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.
We construct a quaternionic-Kähler manifold from a conical special Kähler manifold with a certain type of mutually-local variation of BPS structures. We give global and local explicit formulas for the quaternionic-Kähler metric, and specify under which conditions it is positive-definite. Locally, the metric is a deformation of the 1-loop corrected Ferrara-Sabharval metric obtained via the supergravity c-map. The type of quaternionic-Kähler metrics we obtain are related to work in the physics literature by S. Alexandrov and S. Banerjee, where they discuss the hypermultiplet moduli space metric …
We give an explicit formula for the quaternionic Kähler metrics obtained by the HK/QK correspondence. As an application, we give a new proof of the fact that the Ferrara-Sabharwal metric as well as its one-loop deformation is quaternionic Kähler. A similar explicit formula is given for the analogous (K/K) correspondence between Kähler manifolds endowed with a Hamiltonian Killing vector field. As an example, we apply this formula in the case of an arbitrary conical Kähler manifold.
The hyperKähler-quaternionic Kähler correspondence constructs quaternionic Kähler metrics from hyperKähler metrics with a rotating circle symmetry. We discuss how this may be interpreted as a combination of the twist construction with the concept of elementary deformation, surveying results of our forthcoming paper. We outline how this leads to a uniqueness statement for the above correspondence and indicate how basic examples of c-map constructions may be realised in this context.
We construct explicit left invariant quaternionic contact structures on Lie groups with zero and non-zero torsion, and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of quaternionic contact manifolds not locally quaternionic contact conformal to the quaternionic sphere. We present a left invariant quaternionic contact structure on a seven dimensional non-nilpotent Lie group, and show that this structure is locally quaternionic contact conformal to the flat quaternionic contact structure on the quaternionic Heisenberg group. On the product of a se…
We construct, using harmonic superspace and the quaternionic quotient approach, a quaternionic-Kähler extension of the most general two centres hyper-Kähler metric. It possesses $U(1)\times U(1)$ isometry, contains as special cases the quaternionic-Kähler extensions of the Taub-NUT and Eguchi-Hanson metrics and exhibits an extra one-parameter freedom which disappears in the hyper-Kähler limit. Some emphasis is put on the relation between this class of quaternionic-Kähler metrics and self-dual Weyl solutions of the coupled Einstein-Maxwell equations. The relation between our explicit results an…
The problem of finding new metrics of interest, in the context of SUGRA, is reduced to two stages: first, solving a generalized BPS sigma model with full quaternionic structure proposed by the authors and, second, constructing the hyper-Kähler metric, or suitable deformations of this condition, taking advantage of the correspondence between the quaternionic left-regular potential and the hyper-Kähler metric of the target space. As illustration, new solutions are obtained using generalized Q-sigma model for Wess-Zumino type superpotentials. Explicit solutions analog to the Berger's sphere and A…
We present details of the harmonic space construction of a quaternionic extension of the four-dimensional Taub-NUT metric. As the main merit of the harmonic space approach, the metric is obtained in an explicit form following a generic set of rules. It exhibits $SU(2)\times U(1)$ isometry group and depends on two parameters, Taub-NUT `mass' and the cosmological constant. We consider several limiting cases of interest which correspond to special choices of the involved parameters.
Starting from the generic harmonic superspace action of the quaternion-Kähler sigma models and using the quotient approach we present, in an explicit form, a quaternion-Kähler extension of the double Taub-NUT metric. It possesses $U(1)\times U(1)$ isometry and supplies a new example of non-homogeneous Einstein metric with self-dual Weyl tensor.
We generalise the hyper-Kahler/quaternionic Kahler (HK/QK) correspondence to include para-geometries, and present a new concise proof that the target manifold of the HK/QK correspondence is quaternionic Kahler. As an application, we construct one-parameter deformations of the temporal and Euclidean supergravity c-map metrics and show that they are para-quaternionic Kahler.
