A generalised notion of connection on a fibre bundle E over a manifold M is presented. These connections are characterised by a smooth distribution on E which projects onto a (not necessarily integrable) distribution on M and which, in addition, is `parametrised' in some specific way by a vector bundle map from a prescribed vector bundle over M into TM. Some basic properties of these generalised connections are investigated. Special attention is paid to the class of linear connections over a vector bundle map. It is pointed out that not only the more familiar types of connections encountered in the literature, but also the recently studied Lie algebroid connections, can be recovered as special cases within this more general framework.
This paper studies linear generalised complex structures over vector bundles, as a generalised geometry version of holomorphic vector bundles. In an adapted linear splitting, a linear generalised complex structure on a vector bundle $E\to M$ is equivalent to a $\mathbb C$-multiplication $j$ in the fibers of $TM\oplus E^*$ and $\mathbb C$-Lie algebroid structure on $TM\oplus E^*$. Generalised complex Lie algebroids (or Glanon algebroids) are then studied in this context, and expressed as a pair of complex conjugated Lie bialgebroids.
We develop an alternative view on the concept of connections over a vector bundle map, which consists of a horizontal lift procedure to a prolonged bundle. We further focus on prolongations to an affine bundle and introduce the concept of affineness of a generalised connection.
An analogue of the correspondence between GL(k)-conjugacy classes of matricial polynomials and line bundles is given for K-conjugacy classes, where K is one of the following: maximal parabolic, maximal torus, GL(k-1) embedded diagonally. The generalised Legendre transform construction of hyperkaehler metrics is studied further, showing that many known hyperkaehler metrics (including the ones on coadjoint orbits) arise in this way, and giving a large class of new (pseudo-)hyperkaehler metrics, analogous to monopole metrics.
We study the existence of a natural `linearisation' process for generalised connections on an affine bundle. It is shown that this leads to an affine generalised connection over a prolonged bundle, which is the analogue of what is called a connection of Berwald type in the standard theory of connections. Various new insights are being obtained in the fine structure of affine bundles over an anchored vector bundle and affineness of generalised connections on such bundles.
This is a review article on some applications of generalised parabolic structures to the study of torsion free sheaves and $L$-twisted Hitchin pairs on nodal curves. In particular, we survey on the relation between representations of the fundamental group of a nodal curve and the moduli spaces of generalised parabolic bundles and generalised parabolic $L$-twisted Hitchin pairs on its normalisation as well as on an analogue of the Hitchin map for generalised parabolic $L$-twisted Hitchin pairs.
Adjusting conventional Chern-Simons theory to ${\rm G}_2$-manifolds, one describes ${\rm G}_2$-instantons on bundles over a certain class of $7$-dimensional flat tori which fiber non-trivially over $T^4$, by a pullback argument. Moreover, if $c_2\neq0$, any (generic) deformation of the ${\rm G}_2$-structure away from such a fibred structure causes all instantons to vanish. A brief investigation in the general context of (conformally compatible) associative fibrations $f:Y^7\to X^4$ relates ${\rm G}_2$-instantons on pullback bundles $f^*E\to Y$ and self-dual connections on the bundle $E\to X$ o…
In general terms, Gelfand duality refers to a correspondence between a geometric, topological, or analytical category, and an algebraic category. For example, in smooth differential geometry, Gelfand duality refers to the topological embedding of a smooth manifold in the topological dual of its algebra of smooth functions. This is generalised here in two directions. First, the topological embeddings for manifolds are generalised to the cases of real analytic and Stein manifolds, using a unified cohomological argument. Second, this type of duality is extended to vector bundles, affine bundles, …
In this short paper, we will review the proposal of a correspondence between the doubled geometry of Double Field Theory and the higher geometry of bundle gerbes. Double Field Theory is T-duality covariant formulation of the supergravity limit of String Theory, which generalises Kaluza-Klein theory by unifying metric and Kalb-Ramond field on a doubled-dimensional space. In light of the proposed correspondence, this doubled geometry is interpreted as an atlas description of the higher geometry of bundle gerbes. In this sense, Double Field Theory can be interpreted as a field theory living on th…
We consider various generalisations of the string class of a loop group bundle. The string class is the obstruction to lifting a bundle whose structure group is the loop group $LG$ to one whose structure group is the Kac-Moody central extension of the loop group. We develop a notion of higher string classes for bundles whose structure group is the group of based loops, $ΩG$. In particular, we give a formula for characteristic classes in odd dimensions for such bundles which are associated to characteristic classes for $G$-bundles in the same way that the string class is related to the first Po…
We consider smooth families of Lie groups (group bundles) and connections that are compatible with the group operation. We characterize the space of group connections on a group bundle as an affine space modeled over the vector space of $1$-forms with values cocycles in the Lie algebra bundle of the aforementioned group bundle. We show that group connections satisfy the Ambrose-Singer theorem and that group bundles can be seen as a particular case of associated bundles realizing group connections as associated connections. We give a construction of the Moduli space of group connections with fi…
In this work, generalized principal bundles modelled by Lie group bundle actions are investigated. In particular, the definition of equivariant connections in these bundles, associated to Lie group bundle connections, is provided, together with the analysis of their existence and their main properties. The final part gives some examples. In particular, since this research was initially originated by some problems on geometric reduction of gauge field theories, we revisit the classical Utiyama Theorem from the perspective investigated in the article.
