Let (M,I,J,K) be a compact hypercomplex manifold admitting an HKT-metric. Assume that the canonical bundle of (M,I) is trivial as a holomorphic line bundle. We show that the holonomy of Obata connection on M is contained in SL(n,H). In Appendix we apply these arguments to compact nilmanifolds equipped with abelian hypercomplex structures, showing that such manifolds have holonomy in SL(n,H).
A hypercomplex manifold M is a manifold with a triple I,J,K of complex structure operators satisfying quaternionic relations. For each quaternion L=aI +bJ+cK, L^2=-1, L is also a complex structure operator on M, called an induced complex structure. We are studying compact complex subvarieties of (M,L), when L is a generic induced complex structure. Under additional assumptions (Obata holonomy contained in SL(n,H), existence of an HKT metric), we prove that (M,L) contains no divisors, and all complex subvarieties of codimension 2 are trianalytic (that is, also hypercomplex).
A hypercomplex manifold is a manifold equipped with three complex structures I, J, K satisfying the quaternionic relations. Let M be a 4-dimensional compact smooth manifold equipped with a hypercomplex structure, and E be a vector bundle on M. We show that the moduli space of anti-self-dual connections on E is also hypercomplex, and admits a strong HKT metric. We also study manifolds with (4,4)-supersymmetry, that is, Riemannian manifolds equipped with a pair of strong HKT-structures that have opposite torsion. In the language of Hitchin's and Gualtieri's generalized complex geometry, (4,4)-ma…
A hypercomplex structure on a smooth manifold is a triple of integrable almost complex structures satisfying quaternionic relations. The Obata connection is the unique torsion-free connection that preserves each of the complex structures. The holonomy group of the Obata connection is contained in $GL(n, \mathbb{H})$. There is a well-known construction of hypercomplex structures on Lie groups due to Joyce. In this paper we show that the holonomy of the Obata connection on SU(3) coincides with $GL(2, \mathbb{H})$.
We review some cohomological aspects of complex and hypercomplex manifolds and underline the differences between both realms. Furthermore, we try to highlight the similarities between compact complex surfaces on one hand and compact hypercomplex manifolds of real dimension 8 with holonomy of the Obata connection in SL(2,H) on the other hand.
A nilmanifold is a quotient of a nilpotent group $G$ by a co-compact discrete subgroup. A complex nilmanifold is one which is equipped with a $G$-invariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle. This is used to study hypercomplex nilmanifolds (nilmanifolds with a triple of $G$-invariant complex structures which satisfy quaternionic relations). We prove that a hypercomplex nilmanifold admits an HKT (hyperkahler with torsion) metric if and only if the underlying hypercomplex structure is abelian. Moreover, any $G$-invariant HKT-metric on a nilmanifol…
The tangent bundle of an almost Norden manifold and the complete lift of the Norden metric is considered as a 4n-manifold. It is equipped with an almost hypercomplex Hermitian-Norden structure. It is characterized geometrically. The case when the base manifold is an h-sphere is considered.
A twist construction for manifolds with torus action is described generalising certain T-duality examples and constructions in hypercomplex geometry. It is applied to complex, SKT, hypercomplex and HKT manifolds to construct compact simply-connected examples. In particular, we find hypercomplex manifolds that admit no compatible HKT metric, and HKT manifolds whose Obata connection has holonomy contained in $SL(n,\mathbb H)$.
The tangent bundle as a $4n$-manifold is equipped with an almost hypercomplex pseudo-Hermitian structure and it is characterized with respect to the relevant classifications. A number of 8-dimensional examples of the considered type of manifold are received from the known explicit examples in that manner.
A hypercomplex manifold $M$ is a manifold equipped with three complex structures satisfying quaternionic relations. Such a manifold admits a canonical torsion-free connection preserving the quaternion action, called Obata connection. A quaternionic Hermitian metric is a Riemannian metric on which is invariant with respect to unitary quaternions. Such a metric is called HKT if it is locally obtained as a Hessian of a function averaged with quaternions. HKT metric is a natural analogue of a Kahler metric on a complex manifold. We push this analogy further, proving a quaternionic analogue of Buch…
A hypercomplex manifold is a manifold equipped with a triple of complex structures satisfying the quaternionic relations. A holomorphic Lagrangian variety on a hypercomplex manifold with trivial canonical bundle is a holomorphic subvariety which is calibrated by a form associated with the holomorphic volume form; this notion is a generalization of the usual holomorphic Lagrangian subvarieties known in hyperkaehler geometry. An HKT (hyperkaehler with torsion) metric on a hypercomplex manifold is a metric determined by a local potential, in a similar way to the Kaehler metric. We prove that a ba…
We study compact Sasakian manifolds whose Tondeur connection has holonomy group either trivial or contained in Sp(n). We show that the first condition forces the manifold to be a compact quotient of the Heisenberg Lie group, while in the simply-connected case a Sasakian structure has the holonomy of the Tondeur connection contained in Sp(n) if and only if there exists a transverse hypercomplex structure. This latter result is the "Sasakian version" of a theorem of Verbitsky.
