quaternion-Kahler Research Articles

Decomposition and minimality of lagrangian submanifolds in nearly Kähler manifolds

We show that Lagrangian submanifolds in six-dimensional nearly Kahler (non-Kahler) manifolds and in twistor spaces Z 4n+2 over quaternionic Kahler manifolds Q 4n are minimal. Moreover, we prove that any Lagrangian submanifold L in a nearly Kahler manifold M splits into a product of two Lagrangian submanifolds for which one factor is Lagrangian in the strict nearly Kahler part of M and the other factor is Lagrangian in the Kahler part of M. Using this splitting theorem, we then describe Lagrangian submanifolds in nearly Kahler manifolds of dimensions six, eight, and ten.

New extended superconformal sigma models and quaternion Kähler manifolds

Quaternion Kahler manifolds are known to be the target spaces for matter hypermultiplets coupled to N=2 supergravity. It is also known that there is a one-to-one correspondence between 4n-dimensional quaternion Kahler manifolds and those 4(n+1)-dimensional hyperkahler spaces which are the target spaces for rigid superconformal hypermultiplets. We present a projective-superspace construction to generate a hyperkahler cone M^{4(n+1)}_H of dimension 4(n+1) from a 2n-dimensional real analytic Kahler-Hodge manifold M^{2n}_K. The latter emerges as a maximal Kahler submanifold of the 4n-dimensional quaternion Kahler space M^{4n}_Q such that its Swann bundle coincides with M^{4(n+1)}_H. Our approach should be useful for the explicit construction of new quaternion Kahler metrics. The results obtained are also of interest, e.g., in the context of supergravity reduction N=2 -> N=1, or from the point of view of embedding N=1 matter-coupled supergravity into an N=2 theory.

On positive quaternionic Kähler manifolds with certain symmetry rank

Let M be a positive quaternionic Kahler manifold of real dimension 4m. In this paper we show that if the symmetry rank of M is greater than or equal to [m/2] + 3, then M is isometric to HPm or Gr2(Cm+2). This is sharp and optimal, and will complete the classification result of positive quaternionic Kahler manifolds equipped with symmetry. The main idea is to use the connectedness theorem for quaternionic Kahler manifolds with a group action and the induction arguments on the dimension of the manifold.

Corrected energy of the Reeb distribution of a 3-Sasakian manifold

In this paper we show that the Reeb distribution on a spherical space form which admits a 3-Sasakian structure minimizes the corrected energy. Analogously for the characteristic distribution of the normal complex contact structure on the complex projective space $\mathbb{C}P^{2m+1}$ induced via the Hopf fibration $S^{1}\hookrightarrow S^{4m+3}\to \mathbb{C}P^{2m+1}$. This last result is a consequence of a more general result on the corrected energy of the characteristic distribution of a compact twistor space over a quaternionic-Kahler manifold with positive scalar curvature (equipped with a normal complex contact metric structure).

Spectrum of the Laplacian on quaternionic Kähler manifolds

Spectrum of the Laplacian on quaternionic Kähler manifolds

Twistors and black holes

Motivated by black hole physics in N=2, D=4 supergravity, we study the geometry of quaternionic-Kahler manifolds M obtained by the c-map construction from projective special Kahler manifolds M_s. Improving on earlier treatments, we compute the Kahler potentials on the twistor space Z and Swann space S in the complex coordinates adapted to the Heisenberg symmetries. The results bear a simple relation to the Hesse potential \Sigma of the special Kahler manifold M_s, and hence to the Bekenstein-Hawking entropy for BPS black holes. We explicitly construct the ``covariant c-map'' and the ``twistor map'', which relate real coordinates on M x CP^1 (resp. M x R^4/Z_2) to complex coordinates on Z (resp. S). As applications, we solve for the general BPS geodesic motion on M, and provide explicit integral formulae for the quaternionic Penrose transform relating elements of H^1(Z,O(-k)) to massless fields on M annihilated by first or second order differential operators. Finally, we compute the exact radial wave function (in the supergravity approximation) for BPS black holes with fixed electric and magnetic charges.

A Schur-type theorem on indefinite quaternionic Kahler manifolds

A Schur-type theorem on indefinite quaternionic Kahler manifolds

Off-shellN= 2 tensor supermultiplets

A multiplet calculus is presented for an arbitrary number n of N=2 tensor supermultiplets. For rigid supersymmetry the known couplings are reproduced. In the superconformal case the target spaces parametrized by the scalar fields are cones over (3n-1)-dimensional spaces encoded in homogeneous SU(2) invariant potentials, subject to certain constraints. The coupling to conformal supergravity enables the derivation of a large class of supergravity Lagrangians with vector and tensor multiplets and hypermultiplets. Dualizing the tensor fields into scalars leads to hypermultiplets with hyperkahler or quaternion-Kahler target spaces with at least n abelian isometries. It is demonstrated how to use the calculus for the construction of Lagrangians containing higher-derivative couplings of tensor multiplets. For the application of the c-map between vector and tensor supermultiplets to Lagrangians with higher-order derivatives, an off-shell version of this map is proposed. Various other implications of the results are discussed. As an example an elegant derivation of the classification of 4-dimensional quaternion-Kahler manifolds with two commuting isometries is given.

