(This is a revised version of the paper) - In the present paper we study the geometry of doubly extended Lie groups with their natural biinvariant metric. We describe the curvature, the holonomy and the space of parallel spinors. This is completely done for all simply connected groups with biinvariantmetric of Lorentzian signature $(1,n-1)$, of signature $(2,n-2)$ and of signature $(p,q)$, where $p+q\leq 6$. Furthermore, some special series with higher signature are discussed.
We describe the possible holonomy groups of simply connected irreducible non-locally symmetric pseudo-Riemannian spin manifolds which admit parallel spinors.
We introduce a notion of twisted pure spinor in order to characterize, in a unified way, all the special Riemannian holonomy groups just as a classical pure spinor characterizes the special Kähler holonomy. Motivated by certain curvature identities satisfied by manifolds admitting parallel twisted pure spinors, we also introduce the Clifford monopole equations as a natural geometric generalization of the Seiberg-Witten equations. We show that they restrict to the Seiberg-Witten equations in 4 dimensions, and that they admit non-trivial solutions on manifolds with special Riemannian holonomy.
In this paper we examine the structure of Riemannian manifolds with a special kind of Codazzi tensors. We use them to construct globally hyperbolic Lorentzian manifolds with complete Cauchy hypersurfaces for any weakly irreducible holonomy representation with parallel spinors, i.e. with a holonomy group which is a semidirect product between $\R^{n-2}$ and one of $\1, SU(k), Sp(1), G_2$ and $Spin(7)$.
In this paper we complete the classification of spin manifolds admitting parallel spinors, in terms of the Riemannian holonomy groups. More precisely, we show that on a given n-dimensional Riemannian manifold, spin structures with parallel spinors are in one to one correspondence with lifts to Spin_n of the Riemannian holonomy group, with fixed points on the spin representation space. In particular, we obtain the first examples of compact manifolds with two different spin structures carrying parallel spinors.
We prove that the Cauchy problem for parallel null vector fields on smooth Lorentzian manifolds is well posed. The proof is based on the derivation and analysis of suitable hyperbolic evolution equations given in terms of the Ricci tensor and other geometric objects. Moreover, we classify Riemannian manifolds satisfying the constraint conditions for this Cauchy problem. It is then possible to characterise certain holonomy reductions of globally hyperbolic manifolds with parallel null vector in terms of flow equations for Riemannian special holonomy metrics. For exceptional holonomy groups thes…
We prove that given a pseudo-Riemannian conformal structure whose conformal holonomy representation fixes a totally lightlike subspace of arbitrary dimension, there is, wrt. a local metric in the conformal class defined off a singular set, a parallel, totally lightlike distribution on the tangent bundle which contains the image of the Ricci-tensor. This generalizes results obtained for invariant lightlike lines and planes and closes a gap in the understanding of the geometric meaning of reducibly acting conformal holonomy groups. We show how this result naturally applies to the classification …
We prove that, for M theory or type II, generic Minkowski flux backgrounds preserving $\mathcal{N}$ supersymmetries in dimensions $D\geq4$ correspond precisely to integrable generalised $G_{\mathcal{N}}$ structures, where $G_{\mathcal{N}}$ is the generalised structure group defined by the Killing spinors. In other words, they are the analogues of special holonomy manifolds in $E_{d(d)} \times\mathbb{R}^+$ generalised geometry. In establishing this result, we introduce the Kosmann-Dorfman bracket, a generalisation of Kosmann's Lie derivative of spinors. This allows us to write down the internal…
We investigate the holonomy group of a linear metric connection with skew-symmetric torsion. In case of the euclidian space and a constant torsion form this group is always semisimple. It does not preserve any non-degenerated 2-form or any spinor. Suitable integral formulas allow us to prove similar properties in case of a compact Riemannian manifold equipped with a metric connection of skew-symmetric torsion. On the Aloff-Wallach space N(1,1) we construct families of connections admitting parallel spinors. Furthermore, we investigate the geometry of these connections as well as the geometry o…