Given a complex manifold $M$ equipped with a holomorphic action of a connected complex Lie group $G$, and a holomorphic principal $H$--bundle $E_H$ over $X$ equipped with a $G$--connection $h$, we investigate the connections on the principal $H$--bundle $E_H$ that are (strongly) adapted to $h$. Examples are provided by holomorphic principal $H$--bundles equipped with a flat partial connection over a foliated manifold.
In this paper we introduce a generalisation of the notion of holonomy for connections over a bundle map on a principal fibre bundle. We prove that, as in the standard theory on principal connections, the holonomy groups are Lie subgroups of the structure group of the principle fibre bundle and we also derive a straightforward generalisation of the Reduction Theorem.
We generalize reduction theorems for classical connections to operators with values in $k$-th order natural bundles. Using the first reduction theorem in order two we classify all (0,2)-tensor fields on the cotangent bundle of a manifold with a linear (non-symmetric) connection.
From a certain strongly equivariant bundle gerbe with connection and curving over a smooth manifold on which a Lie group acts, we construct under some conditions a bundle gerbe with connection and curving over the quotient space. In general, the construction requires a choice, and we can consequently obtain distinct stable isomorphism classes of bundle gerbes with connection and curving over the quotient space. A bundle gerbe naturally arising in Chern-Simons theory provides an example of the reduction.
We construct a connection and a curving on a bundle gerbe associated with lifting a structure group of a principal bundle to a central extension. The construction is based on certain structures on the bundle, i.e. connections and splittings. The Deligne cohomology class of the lifting bundle gerbe with the connection and with the curving coincides with the obstruction class of the lifting problem with these structures.
We give a differential geometric description of the Cartan (or tractor) bundle and its canonical connection in CR geometry, thus offering a direct, alternative, definition to the usual abstract approach.
A new construction of a universal connection was given in \cite{BHS}. The main aim here is to explain this construction. A theorem of Atiyah and Weil says that a holomorphic vector bundle $E$ over a compact Riemann surface admits a holomorphic connection if and only if the degree of every direct summand of $E$ is degree. In \cite{AB}, this criterion was generalized to principal bundles on compact Riemann surfaces. This criterion for principal bundles is also explained.
Around 1923, Elie Cartan introduced affine connections on manifolds and definedthe main related concepts: torsion, curvature, holonomy groups. He discussed applications of these concepts in Classical and Relativistic Mechanics; in particular he explained how parallel transport with respect to a connection can be related to the principle of inertia in Galilean Mechanics and, more generally, can be used to model the motion of a particle in a gravitational field. In subsequent papers, Elie Cartan extended these concepts for other types of connections on a manifold: Euclidean, Galilean and Minkows…
We construct Yang-Mills connections on SO(n)-bundles over spheres equipped with the Euclidean metric. We use a cohomogeneity one group action on the bundle to reduce the Yang-Mills-equation to a system of ordinary differential equations. The system is shown to have solutions by variational methods, using ideas from harmonic map theory. Examples include Yang-Mills connections on each of the countably many principal SO(6)-bundles over $S^6$, and countably many Yang-Mills connections on $TS^n$ for $n\in\{5,...,9\}$.
We characterize all LVMB manifolds X such that the holomorphic tangent bundle TX is spanned at the generic point by a family of global holomorphic vector fields, each of them having non-empty zero locus. We deduce that holomorphic connections on semi-stable holomorphic vector bundles over LVMB manifolds with this previous property are always flat.
The aim of this article is to proof a necessary and sufficient condition for the existence of a Cartan connection on a principal bundle. After collecting the essentially well known facts to fix the terminology, soldering forms and geometrizable principal bundles are defined to finally prove the existence criterion.