We show that for all very special quaternionic manifolds a different N=1 reduction exists, defining a Kaehler Geometry which is ``dual'' to the original very special Kaehler geometry with metric G_{a\bar{b}}= - \partial_a \partial_b \ln V (V={1/6}d_{abc}λ^a λ^b λ^c). The dual metric g^{ab}=V^{-2} (G^{-1})^{ab} is Kaehler and it also defines a flat potential as the original metric. Such geometries and some of their extensions find applications in Type IIB compactifications on Calabi--Yau orientifolds.
We show the properties of the triplet Killing potentials of quaternionic Kaehler manifolds which have been missing in the literature. It is done by means of the metric formula of the manifolds. We compute the triplet Killing potentials for the quaternionic Kaehler manifold Sp(n+1)/Sp(n)xSp(1) as an illustration.
We consider a general 4n-dimensional quaternionic Kahler geometry with a free action of the torus T^(n+1). The toric action lifts onto the Swann bundle of the quaternionic Kahler space to a tri-holomorphic action that commutes with the standard H* action on the bundle. By matching Pedersen and Poon's generalized Gibbons-Hawking Ansatz description of the total space with the Swann picture we extract the local geometry of the quaternionic Kahler base. Specifically, we obtain explicit expressions for the quaternionic Kahler metric and Sp(1) connection in terms of a set of reduced Higgs fields and…
We use the harmonic space technique to construct explicitly a quaternionic extension of the Taub-NUT metric. It depends on two parameters, the first being the Taub-NUT `mass' and the second one the cosmological constant.
Starting from the most general harmonic superspace action of self-interacting Q^+ hypermultiplets in the background of N=2 conformal supergravity, we derive the general action for the bosonic sigma model with a generic 4n dimensional quaternionic-Kahler (QK) manifold as the target space. The action is determined by the analytic harmonic QK potential. We find out this action to have two flat limits. One gives the hyper-Kahler sigma model with a 4n dimensional target manifold, while another yields a conformally invariant sigma model with 4(n+1) dimensional hyper-Kahler target. We work out the ha…
We discuss a class of 4-dimensional non-homogeneous quaternionic spaces, which become the two known homogeneous spaces (EAdS_4$ and SU(2,1)/U(2)) in certain limits. These moduli spaces have two regions where the metric is positive definite, separated by a non-physical region where the metric has timelike directions and which contains a curvature singularity. They admit four isometries and we consider their general Abelian gauging. The critical points of the resulting superpotential and hence the possible domain wall solutions differ significantly in the two regions. On one side one can constru…
Special geometry is most known from 4-dimensional N=2 supergravity, though it contains also quaternionic and real geometries. In this review, we first repeat the connections between the various special geometries. Then the constructions are given starting from the superconformal approach. Without going in detail, we give the main underlying principles. We devote special attention to the quaternionic manifolds, introducing the notion of hypercomplex geometry, being manifolds close to hyperkaehler manifolds but without a metric. These are related to supersymmetry models without an action.
We investigate the Einstein equation with a positive cosmological constant for $4n+4$-dimensional metrics on bundles over Quaternionic Kähler base manifolds whose fibers are 4-dimensional Bianchi IX manifolds. The Einstein equations are reduced to a set of non-linear ordinary differential equations. We numerically find inhomogeneous compact Einstein spaces with orbifold singularity.