In this note it is shown that the Maslov Index for pairs of Lagrangian Paths as introduced by Cappell, Lee and Miller appears by parallel transporting elements of (a certain complex line-subbundle of) the symplectic spinorbundle over Euclidean space, when pulled back to an (embedded) Lagrangian submanifold $L$, along closed or non-closed paths therein. In especially, the CLM-Index mod 4 determines the holonomy group of this line bundle w.r.t. the Levi-Civita-connection on $L$, hence its vanishing mod 4 is equivalent to the existence of a trivializing parallel section. Moreover, it is shown tha…
We study the full holonomy group of Lorentzian manifolds with a parallel null line bundle. We prove several results that are based on the classification of the restricted holonomy groups of such manifolds and provide a construction method for manifolds with disconnected holonomy which starts from a Riemannian manifold and a properly discontinuous group of isometries. Most of our examples are quotients of pp-waves with disconnected holonomy and without parallel vector field. Furthermore, we classify the full holonomy groups of solvable Lorentzian symmetric spaces and of Lorentzian manifolds wit…
On a closed connected oriented manifold $M$ we study the space $\mathcal{M}_\|(M)$ of all Riemannian metrics which admit a non-zero parallel spinor on the universal covering. Such metrics are Ricci-flat, and all known Ricci-flat metrics are of this form. We show the following: The space $\mathcal{M}_\|(M)$ is a smooth submanifold of the space of all metrics, and its premoduli space is a smooth finite-dimensional manifold. The holonomy group is locally constant on $\mathcal{M}_\|(M)$. If $M$ is spin, then the dimension of the space of parallel spinors is a locally constant function on $\mathcal…
The existence of a recurrent spinor field on a pseudo-Riemannian spin manifold $(M,g)$ is closely related to the existence of a parallel 1-dimensional complex subbundle of the spinor bundle of $(M,g)$. We characterize the following simply connected pseudo-Riemannian manifolds admitting such subbundles in terms of their holonomy algebras: Riemannian manifolds; Lorentzian manifolds; pseudo-Riemannian manifolds with irreducible holonomy algebras; pseudo-Riemannian manifolds of neutral signature admitting two complementary parallel isotropic distributions.
We describe, by their holonomy groups, all complete simply connected irreducible non-locally symmetric pseudo-Riemannian SpinC manifolds which admit parallel spinors. So we generalize the Riemannian SpinC case and the pseudo-Riemannian Spin one.
On a Lorentzian manifold the existence of a parallel null vector field implies certain constraint conditions on the induced Riemannian geometry of a space-like hypersurface. We will derive these constraint conditions and, conversely, show that every real analytic Riemannian manifold satisfying the constraint conditions can be extended to a Lorentzian manifold with a parallel null vector field. Similarly, every parallel null spinor on a Lorentzian manifold induces an imaginary generalised Killing spinor on a space-like hypersurface. Then, based on the fact that a parallel spinor field induces a…
We define intrinsic torsion in generalised geometry and use it to introduce a new notion of generalised special holonomy. We then consider generic warped supersymmetric flux compactifications of M theory and Type II of the form $\mathbb{R}^{D-1,1}\times M$. Using the language of $E_{d(d)}\times\mathbb{R}^+$ generalised geometry, we show that, for $D\geq 4$, preserving minimal supersymmetry is equivalent to the manifold $M$ having generalised special holonomy and list the relevant holonomy groups. We conjecture that this result extends to backgrounds preserving any number of supersymmetries. As…
In this paper, a survey of the recent results about the classification of the connected holonomy groups of the Lorentzian manifolds is given. A simplification of the construction of the Lorentzian metrics with all possible connected holonomy groups is obtained. As the applications, the Einstein equation, Lorentzian manifolds with parallel and recurrent spinor fields, conformally flat Walker metrics and the classification of 2-symmetric Lorentzian manifolds are considered.