We show how the amplitude of holonomies on a vector bundle can be controlled by the integral of the curvature of the connection on a surface enclosed by the curve.
In this paper we present a method by which is obtained a sequence of $k$-semisprays and two sequences of nonlinear connections on the $k$-tangent bundle $T^kM$, starting from a given one. Interesting particular cases appear for Lagrange and Finsler spaces of order $k$.
We investigate G-invariant symplectic structures on the cotangent bundle T*P of a principal G-bundle P(M,G) which are canonically related to automorphisms of the tangent bundle TP covering the identity map of P and commuting with the action of TG on TP. The symplectic structures corresponding to connections on P(M,G) are also investigated. The Marsden-Weinstein reduction procedure for these symplectic structures is discussed.
Adopting the global approach to tangent bundles of order two established in[1], we develop this approach to find new results. We also generalize various results of [3], [4] and [6] to the geometry of tangent bundles of order two.
A diffeological connection on a diffeological vector pseudo-bundle is defined just the usual one on a smooth vector bundle; this is possible to do, because there is a standard diffeological counterpart of the cotangent bundle. On the other hand, there is not yet a standard theory of tangent bundles, although there are many suggested and promising versions, such as that of the internal tangent bundle, so the abstract notion of a connection on a diffeological vector pseudo-bundle does not automatically provide a counterpart notion for Levi-Civita connections. In this paper we consider the dual o…
In this article we consider the continuity of the eigenvalues of the connection Laplacian of $G$-connections on vector bundles over Riemannian manifolds. To show it, we introduce the notion of the asymptotically $G$-equivariant measured Gromov-Hausdorff topology on the space of metric measure spaces with isometric $G$-actions, and apply it to the total spaces of principal $G$-bundles equipped with $G$-connections over Riemannian manifolds.
In this article we present a study of the subspaces of the manifold OscM, the total space of the osculator bundle of a real manifold M. We obtain the induced connections of the canonical metrical N-linear connection determined by the homogeneous prolongation of a Finsler metric to the manifold OscM. We present the relation between the induced and intrinsic geometric objects of the associated osculator submanifold.
Connections on principal bundles play a fundamental role in expressing the equations of motion for mechanical systems with symmetry in an intrinsic fashion. A discrete theory of connections on principal bundles is constructed by introducing the discrete analogue of the Atiyah sequence, with a connection corresponding to the choice of a splitting of the short exact sequence. Equivalent representations of a discrete connection are considered, and an extension of the pair groupoid composition, that takes into account the principal bundle structure, is introduced. Computational issues, such as the…
Generalised geometry studies structures on a d-dimensional manifold with a metric and 2-form gauge field on which there is a natural action of the group SO(d,d). This is generalised to d-dimensional manifolds with a metric and 3-form gauge field on which there is a natural action of the group $E_{d}$. This provides a framework for the discussion of M-theory solutions with flux. A different generalisation is to d-dimensional manifolds with a metric, 2-form gauge field and a set of p-forms for $p$ either odd or even on which there is a natural action of the group $E_{d+1}$. This is useful for ty…
Motivated to study the geometry of the exotic spheres constructed in [5], we derive a necessary condition for non-negative sectional curvature in certain total spaces of Riemannian submersions with totally geodesic fibers. In particular, we prove that the bundles in [5] and [1] have sections of negative curvature.
Multiplicative bundle gerbes are gerbes over a Lie group which are compatible with the group structure. In this article connections on such bundle gerbes are introduced and studied. It is shown that multiplicative bundle gerbes with connection furnish geometrical constructions of the following objects: smooth central extensions of loop groups, Chern-Simons actions for arbitrary gauge groups, and symmetric bi-branes for WZW models with topological defect lines.
Local connection forms provide a very useful tool for handling connections on principal bundles, because they ignore any complexities of the total space and, essentially, involve only two fundamental features of the structure group, namely the adjoint representation and the left (logarithmic) differential. The main results of this note characterize connections related together by bundle morphisms, while applications (taken from various sources) refer to connections on (Banach) associated bundles, in particular vector bundles, and connections on inverse limit bundles (in the Fréchet framework).…
Let $FM,\mathcal{M}_M$ be the bundles of linear frames and Riemannian metrics of a manifold $M$, respectively. The existence of a unique $\mathrm{Diff}M$-invariant connection form on $J^1\mathcal{M}_M\times_MFM\to J^1\mathcal{M}_M$, which is Riemannian with respect to the universal metric on $J^1\mathcal{M}_M\times_MTM$, is proved. Aplications to the construction of universal Pontryagin and Euler forms, are given.