In this paper we provide examples of hypercomplex manifolds which do not carry HKT structure. We also prove that the existence of HKT structure is not stable under small deformations. Similarly we provide examples of compact complex manifolds with vanishing first Chern class which do not admit a Hermitian structure with restricted holonomy of its Bismut connection in SU(n). Again we prove that such property is not stable under small deformations.
In this paper we show how hypercomplex function theoretical objects can be used to construct explicitly self-dual SU(2)-Yang-Mills instanton solutions on certain classes of conformally flat 4-manifolds. We use a hypercomplex argument principle to establish a natural link between the fundamental solutions of $D Δf = 0$ and the second Chern class of the SU(2) principal bundles over these manifolds. The considered base manifolds of the bundles are not simply-connected, in general. Actually, this paper summarizes an extension of the corresponding results of Gürsey and Tze on a hyper-complex analyt…
Unit tangent bundles $UM$ of semi-Riemannian manifolds $M$ are shown to be examples of dynamical Legendrian contact structures, which were defined in recent work [25] of Sykes-Zelenko to generalize leaf spaces of 2-nondegenerate CR manifolds. In doing so, Sykes-Zelenko extended the classification in Porter-Zelenko [20] of regular, 2-nondegenerate CR structures to those that can be recovered from their leaf space. The present paper treats dynamical Legendrian contact structures associated with 2-nondegenerate CR structures which were called "strongly regular" in Porter-Zelenko, named "L-contact…
We give a new construction of the holonomy groupoid of a regular foliation in terms of a partial connection on a diffeological principal bundle of germs of transverse parametrisations. We extend these ideas to construct a novel holonomy groupoid for any foliated bundle, which we prove sits at the top of a hierarchy of diffeological jet holonomy groupoids associated to the foliated bundle.
We show that on an HKT manifold the holonomy of the Obata connection is contained in SL(n,H) if and only if the Lee form is an exact one form. As an application, we show compact HKT manifolds with holomorphically trivial canonical bundle which are not balanced. A simple criterion for non-existence of HKT metric on hypercomplex manifold is given in terms of the Ricci-type tensors of the Obata connection.
We define the equivariant holonomy of an invariant connection on a principal U(1)-bundle. The properties of the ordinary holonomy are generalized to the equivariant setting. In particular, equivariant U(1)-bundles with connection are shown to be classified by its equivariant holonomy modulo isomorphisms. We also show that the equivariant holonomy can be used to obtain results about equivariant prequantization and anomaly cancellation.
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 investigate geometric properties of indecomposable but non-irreducible Lorentzian manifolds, which are total spaces of circle bundles. We investigate under which conditions these manifolds are complete and give examples which fulfill the obtained conditions. In particular we investigate the Einstein equation for these spaces yielding examples for complete compact Ricci flat Lorentzian manifolds and manifolds with timelike Killing vector fields. Finally we study their holonomy and obtain in particular complete examples for Lorentzian manifolds with holonomy of so called type 4.
This paper is concerned with the holonomy of a class of spaces which includes Landsberg spaces of Finsler geometry. The methods used are those of Lie groupoids and algebroids as developed by Mackenzie. We prove a version of the Ambrose-Singer Theorem for such spaces. The paper ends with a discussion of how the results may be extended to Finsler spaces and homogeneous nonlinear connections in general.
We completely explore the system of ODE's which is equivalent to the existence of a parallel $Spin(7)$-structure on the cone over a 7-dimensional 3-Sasakian manifold. The one-dimensional family of solutions of this system is constructed. The solutions of this family correspond to metrics with holonomy SU(4) which generalize the Calabi metrics.
We define a notion of connection in a fibre bundle that is compatible with a singular foliation of the base. Fibre bundles equipped with such connections are in plentiful supply, arising naturally for any Lie groupoid-equivariant bundle, and simultaneously generalising regularly foliated bundles in the sense of Kamber-Tondeur and singular foliations. We define hierarchies of diffeological holonomy groupoids associated to such bundles, which arise from the parallel transport of jet/germinal conservation laws. We show that the groupoids associated in this manner to trivial singularly foliated bu…
We show that the holonomy invariance of a function on the tangent bundle of a manifold, together with very mild regularity conditions on the function, is equivalent to the existence of local parallelisms compatible with the function in a natural way. Thus, in particular, we obtain a characterization of generalized Berwald manifolds. We also construct a simple example of a generalized Berwald manifold which is not Berwald.