A set of canonical parahermitian connections on an almost paraHermitian manifold is defined. ParaHermitian version of the Apostolov-Gauduchon generalization of the Goldberg-Sachs theorem in General Relativity is given. It is proved that the Nijenhuis tensor of a Nearly paraKähler manifolds is parallel with respect to the canonical connection. Salamon's twistor construction on quaternionic manifold is adapted to the paraquaternionic case. A hyper-paracomplex structure is constructed on Kodaira-Thurston (properly elliptic) surfaces as well as on the Inoe surfaces modeled on $Sol^4_1$. A locally …
In the present work some examples of toric hyperkahler metrics in eight dimensions are constructed. First it is described how the Calderbank-Pedersen metrics arise as a consequence of the Joyce description of selfdual structures in four dimensions, the Jones-Tod correspondence and a result due to Tod and Przanowski. It is also shown that any quaternionic Kahler metric with $T^2$ isometry is locally isometric to a Calderbank-Pedersen one. The Swann construction of hyperkahler metrics in eight dimensions is applied to them to find hyperkahler examples with $U(1)\times U(1)$ isometry. The connect…
Sachs has derived quaternion field equations that fully exploit the underlying symmetry of the principle of general relativity, one in which the fundamental 10 component metric field is replaced by a 16 component four-vector quaternion. Instead of the 10 field equations of Einstein's tensor formulation, these equations are 16 in number corresponding to the 16 analytic parametric functions {\partial}x^{μ'}/{\partial}x^{ν} of the Einstein Lie Group. The difference from the Einstein equations is that these equations are not covariant with respect to reflections in space-time, as a consequence of …
We determine the one-instanton corrections to the universal hypermultiplet moduli space coming both from Euclidean membranes and NS-fivebranes wrapping the cycles of a (rigid) Calabi-Yau threefold. These corrections are completely encoded by a single function characterizing a generic four-dimensional quaternion-Kahler metric without isometries. We give explicit solutions for this function describing all one-instanton corrections, including the fluctuations around the instanton to all orders in the string coupling constant. In the semi-classical limit these results are in perfect agreement with…
We prove that the supergravity r- and c-maps preserve completeness. As a consequence, any component H of a hypersurface {h=1} defined by a homogeneous cubic polynomial such that -d^2 h is a complete Riemannian metric on H defines a complete projective special Kahler manifold and any complete projective special Kahler manifold defines a complete quaternionic Kahler manifold of negative scalar curvature. We classify all complete quaternionic Kahler manifolds of dimension less or equal to 12 which are obtained in this way and describe some complete examples in 16 dimensions.
The total space of the spinor bundle on the four dimensional sphere S^4 is a quaternionic line bundle that admits a metric of Spin(7) holonomy. We consider octonionic Yang-Mills instanton on this eight dimensional gravitational instanton. This is a higher dimensional generalization of (anti-)self-dual instanton on the Eguchi-Hanson space. We propose an ansatz for Spin(7) Yang-Mills field and derive a system of non-linear ordinary differential equations. The solutions are classified according to the asymptotic behavior at infinity. We give a complete solution, when the gauge group is reduced to…
The earlier proposed conditions of (bi)quaternionic differentiability are nonlinear, give rize to the 2-spinor and the self-dual gauge structures and may be considered as the it generating system of equations (GSE) with respect to the source-free Maxwell, Yang-Mills and eikonal equations. We present the general solution of the GSE in terms of twistor variables, analize its rather specific gauge symmetry and demonstrate the relation of GSE to the equations of shear-free null congruences and, consequently, - to effective metrics of Kerr- Shild type. The concept of particles as singularities of p…
We show that the one-loop quantum deformation of the universal hypermultiplet provides a family of complete $1/4$-pinched negatively curved quaternionic Kähler (i.e. half conformally flat Einstein) metrics $g^c$, $c\ge 0$, on $\mathbb R^4$. The metric $g^0$ is the complex hyperbolic metric whereas the family $(g^c)_{c>0}$ is equivalent to a family of metrics $(h^b)_{b>0}$ depending on $b=1/c$ and smoothly extending to $b=0$ for which $h^0$ is the real hyperbolic metric. In this sense the one-loop deformation interpolates between the real and the complex hyperbolic metrics. We also determine th…
Some instanton corrections to the universal hypermultiplet moduli space metric of the type-IIA string theory compactified on a Calabi-Yau threefold arise due to multiple wrapping of BPS membranes and fivebranes around certain cycles of Calabi-Yau. The classical universal hypermultipet metric is locally equivalent to the Bergmann metric of the symmetric quaternionic space SU(2,1)/U(2), whereas its generic quaternionic deformations are governed by the integrable SU(infinity) Toda equation. We calculate the exact (non-perturbative) UH metrics in the special cases of (i) the D-instantons (the wrap…
The classification of homogeneous quaternionic manifolds has been done by Alekseevskii, Wolf et al using transitive solvable group of isometries. These manifolds are not generically symmetric, but there is a subset of quaternionic manifolds that are symmetric and Einstein. A further subset of these manifolds are the magic square manifolds. We show that all the symmetric quaternionic manifolds including the magic square can be succinctly classified by constrained instantons. These instantons are mostly semilocal, and their constructions for the magic square can be done from the corresponding Se…
We describe a quaternionic-based Ansatz generalizing the Gibbons-Hawking Ansatz to a class of hyperkähler metrics with hidden symmetries. We then apply it to obtain explicit expressions for gravitational instanton metrics of type $D_k$.