In this expository article we discuss the relations between Sasakian geometry, reduced holonomy and supersymmetry. It is well known that the Riemannian manifolds other than the round spheres that admit real Killing spinors are precisely Sasaki-Einstein manifolds, 7-manifolds with a nearly parallel G2 structure, and nearly Kaehler 6-manifolds. We then discuss the relations between the latter two and Sasaki-Einstein geometry.
We study the geometric structure of Lorentzian spin manifolds, which admit imaginary Killing spinors. The discussion is based on the cone construction and a normal form classification of skew-adjoint operators in signature $(2,n-2)$. Derived geometries include Brinkmann spaces, Lorentzian Einstein-Sasaki spaces and certain warped product structures. Exceptional cases with decomposable holonomy of the cone are possible.
We describe, by their holonomy groups, all simply connected irreducible non-locally symmetric pseudo-Riemannian SpinC manifolds which admit parallel spinors. So we generalise the Riemannian case and the pseudo-Riemannian one.
The holonomy group of an (n+2)-dimensional simply-connected, indecomposable but non-irreducible Lorentzian manifold (M,h) is contained in the parabolic group $(\mathbb{R} \times SO(n))\ltimes \mathbb{R}^n$. The main ingredient of such a holonomy group is the SO(n)--projection $G:=pr_{SO(n)}(Hol_p(M,h))$ and one may ask whether it has to be a Riemannian holonomy group. In this paper we show that this is always the case, completing our results of the first part math.DG/0305139. We draw consequences for the existence of parallel spinors on Lorentzian manifolds.
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 demonstrat…
Inspired by the recent work of Physicists Hertog-Horowitz-Maeda, we prove two stability results for compact Riemannian manifolds with nonzero parallel spinors. Our first result says that Ricci flat metrics which also admits nonzero parallel spinors are stable (in the direction of changes in conformal structures) as the critical points of the total scalar curvature functional. In fact, we show that the Lichnerowicz Laplacian, which governs the second variation, is the square of a twisted Dirac operator. Our second result, which is a local version of the first one, shows that any metrics of posi…
Holonomy algebras of Lorentzian Weyl spin manifolds with weighted parallel spinors are found. For Lorentzian Weyl manifolds admitting recurrent null vector fields are introduced special local coordinates similar to Kundt and Walker ones. Using that, the local form of all Lorentzian Weyl spin manifolds with weighted parallel spinors is given. The Einstein-Weyl equation for the obtained Weyl structures is analyzed and examples of Einstein-Weyl spaces with weighted parallel spinors are given.
We show that a 7-dimensional non-compact Ricci-flat Riemannian manifold with Riemannian holonomy G_2 can admit non-integrable G_2 structures of type R + S^2_0(R^7) + R^7 in the sense of Fernández and Gray. This relies on the construction of some G_2 solvmanifolds, whose Levi-Civita connection is known to give a parallel spinor, admitting a 2-parameter family of metric connections with non-zero skew-symmetric torsion that has parallel spinors as well. The family turns out to be a deformation of the Levi-Civita connection. This is in contrast with the case of compact scalar-flat Riemannian spin …
We study the intrinsic geometry of hypersurfaces in Calabi-Yau manifolds of real dimension 6 and, more generally, SU(2)-structures on 5-manifolds defined by a generalized Killing spinor. We prove that in the real analytic case, such a 5-manifold can be isometrically embedded as a hypersurface in a Calabi-Yau manifold in a natural way. We classify nilmanifolds carrying invariant structures of this type, and present examples of the associated metrics with holonomy SU(3).
