quaternion-Kahler Research Articles

Harmonic Flow of Quaternion-Kähler Structures

We formulate the gradient Dirichlet flow of Sp(2)Sp(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Sp}(2)\ extrm{Sp}(1)$$\\end{document}-structures on 8-manifolds, as the first systematic study of a geometric quaternion-Kähler (QK) flow. Its critical condition of harmonicity is especially relevant in the QK setting, since torsion-free structures are often topologically obstructed. We show that the conformally parallel property implies harmonicity, extending a result of Grigorian in the G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document} case. We also draw several comparisons with Spin(7)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Spin}(7)$$\\end{document}-structures. Analysing the QK harmonic flow, we prove an almost-monotonicity formula, which implies to long-time existence under small initial energy, via ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document}-regularity.We set up a theory of harmonic QK solitons, constructing a non-trivial steady example. We produce explicit long-time solutions: one, converging to a torsion-free limit on the hyperbolic plane; and another, converging to a limit which is harmonic but not torsion-free, on the manifold SU(3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{SU}(3)$$\\end{document}. We also study compactness and the formation of singularities.

Horizontally Conformal Submersions from CR-Submanifolds of Locally Conformal Quaternionic Kaehler Manifolds

Horizontally Conformal Submersions from CR-Submanifolds of Locally Conformal Quaternionic Kaehler Manifolds

Global symmetries of Quaternion-K\xe4hler mathcal{N} = 4 supersymmetric mechanics

We analyze the global symmetries of mathcal{N} = 4 supersymmetric mechanics in- volving 4n-dimensional Quaternion-Kähler (QK) 1D sigma models on projective spaces ℍHn and ℍPn as the bosonic core. All Noether charges associated with global worldline symmetries are shown to vanish as a result of equations of motion, which implies that we deal with a severely constrained hamiltonian system. The complete hamiltonian analysis of the bosonic sector is performed.

A generalized Wintgen inequality for quaternionic CR-submanifolds

The aim of this paper is to extend the classical DDVV inequality to CR-submanifolds of quaternionic Kahler manifolds of constant quaternionic sectional curvature. We first obtain a more general inequality involving the normalized scalar normal curvature $$\rho _N$$ (defined from the second fundamental form) and then derive a DDVV-type inequality involving the normalized normal scalar curvature $$\rho ^{\perp }$$ (defined from the normal curvature tensor) for CR-submanifolds in quaternionic ambient space. We also characterize the second fundamental form of those submanifolds for which the equality case holds and give a nontrivial example of submanifold satisfying the equality case identically.

Examples of transversally complex submanifolds of the associative Grassmann manifold

We consider the Grassmann manifold of associative subspaces in the space of imaginary octonions, which we call the associative Grassmann manifold. It is known that the associative Grassmann manifold is an eight-dimensional compact symmetric quaternionic Kahler manifold. We construct interesting examples of four-dimensional complex submanifolds of the associative Grassmann manifold.

Projective geometry and the quaternionic Feix–Kaledin construction

Starting from a complex manifold S with a real-analytic c-projective structure whose curvature has type (1,1), and a complex line bundle L with a connection whose curvature has type (1,1), we construct the twistor space Z of a quaternionic manifold M with a quaternionic circle action which contains S as a totally complex submanifold fixed by the action. This extends a construction of hypercomplex manifolds, including hyperkaehler metrics on cotangent bundles, obtained independently by B. Feix and D. Kaledin. When S is a Riemann surface, M is a self-dual conformal 4-manifold, and the quotient of M by the circle action is an Einstein-Weyl manifold with an asymptotically hyperbolic end, and our construction coincides with a construction presented by the first author in a previous paper. The extension also applies to quaternionic Kaehler manifolds with circle actions, as studied by A. Haydys and N. Hitchin.

New deformations of N = 4 and N = 8 supersymmetric mechanics

This is a review of two different types of the deformed N = 4 and N = 8 supersymmetric mechanics. The first type is associated with the worldline realizations of the supergroups SU(2|1) (four supercharges), as well as of SU(2|2) and SU(4|1) (eight supercharges). The second type is the quaternion- Kähler (QK) deformation of the hyper-Kähler (HK) N = 4 mechanics models. The basic distinguishing feature of the QK models is a local N = 4 supersymmetry realized in d = 1 harmonic superspace.