We study general linear perturbations of a class of 4d real-dimensional hyperkahler manifolds obtainable by the (generalized) Legendre transform method. Using twistor methods, we show that deformations can be encoded in a set of holomorphic functions of 2d+1 variables, as opposed to the functions of d+1 variables controlling the unperturbed metric. Such deformations generically break all tri-holomorphic isometries of the unperturbed metric. Geometrically, these functions generate the symplectomorphisms which relate local complex Darboux coordinate systems in different patches of the twistor sp…
We obtain a simple explicit expression for the hyper-Kahler Calabi metric on the co-tangent bundle of CP^{n+1}, for all n, in which it is constructed as a metric of cohomogeneity one with SU(n+2)/U(n) principal orbits. These results enable us to obtain explicit expressions for an L^2-normalisable harmonic 4-form in D=8, and an L^2-normalisable harmonic 6-form in D=12. We use the former in order to obtain an explicit resolved M2-brane solution, and we show that this solution is invariant under all three of the supersymmetries associated with the covariantly-constant spinors in the 8-dimensional…
In the present work the Calderbank-Pedersen description of four dimensional manifolds with self-dual Weyl tensor is used to obtain examples of quaternionic-kahler metrics with two commuting isometries. The eigenfunctions of the hyperbolic laplacian are found by use of Backglund transformations acting over solutions of the Ward monopole equation. The Bryant-Salamon construction of $G_2$ holonomy metrics arising as $R^3$ bundles over quaternionic-kahler base spaces is applied to this examples to find internal spaces of the M-theory that leads to an N=1 supersymmetry in four dimensions. Type IIA …
Using the natural connection equivalent to the SU(2) Yang-Mills instanton on the quaternionic Hopf fibration of $S^7$ over the quaternionic projective space ${\bf HP}^1\simeq S^4$ with an $SU(2)\simeq S^3$ fiber the geometry of entanglement for two qubits is investigated. The relationship between base and fiber i.e. the twisting of the bundle corresponds to the entanglement of the qubits. The measure of entanglement can be related to the length of the shortest geodesic with respect to the Mannoury-Fubini-Study metric on ${\bf HP}^1$ between an arbitrary entangled state, and the separable state…
We present theories of N=2 hypermultiplets in four spacetime dimensions that are invariant under rigid or local superconformal symmetries. The target spaces of theories with rigid superconformal invariance are (4n)-dimensional {\it special} hyper-Kähler manifolds. Such manifolds can be described as cones over tri-Sasakian metrics and are locally the product of a flat four-dimensional space and a quaternionic manifold. The latter manifolds appear in the coupling of hypermultiplets to N=2 supergravity. We employ local sections of an Sp$(n)\times{\rm Sp}(1)$ bundle in the formulation of the Lagra…
A 7-manifold with G_2 holonomy can be constructed as a R^3 bundle over a quaternionic space. We consider a quaternionic base space which is singular and its metric depends on three parameters, where one of them corresponds to an interpolation between S^4 and CP^2 or its non-compact analogs. This 4-d Einstein space has four isometries and the fixed point set of a generic Killing vector is discussed. When embedded into M-theory the compactification over a given Killing vector gives intersecting 6-branes as IIA configuration and we argue that membrane instantons may resolve the curvature singular…
N = 2 supersymmetry in four space-time dimensions is intimately related to hyperkahler and quaternionic Kahler geometries. On one hand, the target spaces for rigid supersymmetric sigma-models are necessarily hyperkahler manifolds. On the other hand, when coupled to N = 2 supergravity, the sigma-model target spaces must be quaternionic Kahler. It is known that such manifolds of restricted holonomy are difficult to generate explicitly. Projective superspace is a field-theoretic approach to constructing general N = 2 supersymmetric nonlinear sigma-models, and hence to generate new hyperkahler and…
In this note we reconsider linearised metric perturbations in the one-brane Randall-Sundrum Model. We present a simple formalism to describe metric perturbations caused by matter perturbations on the brane and remedy some misconceptions concerning the constraints imposed on the metric and matter perturbations by the presence of the brane.