We determine the holonomy of generalized Killing spinor covariant derivatives of the form $D= \nabla + Ω$ on pseudo-Riemannian reductive homogeneous spaces in a purely algebraic and algorithmic way, where $Ω: TM \rightarrow Λ^*(TM)$ is a left-invariant homomorphism. This is essentially an application of the theory of invariant principal bundle connections defined over homogeneous principal bundles. Moreover, the algorithm allows a computation of the associated Killing superalgebra in certain cases. The procedure is demonstrated by determining the supersymmetries of certain homogeneous M2 duals…
We show that given a conformal structure whose holonomy representation fixes a totally lightlike subspace of arbitrary dimension, there is always a local metric in the conformal class off a singular set which is Ricci-isotropic and gives rise to a parallel, totally lightlike distribution on the tangent bundle. This naturally applies to parallel spin tractors resp. twistor spinors on conformal spin manifolds and clarifies which twistor spinors are locally equivalent to parallel spinors. Moreover, we study the zero set of a twistor spinor using the curved orbit decomposition for parabolic geomet…
We construct the space of infinitesimal variations for the Strominger system and an obstruction space to integrability, using elliptic operator theory. We initiate the study of the geometry of the moduli space, describing the infinitesimal structure of a natural foliation on this space. The associated leaves are related to generalized geometry and correspond to moduli spaces of solutions of suitable Killing spinor equations on a Courant algebroid. As an application, we propose a unifying framework for metrics with holonomy $\SU(3)$ and solutions of the Strominger system.
In 1981, covariantly constant spinors were introduced into Kaluza-Klein theory as a way of counting the number of supersymmetries surviving compactification. These are related to the holonomy group of the compactifying manifold. The first non-trivial example was provided in 1982 by D=11 supergravity on the squashed S7, whose G2 holonomy yields N=1 in D=4. In 1983, another example was provided by D=11 supergravity on K3, whose SU(2) holonomy yields half the maximum supersymmetry. In 2002, G2 and K3 manifolds continue to feature prominently in the full D=11 M-theory and its dualities. In particu…
We determine the geometry of supersymmetric heterotic string backgrounds for which all parallel spinors with respect to the connection $\hat\nabla$ with torsion $H$, the NS$\otimes$NS three-form field strength, are Killing. We find that there are two classes of such backgrounds, the null and the timelike. The Killing spinors of the null backgrounds have stability subgroups $K\ltimes\bR^8$ in $Spin(9,1)$, for $K=Spin(7)$, SU(4), $Sp(2)$, $SU(2)\times SU(2)$ and $\{1\}$, and the Killing spinors of the timelike backgrounds have stability subgroups $G_2$, SU(3), SU(2) and $\{1\}$. The former admit…
For a Riemannian submersion from a simple compact Lie group with a bi-invariant metric, we prove the action of its holonomy group on the fibers is transitive. As a step towards classifying Riemannian submersions with totally geodesic fibers, we consider the parameterized surface induced by lifting a base geodesic to points along a geodesic in a fiber. Such a surface is "doubly ruled" (it is ruled by horizontal geodesics and also by vertical geodesics). Its characterizing properties allow us to define "doubly ruled parameterized surfaces" in any Riemannian manifold, independent of Riemannian su…
In this review, basic definitions of spin geometry are given and some of its applications to supersymmetry, supergravity and condensed matter physics are summarized. Clifford algebras and spinors are defined and the first-order differential operators on spinors which lead to the definitions of twistor and Killing spinors are discussed. Holonomy classification for manifolds admitting parallel and Killing spinors are given. Killing-Yano and conformal Killing-Yano forms resulting from the spinor bilinears of Killing and twistor spinors are introduced and the symmetry operators of special spinor e…
We define new Riemannian structures on 7-manifolds by a differential form of mixed degree which is the critical point of a (possibly constrained) variational problem over a fixed cohomology class. The unconstrained critical points generalise the notion of a manifold of holonomy $G_2$, while the constrained ones give rise to a new geometry without a classical counterpart. We characterise these structures by the means of spinors and show the integrability conditions to be equivalent to the supersymmetry equations on spinors in supergravity theory of type IIA/B with bosonic background fields. In …