String loop corrected hypermultiplet moduli spaces

Using constraints from supersymmetry and string perturbation theory, we determine the string loop corrections to the hypermultiplet moduli space of type II strings compactified on a generic Calabi-Yau threefold. The corresponding quaternion-Kahler manifolds are completely encoded in terms of a single function. The latter receives a one-loop correction and, using superspace techniques, we argue for the existence of a non-renormalization theorem excluding higher loop contributions.

D-term cosmic strings fromN= 2 supergravity

We describe new half-BPS cosmic string solutions in N=2, d=4 supergravity coupled to one vector multiplet and one hypermultiplet. They are closely related to D-term strings in N=1 supergravity. Fields of the N=2 theory that are frozen in the solution contribute to the triplet moment map of the quaternionic isometries and leave their trace in N=1 as a constant Fayet-Iliopoulos term. The choice of U(1) gauging and of special geometry are crucial. The construction gives rise to a non-minimal Kaehler potential and can be generalized to higher dimensional quaternionic-Kaehler manifolds.

The Map Between Conformal Hypercomplex/ Hyper-Kähler and Quaternionic(-Kähler) Geometry

We review the general properties of target spaces of hypermultiplets, which are quaternionic-like manifolds, and discuss the relations between these manifolds and their symmetry generators. We explicitly construct a one-to-one map between conformal hypercomplex manifolds (i.e. those that have a closed homothetic Killing vector) and quaternionic manifolds of one quaternionic dimension less. An important role is played by '\xi-transformations', relating complex structures on conformal hypercomplex manifolds and connections on quaternionic manifolds. In this map, the subclass of conformal hyper-Kaehler manifolds is mapped to quaternionic-Kaehler manifolds. We relate the curvatures of the corresponding manifolds and furthermore map the symmetries of these manifolds to each other.

Brane inflation, solitons and cosmological solutions: I

In this paper we study various cosmological solutions for a D3/D7 system directly from M-theory with fluxes and M2-branes. In M-theory, these solutions exist only if we incorporate higher derivative corrections from the curvatures as well as G-fluxes. We take these corrections into account and study a number of toy cosmologies, including one with a novel background for the D3/D7 system whose supergravity solution can be completely determined. This new background preserves all the good properties of the original model and opens up avenues to investigate cosmological effects from wrapped branes and brane-antibrane annihilation, to name a few. We also discuss in some detail semilocal defects with higher global symmetries, for example exceptional ones, that could occur in a slightly different regime of our D3/D7 model. We show that the D3/D7 system does have the required ingredients to realise these configurations as non-topological solitons of the theory. These constructions also allow us to give a physical meaning to the existence of certain underlying homogeneous quaternionic Kahler manifolds.

Quaternionic maps between quaternionic Kähler manifolds

In this paper, we consider quaternionic maps between quaternionic Kahler manifolds. We will derive a monotonicity inequality, a Bochner type formula and small energy regularity for quaternionic maps.

A twistor construction of Kähler submanifolds of a quaternionic Kähler manifold

A class of minimal almost complex submanifolds of a Riemannian manifold \(\tilde{M}^{4n}\) with a parallel quaternionic structure Q, in particular of a 4-dimensional oriented Riemannian manifold, is studied. A notion of Kahler submanifold is defined. Any Kahler submanifold is pluriminimal. In the case of a quaternionic Kahler manifold \(\tilde{M}^{4n}\) of non zero scalar curvature, in particular, when \(\tilde{M}^4\) is an Einstein, non Ricci-flat, anti-self-dual 4-manifold, we give a twistor construction of Kahler submanifolds M2n of maximal possible dimension 2n. More precisely, we prove that any such Kahler submanifold M2n of \(\tilde{M}^{4n}\) is the projection of a holomorphic Legendrian submanifold \(L^{2n} \subset\mathcal{Z}\) of the twistor space \((\mathcal{Z},\mathcal{H})\) of \(\tilde{M}^{4n}\), considered as a complex contact manifold with the natural holomorphic contact structure \(\mathcal{H} \subset{T}\mathcal{Z}\). Any Legendrian submanifold of the twistor space \(\mathcal{Z}\) is defined by a generating holomorphic function. This is a natural generalization of Bryant’s construction of superminimal surfaces in S4=ℍP1.