Tubular neighborhoods play an important role in differential topology. We have applied these constructions to geometry of almost Hermitian manifolds. At first, we consider deformations of tensor structures on a normal tubular neighborhood of a submanifold in a Riemannian manifold.Further, an almost hyperHermitian structure has been constructed on the tangent bundle TM with help of the Riemannian connection of an almost Hermitian structure on a manifold M then, we consider an embedding of the almost Hermitian manifold M in the corresponding normal tubular neighborhood of the null section in the…
This paper is a continuation of math.DG/0408005. We first construct special Lagrangian submanifolds of the Ricci-flat Stenzel metric (of holonomy SU(n)) on the cotangent bundle of S^n by looking at the conormal bundle of appropriate submanifolds of S^n. We find that the condition for the conormal bundle to be special Lagrangian is the same as that discovered by Harvey-Lawson for submanifolds in R^n in their pioneering paper. We also construct calibrated submanifolds in complete metrics with special holonomy G_2 and Spin(7) discovered by Bryant and Salamon on the total spaces of appropriate bun…
For the ``Hopf bundle'' $S^1\to S^{2n,1} \to {\mathbb H}^n$, horizontal lifts of simple closed curves are studied. Let $γ$ be a piecewise smooth, simple closed curve on a complete totally geodesic surface $S$ in the base space. Then the holonomy displacement along $γ$ is given by $$ V(γ)=e^{λA(γ) i} $$ where $A(γ)$ is the area of the region on the surface $S$ surrounded by $γ$; $λ=1/2 $ or 0 depending on whether $S$ is a complex submanifold or not. We also carry out a similar investigation for the complex Heisenberg group ${\mathbb R} \to {\mathcal H}^{2n+1} \to {\mathbb C}^n$.
This paper is a step towards realizing T-duality and Hori formulae for loop spaces. Here we prove T-duality and Hori formulae for winding q-loop spaces, which are infinite dimensional subspaces of loop spaces.
Hermitian bundle gerbes with connection are geometric objects for which a notion of surface holonomy can be defined for closed oriented surfaces. We systematically introduce bundle gerbes by closing the pre-stack of trivial bundle gerbes under descent. Inspired by structures arising in a representation theoretic approach to rational conformal field theories, we introduce geometric structure that is appropriate to define surface holonomy in more general situations: Jandl gerbes for unoriented surfaces, D-branes for surfaces with boundaries, and bi-branes for surfaces with defect lines.
Our main aim is to associate a holonomy Lie groupoid to the connective structure of an abelian gerbe. The construction has analogies with a procedure for the holonomy Lie groupoid of a foliation, in using a locally Lie groupoid and a globalisation procedure. We show that path connections and 2-holonomy on line bundles may be formulated using the notion of a connection pair on a double category, due to Brown-Spencer, but now formulated in terms of double groupoids using the thin fundamental groupoids introduced by Caetano-Mackaay-Picken. To obtain a locally Lie groupoid to which globalisation a…
We show that the Igusa-Klein topological torsion and the Bismut-Lott analytic torsion are equivalent for any flat vector bundle whose holonomy is a finite subgroup of $\mathrm{GL}_n(\mathbb{Q})$. Our proof uses Artin's induction theorem in representation theory to reduce the problem to the special case of trivial flat line bundles, which is a recent result of Puchol, Zhu and the second author. The idea of using Artin's induction theorem appeared in a paper of Ohrt on the same topic, of which our present work is an improvement.
An associated Nijenhuis tensor of endomorphisms in the tangent bundle is introduced. Special attention is paid to such tensors for an almost hypercomplex structure and the metric of Hermitian-Norden type. There are studied relations between the six associated Nijenhuis tensors as well as their vanishing. It is given a geometric interpretation of the vanishing of these tensors as a necessary and sufficient condition for the existence of a unique connection with totally skew-symmetric torsion tensor. Similar idea is used in the paper of T. Friedrich and S. Ivanov in Asian J. Math. (2002) for som…
We construct examples of compact hyperkaehler manifolds with torsion (HKT manifolds) which are not homogeneous and not locally conformal hyperkaehler. Consider a total space T of a tangent bundle over a hyperkaehler manifold M. The manifold T is hypercomplex, but it is never hyperkaehler, unless M is flat. We show that T admits an HKT-structure. We also prove that a quotient of T by a $\Z$-action $v \arrow q^n v$ is HKT, for any real number $q\in \R$, $q>1$. This quotient is compact, if M is compact. A more general version of this construction holds for all hyperholomorphic bundles with holono…
The Wess-Zumino term in two-dimensional conformal field theory is best understood as a surface holonomy of a bundle gerbe. We define additional structure for a bundle gerbe that allows to extend the notion of surface holonomy to unoriented surfaces. This provides a candidate for the Wess-Zumino term for WZW models on unoriented surfaces. Our ansatz reproduces some results known from the algebraic approach to WZW models.