Quaternion-K\xe4hler mathcal{N} = 4 supersymmetric mechanics

Using the mathcal{N} = 4, 1D harmonic superspace approach, we construct a new type of mathcal{N} = 4 supersymmetric mechanics involving 4n-dimensional Quaternion-Kähler (QK) 1D sigma models as the bosonic core. The basic ingredients of our construction are local mathcal{N} = 4,1D supersymmetry realized by the appropriate transformations in 1D harmonic superspace, the general mathcal{N} = 4, 1D superfield vielbein and a set of 2(n + 1) analytic “matter” superfields representing (n + 1) off-shell supermultiplets (4, 4, 0). Both superfield and component actions are given for the simplest QK models with the manifolds ℍHn = Sp(1, n)/[Sp(1) × Sp(n)] and ℍPn = Sp(1 + n)/[Sp(1) × Sp(n)] as the bosonic targets. For the general case the relevant superfield action and constraints on the (4, 4, 0) “matter” superfields are presented. Further generalizations are briefly discussed.

Moduli spaces of AdS5 vacua in N $$ \mathcal{N} $$ = 2 supergravity

We determine the conditions for maximally supersymmetric AdS_5 vacua of five-dimensional gauged N=2 supergravity coupled to vector-, tensor- and hypermultiplets charged under an arbitrary gauge group. In particular, we show that the unbroken gauge group of the AdS_5 vacua has to contain an U(1)_R-factor. Moreover we prove that the scalar deformations which preserve all supercharges form a Kahler submanifold of the ambient quaternionic Kahler manifold spanned by the scalars in the hypermultiplets.

Manifolds with holonomy $$U^*(2m)$$ U ∗ ( 2 m )

We consider the geometry determined by a torsion-free affine connection whose holonomy lies in the subgroup \(U^*(2m)\), a real form of \(GL(2m,\mathbf {C})\), otherwise denoted by \(SL(m,\mathbf {H}) \cdot U(1)\). We show in particular how examples may be generated from quaternionic Kahler or hyperkahler manifolds with a circle action.

Quaternionic Kähler manifolds with Hermitian and Norden metrics

Almost hypercomplex manifolds with Hermitian and Norden metrics and more specially the corresponding quaternionic Kaehler manifolds are considered. Some necessary and sufficient conditions the investigated manifolds be isotropic hyper-Kaehlerian and flat are found. It is proved that the quaternionic Kaehler manifolds with the considered metric structure are Einstein for dimension at least 8. The class of the non-hyper-Kaehler quaternionic Kaehler manifold of the considered type is determined.

ON POSITIVE QUATERNIONIC KÄHLER MANIFOLDS WITH b_4=1

ON POSITIVE QUATERNIONIC KÄHLER MANIFOLDS WITH b_4=1

Blow-ups of Locally Conformally Kähler Manifolds

A locally conformally Khler (LCK) manifold is a complex manifold which admits a covering endowed with a Kahler metric with respect to which the covering group acts through homotheties. We show that the blow-up of a compact LCKmanifold along a complex submanifold admits an LCK structure if and only if this submanifold is globally conformally Kahler. We also prove that a twistor space (of a compact four-manifold, a quaternionKahler manifold, or a Riemannian manifold) cannot admit an LCK metric, unless it is Kahler.

Covariantly constant curvature tensors andD=3,N=4, 5, 8 Chern-Simons matter theories

We construct some examples of $D=3$, $\mathcal{N}=4$ GW theory and $\mathcal{N}=5$ superconformal Chern-Simons matter theory by using the covariantly constant curvature of a quaternionic-Kahler manifold to construct the symplectic 3-algebra in the theories. Comparing with the previous theories, the $\mathcal{N}=4$, 5 theories constructed in this way possess a local $Sp(2n)$ symmetry and a diffeomorphism symmetry associated with the quaternionic-Kahler manifold. We also construct a generalized $\mathcal{N}=8$ BLG theory by utilizing the dual curvature operator of a maximally symmetric space of dimension 4 to construct the Nambu 3-algebra. Comparing with the previous $\mathcal{N}=8$ BLG theory, the theory has a diffeomorphism invariance and a local $SO(4)$ invariance associated with the symmetric space.

Homogeneous nearly Kähler manifolds

The structure of nearly Kahler manifolds was studied by Gray in several articles, mainly in Gray (Math Ann 223:233–248, 1976). More recently, a relevant progress on the subject has been done by Nagy. Among other results, he proved that a complete strict nearly Kahler manifold is locally a Riemannian product of homogeneous nearly Kahler spaces, twistor spaces over quaternionic Kahler manifolds and six-dimensional (6D) nearly Kahler manifolds, where the homogeneous nearly Kahler factors are also 3-symmetric spaces. In the present article, we show some further properties relative to the structure of nearly Kahler manifolds and, using the lists of 3-symmetric spaces given by Wolf and Gray, we display the exhaustive list of irreducible simply connected homogeneous strict nearly Kahler manifolds. For such manifolds, we give details relative to the intrinsic torsion and the Riemannian curvature.