Effective supergravity actions for flop transitions

We construct a family of five-dimensional gauged supergravity actions which describe flop transitions of M-theory compactified on Calabi-Yau threefolds. While the vector multiplet sector can be treated exactly, we use the Wolf spaces X(1+N) = U(1+N,2)/(U(1+N) x U(2)) to model the universal hypermultiplet together with N charged hypermultiplets corresponding to winding states of the M2-brane. The metric, the Killing vectors and the moment maps of these spaces are obtained explicitly by using the superconformal quotient construction of quaternion-Kahler manifolds. The inclusion of the extra hypermultiplets gives rise to a non-trivial scalar potential which is uniquely fixed by M-theory physics.

Flows on Quaternionic-K�hler and Very Special Real Manifolds

BPS solutions of 5-dimensional supergravity correspond to certain gradient flows on the product M x N of a quaternionic-Kaehler manifold M of negative scalar curvature and a very special real manifold N of dimension n >=0. Such gradient flows are generated by the `energy function' f = P^2, where P is a (bundle-valued) moment map associated to n+1 Killing vector fields on M. We calculate the Hessian of f at critical points and derive some properties of its spectrum for general quaternionic-Kaehler manifolds. For the homogeneous quaternionic-Kaehler manifolds we prove more specific results depending on the structure of the isotropy group. For example, we show that there always exists a Killing vector field vanishing at a point p in M such that the Hessian of f at p has split signature. This generalizes results obtained recently for the complex hyperbolic plane (universal hypermultiplet) in the context of 5-dimensional supergravity. For symmetric quaternionic-Kaehler manifolds we show the existence of non-degenerate local extrema of f, for appropriate Killing vector fields. On the other hand, for the non-symmetric homogeneous quaternionic-Kaehler manifolds we find degenerate local minima.

ON HYPERSURFACES OF MANIFOLDS EQUIPPED WITH A HYPERCOSYMPLECTIC 3-STRUCTURE

Hypersurfaces of a Riemannian manifold equipped with a hypercosymplectic 3-structure are studied. Integrability condi- tions for certain distributions on the hypersurface are investigated. Geometry of leaves of certain distribution are also studied. The quaternionic analog of almost complex structures is the almost quaternion (hypercomplex) which is defined by three local (global) al- most complex structures which satisfy the quaternionic relations as the imaginary quaternions satisfy ((3)). Quaternion Kahler manifolds and hyper-Kahler manifolds are special and interesting cases of Riemannian manifolds with almost quaternion and almost hypercomplex structure, respectively. Quaternion Kahler manifolds are Einstein, while hyper- Kahler manifolds are Ricci flat. An almost contact 3-structure was defined by Kuo ((5)) and it is closely related to both almost quaternion and almost hypercomplex structures. Hypersurfaces of manifolds with almost hypercomplex struc- ture inherit naturally three almost contact structures which constitute an almost contact 3-structure. An almost contact metric 3-structure manifold is always (4m + 3)-dimensional. The structural group of the tangent bundle of a (4m + 3)-dimensional manifold equipped with an almost contact 3-structure is reducible to Sp(m) £ I3. In particular, if each almost contact metric structure of an almost contact metric 3-structure is Sasakian, then this structure is called a Sasakian 3-structure. Riemannian manifolds with Sasakian 3-structure

Vanishing theorems for quaternionic Kähler manifolds

In this article we discuss a peculiar interplay between the representation theory of the holonomy group of a Riemannian manifold, the Weitzenbformula for the Hodge-Laplace operator on forms and the Lichnerowicz formula for twisted Dirac op- erators. For quaternionic Kahler manifolds this leads to simple proofs of eigenvalue estimates for Dirac and Laplace operators. Moreover, it enables us to determine which representations can contribute to harmonic forms. As a corollary we prove the vanish- ing of certain odd Betti numbers on compact quaternionic Kahler manifolds of negative scalar curvature. We simplify the proofs of several related results in the positive case.

Hyper-Hermitian Quaternionic Kähler Manifolds

We call a quaternionic Kahler manifold with nonzero scalar curvature, whosequaternionic structure is trivialized by a hypercomplex structure, ahyper-Hermitian quaternionic Kahler manifold. We prove that every locallysymmetric hyper-Hermitian quaternionic Kahler manifold is locally isometricto the quaternionic projective space or to the quaternionic hyperbolic space.We describe locally the hyper-Hermitian quaternionic Kahler manifolds withclosed Lee form and show that the only complete simply connected suchmanifold is the quaternionic hyperbolic space.

Classification of positive quaternion-Kähler [formula omitted]-manifolds

We prove that the 12-dimensional complete quaternion-Kähler manifolds with positive scalar curvature belong to the list of symmetric spaces given by Wolf [12]. To cite this article: H. Herrera, R. Herrera, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 43–46