group. We call the aforementioned structure a \textit{$t$-symplectic Lie groupoid}; the $t$ is motivated by the fact that each target fiber of a $t$-symplectic Lie groupoid is a symplectic manifold. For a Lie groupoid $\mathcal{G}\rightrightarrows M$, we show that there is a one-to-one correspondence between quasi-Frobenius Lie algebroid structures on $A\mathcal{G}$ (the associated Lie algebroid) and $t$-symplectic Lie groupoid structures on $\mathcal{G}\rightrightarrows M$. In addition, we also introduce the notion of a \textit{symplectic Lie group bundle} (SLGB) which is a special case of both a $t$-symplectic Lie groupoid and a Lie group bundle. The basic properties of SLGBs are explored.

Koszul complexes and spectral sequences associated with Lie algebroids

We study some spectral sequences associated with a locally free {{mathscr {O}}}_X-module {{mathscr {A}}} which has a Lie algebroid structure. Here X is either a complex manifold or a regular scheme over an algebraically closed field k. One spectral sequence can be associated with {{mathscr {A}}} by choosing a global section V of {{mathscr {A}}}, and considering a Koszul complex with a differential given by inner product by V. This spectral sequence is shown to degenerate at the second page by using Deligne’s degeneracy criterion. Another spectral sequence we study arises when considering the Atiyah algebroid {{{mathscr {D}}}_{{{mathscr {E}}}}} of a holomolorphic vector bundle {{mathscr {E}}} on a complex manifold. If V is a differential operator on {{mathscr {E}}} with scalar symbol, i.e, a global section of {{{mathscr {D}}}_{{{mathscr {E}}}}}, we associate with the pair ({{mathscr {E}}},V) a twisted Koszul complex. The first spectral sequence associated with this complex is known to degenerate at the first page in the untwisted ({{mathscr {E}}}=0) case.