An SU(3)- or SU(1,2)-structure on a 6-dimensional manifold N^6 can be defined as a pair of a 2-form omega and a 3-form rho. We prove that any analytic SU(3)- or SU(1,2)-structure on N^6 with d omega^2 =0 can be extended to a parallel Spin(7)- or Spin_0(3,4)-structure Phi that is defined on the trivial disc bundle N^6\times B_epsilon(0) for a sufficiently small epsilon>0. Furthermore, we show by an example that Phi is not uniquely determined by (omega,rho) and discuss if our result can be generalized to non-trivial bundles.
Recently twisted K-theory has received much attention due to its applications in string theory and the announced result by Freed, Hopkins and Telemann relating the twisted equivariant K-theory of a compact Lie group to its Verlinde algebra. Rather than considering gerbes as separate objects, in twisted K-theory one considers a gerbe as being part of the data for a twisted vector bundle. There is also a notion of a connection in a twisted vector bundle and Kapustin has studied some aspects of the holonomy of such connections. In this note I study the holonomy of connections in twisted principal…
The motivation for this paper stems \cite{CR} from the need to construct explicit isomorphisms of (possibly nontrivial) principal $G$-bundles on the space of loops or, more generally, of paths in some manifold $M$, over which I consider a fixed principal bundle $P$; the aforementioned bundles are then pull-backs of $P$ w.r.t. evaluation maps at different points. The explicit construction of these isomorphisms between pulled-back bundles relies on the notion of {\em parallel transport}. I introduce and discuss extensively at this point the notion of {\em generalized gauge transformation between…
Given a hypercomplex manifold with a rotating vector field (and additional data), we construct a conical hypercomplex manifold. As a consequence, we associate a quaternionic manifold to a hypercomplex manifold of the same dimension with a rotating vector field. This is a generalization of the HK/QK-correspondence. As an application, we show that a quaternionic manifold can be associated to a conical special complex manifold of half its dimension. Furthermore, a projective special complex manifold (with a canonical c-projective structure) associates with a quaternionic manifold. The latter is a…
A pp-wave is a Lorentzian manifold with a parallel light-like vector field satisfying a certain curvature condition. We introduce generalisations of pp-waves, on one hand by allowing the vector field to be recurrent and on the other hand by weakening the curvature condition. These generalisations are related to the screen holonomy of the Lorentzian manifold. While pp-waves have a trivial screen holonomy there are no restrictions on the screen holonomy of the manifolds with the weaker curvature condition.
We introduce horizontal holonomy groups, which are groups defined using parallel transport only along curves tangent to a given subbundle $D$ of the tangent bundle. We provide explicit means of computing these holonomy groups by deriving analogues of Ambrose-Singer's and Ozeki's theorems. We then give necessary and sufficient conditions in terms of the horizontal holonomy groups for existence of solutions of two problems on foliated manifolds: determining when a foliation can be either (a) totally geodesic or (b) endowed with a principal bundle structure. The subbundle $D$ plays the role of an…
We study the uniqueness of minimal submanifolds and the stability of the mean curvature flow in several well-known model spaces of manifolds of special holonomy. These include the Stenzel metric on the cotangent bundle of spheres, the Calabi metric on the cotangent bundle of complex projective spaces, and the Bryant--Salamon metrics on vector bundles over certain Einstein manifolds. In particular, we show that the zero sections, as calibrated submanifolds with respect to their respective ambient metrics, are unique among compact minimal submanifolds and are dynamically stable under the mean cu…
We construct propose an anzatz for Spin(7) metrics as an R-bundle over closed G2 structures. These G2 structures are R3 bundles over 4-dimensional compact quaternion Kahler spaces. The inspiration for the anzatz metric comes from the Bryant-Salamon construction of G2 holonomy metrics and from the fact that the twistor space of any compact quaternion Kahler space is Kahler-Einstein. The reduction of the holonomy to a subgroup of Spin(7) gives non linear system relating three unknown functions of one variable. We obtain a particular solution and we find that the resulting metric is a Calabi-Yau …
This paper generalizes Bismut's equivariant Chern character to the setting of abelian gerbes. In particular, associated to an abelian gerbe with connection, an equivariantly closed differential form is constructed on the space of maps of a torus into the manifold. These constructions are made explicit using a new local version of the higher Hochschild complex, resulting in differential forms given by iterated integrals. Connections to two dimensional topological field theories are indicated. Similarly, this local higher Hochschild complex is used to calculate the 2-holonomy of an abelian gerbe…
We construct complete noncompact Riemannian metrics with $G_2$-holonomy on noncompact orbifolds that are $\Bbb R^3$-bundles with the twistor space $\mathcal{Z}$ as a spherical fiber.