Wall-crossing, Rogers dilogarithm, and the QK/HK correspondence

When formulated in twistor space, the D-instanton corrected hypermultiplet moduli space in N=2 string vacua and the Coulomb branch of rigid N=2 gauge theories on $R^3 \times S^1$ are strikingly similar and, to a large extent, dictated by consistency with wall-crossing. We elucidate this similarity by showing that these two spaces are related under a general duality between, on one hand, quaternion-Kahler manifolds with a quaternionic isometry and, on the other hand, hyperkahler manifolds with a rotational isometry, further equipped with a hyperholomorphic circle bundle with a connection. We show that the transition functions of the hyperholomorphic circle bundle relevant for the hypermultiplet moduli space are given by the Rogers dilogarithm function, and that consistency across walls of marginal stability is ensured by the motivic wall-crossing formula of Kontsevich and Soibelman. We illustrate the construction on some simple examples of wall-crossing related to cluster algebras for rank 2 Dynkin quivers. In an appendix we also provide a detailed discussion on the general relation between wall-crossing and the theory of cluster algebras.

A prolongation of the conformal-Killing operator on quaternionic-Kähler manifolds

A 2-form on a quaternionic-Kahler manifold (M, g) of dimension 4n ≥ 8 is called compatible (with the quaternionic structure) if it is a section of the direct sum bundle \({S^{2}H \oplus S^{2}E}\) . We construct a prolongation \({\mathcal D}\) of the conformal-Killing operator acting on compatible 2-forms. We show that \({\mathcal D}\) is flat if and only if the quaternionic-Weyl tensor of (M, g) vanishes. Consequences of this result are developed. We construct a skew-symmetric multiplication on the space of conformal-Killing 2-forms on (M, g), and we study its properties in connection with the subspace of compatible conformal-Killing 2-forms.

Quaternionic Kähler Detour Complexes and $${\mathcal{N} = 2}$$ Supersymmetric Black Holes

We study a class of supersymmetric spinning particle models derived from the radial quantization of stationary, spherically symmetric black holes of four dimensional N= 2 supergravities. By virtue of the c-map, these spinning particles move in quaternionic Kaehler manifolds. Their spinning degrees of freedom describe mini-superspace-reduced supergravity fermions. We quantize these models using BRST detour complex technology. The construction of a nilpotent BRST charge is achieved by using local (worldline) supersymmetry ghosts to generating special holonomy transformations. (An interesting byproduct of the construction is a novel Dirac operator on the superghost extended Hilbert space.) The resulting quantized models are gauge invariant field theories with fields equaling sections of special quaternionic vector bundles. They underly and generalize the quaternionic version of Dolbeault cohomology discovered by Baston. In fact, Baston's complex is related to the BPS sector of the models we write down. Our results rely on a calculus of operators on quaternionic Kaehler manifolds that follows from BRST machinery, and although directly motivated by black hole physics, can be broadly applied to any model relying on quaternionic geometry.

On the classification of positive quaternionic Kähler manifolds with b 4 = 1

Let M be a positive quaternionic Kahler manifold of dimension 4m. We already showed that if the symmetry rank is greater than or equal to \( \left[ {\tfrac{m} {2}} \right] + 2 \) and the fourth Betti number b4 is equal to one, then M is isometric to ℍPm. The goal of this paper is to report that we can improve the lower bound of the symmetry rank by one for higher even-dimensional positive quaternionic Kahler manifolds. Namely, it is shown in this paper that if the symmetry rank of M with b4(M) = 1 is greater than or equal to \( \tfrac{m} {2} + 1 \) for m ≥ 10, then M is isometric to ℍPm. One of the main strategies of this paper is to apply a more delicate argument of Frankel type to positive quaternionic Kahler manifolds with certain symmetry rank.

Linear Perturbations of Quaternionic Metrics

We extend the twistor methods developed in our earlier work on linear deformations of hyperkahler manifolds [arXiv:0806.4620] to the case of quaternionic-Kahler manifolds. Via Swann's construction, deformations of a 4d-dimensional quaternionic-Kahler manifold $M$ are in one-to-one correspondence with deformations of its $4d+4$-dimensional hyperkahler cone $S$. The latter can be encoded in variations of the complex symplectomorphisms which relate different locally flat patches of the twistor space $Z_S$, with a suitable homogeneity condition that ensures that the hyperkahler cone property is preserved. Equivalently, we show that the deformations of $M$ can be encoded in variations of the complex contact transformations which relate different locally flat patches of the twistor space $Z_M$ of $M$, by-passing the Swann bundle and its twistor space. We specialize these general results to the case of quaternionic-Kahler metrics with $d+1$ commuting isometries, obtainable by the Legendre transform method, and linear deformations thereof. We illustrate our methods for the hypermultiplet moduli space in string theory compactifications at tree- and one-loop level.