Narasihman and Ramanan proved that an arbitrary connection in a vector bundle over a base space B can be obtained as the pull-back (via a correctly chosen classifying map from B into the appropriate Grassmannian) of the universal connection in the universal bundle over the Grassmannian. The purpose of this paper is to relate geometric properties of the classifying map to geometric properties of the pulled-back connection. More specifically, we describe conditions on the classifying map under which the pulled-back connection: (1) is fat (in the sphere bundle), (2) has a parallel curvature tenso…
In this paper, we will provide an alternative definition for the singular Hermitian metric on a vector bundle. Moreover, we discuss the Griffiths and Nakano positivities under this circumstance and prove a generalised Griffiths' vanishing theorem.
We show that generalised geometry gives a unified description of bosonic eleven-dimensional supergravity restricted to a $d$-dimensional manifold for all $d\leq7$. The theory is based on an extended tangent space which admits a natural $E_{d(d)} \times \mathbb{R}^+$ action. The bosonic degrees of freedom are unified as a "generalised metric", as are the diffeomorphism and gauge symmetries, while the local $O(d)$ symmetry is promoted to $H_d$, the maximally compact subgroup of $E_{d(d)}$. We introduce the analogue of the Levi--Civita connection and the Ricci tensor and show that the bosonic act…
This is the second in a series of papers discussing in the framework of gerbe theory canonical and geometric aspects of the 2d nonlinear sigma model in the presence of conformal defects in the worldsheet. Employing the formal tools worked out in the first paper of the series, 1101.1126 [hep-th], a thorough analysis of rigid symmetries of the sigma model is carried out with emphasis on algebraic structures on generalised tangent bundles over the target space of the theory and over its state space that give rise to a realisation of the symmetry algebra on states. The analysis leads to a proposal…
The toral rank conjecture speculates that the sum of the Betti numbers of a compact manifold admitting a free action of a torus of rank $r$ is bounded from below by $2^r$. Clearly, such an action yields a torus bundle, and, more generally, the same cohomological bound is conjectured for total spaces of suitable topological torus fibrations by Félix--Oprea--Tanré. In this article we show that this generalised toral rank conjecture cannot hold by providing various different counter-examples to it (for each rank $r\geq 5$). In particular, we show that there are sequences of smooth nilpotent fibre…
We reformulate ten-dimensional type II supergravity as a generalised geometrical analogue of Einstein gravity, defined by an $O(9,1)\times O(1,9)\subset O(10,10)\times\mathbb{R}^+$ structure on the generalised tangent space. Using the notion of generalised connection and torsion, we introduce the analogue of the Levi-Civita connection, and derive the corresponding tensorial measures of generalised curvature. We show how, to leading order in the fermion fields, these structures allow one to rewrite the action, equations of motion and supersymmetry variations in a simple, manifestly $\mathit{Spi…
Gravity and generalised geometry from a Lie 2-algebroid perspective
Rudjer Bošković Institute（鲁德·博什科维奇研究所）；Imperial College（帝国理工学院）；London Institute for Mathematical Sciences, Royal Institution（伦敦数学科学研究所，皇家学会）；Dublin Institute for Advanced Studies（都柏林高等研究院）；Galatasaray Üniversitesi（加拉塔萨雷大学）；Constructor University（康斯特克特大学）
We put together some of the efforts by several people of making aspects of fibre bundle theory into algebra. The initiator of these efforts was Charles Ehresmann, who put the notion of groupoid and groupoid action in the focus of fibre bundle theory in general, and in connection theory in particular.
We construct and study general connections on Lie groupoids and differentiable stacks as well as on principal bundles over them using Atiyah sequences associated to transversal tangential distributions.
The goal of this paper is to introduce the lifting theory that has an important role in geometry. Therefore, using the lifts of differential geometric structures we show that tangent bundle TM of paracomplex manifold M admits para-complex torsion-free affine connection.
A linear connection is associated to a nonlinear connection on a vector bundle by a linearization procedure. Our definition is intrinsic in terms of vector fields on the bundle. For a connection on an affine bundle our procedure can be applied after homogenization and restriction. Several applications in Classical Mechanics are provided.
Let $X$ be a smooth projective variety over $\mathbb C$. We prove that a twisted Higgs vector bundle $(\calE\, ,θ)$ on $X$ admits an Einstein--Hermitian connection if and only if $(\calE\, ,θ)$ is polystable. A similar result for twisted vector bundles (no Higgs fields) was proved by S. Wang in \cite{Wa}. Our approach is simpler.