Given a principal bundle with a connection, we look for an asymptotic expansion of the holonomy of a loop in terms of its length. This length is defined relative to some Riemannian or sub-Riemannian structure. We are able to give an asymptotic formula that is independent of choice of gauge.
We prove that Fefferman spaces, associated to non--degenerate CR structures of hypersurface type, are characterised, up to local conformal isometry, by the existence of a parallel orthogonal complex structure on the standard tractor bundle. This condition can be equivalently expressed in terms of conformal holonomy. Extracting from this picture the essential consequences at the level of tensor bundles yields an improved, conformally invariant analogue of Sparling's characterisation of Fefferman spaces.
We consider a class of $S^{1}$-bundles whose total space admits a nowhere vanishing recurrent lightlike vector field with respect to a Lorentzian metric. This metric can be modified such that its restricted holonomy group is indecomposable and reducible. We apply Hodge theory to construct examples with Hermitian screen holonomy. Finally, we construct complete pp-waves.
Consider the principal $U(n)$ bundles over Grassmann manifolds $U(n)\rightarrow U(n+m)/U(m) \stackrelπ\rightarrow G_{n,m}$. Given $X \in U_{m,n}(\mathbb{C})$ and a 2-dimensional subspace $\mathfrak{m}' \subset \mathfrak{m} $ $ \subset \mathfrak{u}(m+n), $ assume either $\mathfrak{m}'$ is induced by $X,Y \in U_{m,n}(\mathbb{C})$ with $X^{*}Y = μI_n$ for some $μ\in \mathbb{R}$ or by $X,iX \in U_{m,n}(\mathbb{C})$. Then $\mathfrak{m}'$ gives rise to a complete totally geodesic surface $S$ in the base space. Furthermore, let $γ$ be a piecewise smooth, simple closed curve on $S$ parametrized by $0\…
We describe the infinitesimal moduli space of pairs $(Y, V)$ where $Y$ is a manifold with $G_2$ holonomy, and $V$ is a vector bundle on $Y$ with an instanton connection. These structures arise in connection to the moduli space of heterotic string compactifications on compact and non-compact seven dimensional spaces, e.g. domain walls. Employing the canonical $G_2$ cohomology developed by Reyes-Carrión and Fernández and Ugarte, we show that the moduli space decomposes into the sum of the bundle moduli $H^1_{\check{d}_A}(Y,\mathrm{End}(V))$ plus the moduli of the $G_2$ structure preserving the i…
Narasimhan and Ramadas showed that the Gribov ambiguity was maximal for the product SU(2) bundle over S^3. Specifically they showed that the holonomy group of the Coulomb connection is dense in the connected component of the identity of the gauge group. Instead of base manifold S^3, we consider here a base manifold with a boundary. In this with-boundary case we must include boundary conditions on the connection forms. We will use the so-called conductor (a.k.a Dirichlet) boundary conditions on connections. With these boundary conditions, we will first show that the space of connections is a C^…
A classic theorem in the theory of connections on principal fiber bundles states that the evaluation of all holonomy functions gives enough information to characterize the bundle structure (among those sharing the same structure group and base manifold) and the connection up to a bundle equivalence map. This result and other important properties of holonomy functions has encouraged their use as the primary ingredient for the construction of families of quantum gauge theories. However, in these applications often the set of holonomy functions used is a discrete proper subset of the set of holon…
Let $π:P\to B$ be a smooth $G$-bundle over a compact Riemannian manifold $B$ and $c$ a smooth loop in $B$ of constant seed $a(>0)$, where $G$ is compact semi-simple Lie group. In this paper, we prove that the holonomy map ${\rm hol}_c:\mathcal A_P^{H^s}\to G$ is a homothetic submersion of coefficient $a$, where $s$ is a non-negative integer, $\mathcal A_P^{H^s}$ is the Hilbert space of all $H^s$-connections of the bundle $P$. In particular, we prove that, if $s=0$, then ${\rm hol}_c$ has minimal regularizable fibres. From this fact, we can derive that each component of the inverse image of any…