We study the behavior of metric perturbations in a recently proposed model of phantom dark energy with tachyonic instability at long wavelengths. We find that metric perturbations exponentially grow in time, starting from very small values determined by vacuum fluctuations, and may become sizeable at late times. This property may be of interest for phenomenology.
Theories in 5 dimensions with minimal supersymmetry are studied for domain-wall solutions and in the context of the AdS/CFT correspondence. The scalar manifold is a product of a very special real manifold and a quaternionic-Kaehler manifold. Superconformal methods can clarify the structure of these manifolds, which are part of the family of special manifolds. BPS solutions depending on the scalars and a warp factor of the 5-dimensional metric with a flat 4-dimensional metric can interpolate between critical points determined by algebraic attractor equations. The mixing of vector and hypermulti…
We consider a scalar field interacting with a quantized metric varying on a submanifold (e.g. a scalar field interacting with a quantized gravitational wave). We explicitly sum up the perturbation series for the time-ordered vacuum expectation values of the scalar field. As a result we obtain a modified non-canonical short distance behavior.
We derive the equations of motion for scalar metric perturbations in a particular nonsingular bouncing cosmology, where the big bang singularity is replaced by a spacetime defect with a degenerate metric. The adiabatic perturbation solution is obtained for nonrelativistic hydrodynamic matter. We get the same result by working with conformal coordinates. This last method is also valid for vector and tensor metric perturbations, and selected results are presented. We, finally, discuss several new effects from the linear perturbations of this nonsingular bouncing cosmology, such as across-bounce …
We present a well-posed constraint-preserving scheme for evolving first-order metric perturbations on an arbitrary background with arbitrary source. We use this scheme to evolve the leading-order metric perturbation in order-reduced dynamical Chern-Simons gravity (dCS) on a Kerr background. In particular we test the stability of stationary dCS data on a Kerr background with stationary first-order dCS scalar field source. We find that the leading-order metric perturbation numerically exhibits linear growth, but that the level of this growth converges to zero with numerical resolution. This anal…
We determine the general form of the first order linear symmetry operators for the linearized field equation of metric perturbations in the spacetimes of dimension D>=4. Apart from the part derived easily from the invariance under general coordinate transformations, we find a part consisting of a Killing-Yano 3-form.
We study axial (or odd-parity) perturbations about static and spherically symmetric hairy black hole (BH) solutions in shift-symmetric DHOST (Degenerate Higher-Order Scalar-Tensor) theories. We first extend to the family of DHOST theories the first-order formulation that we recently developed for Horndeski theories. Remarkably, we find that the dynamics of DHOST axial perturbations is equivalent to that of axial perturbations in general relativity (GR) evolving in a, distinct, effective metric. In the particular case of quadratic DHOST theories, this effective metric is derived from the backgr…
Linear perturbation theory is a powerful toolkit for studying black hole spacetimes. However, the perturbation equations are hard to solve unless we can use separation of variables. In the Kerr spacetime, metric perturbations do not separate, but curvature perturbations do. The cost of curvature perturbations is a very complicated metric-reconstruction procedure. This procedure can be avoided using a symmetry-adapted choice of basis functions in highly symmetric spacetimes, such as near-horizon extremal Kerr. In this paper, we focus on this spacetime, and (i) construct the symmetry-adapted bas…
The decomposition of the linear-order metric perturbation is discussed in the context of the higher-order gauge-invariant perturbation theory. We show that the linear order metric perturbation is decomposed into gauge-invariant and gauge-variant parts on the general background spacetime which admits ADM decomposition. This decomposition was an important premise of the general framework of the higher order gauge-invariant perturbation theory proposed in the papers [K. Nakamura, Prog. Theor. Phys. vol.110 (2003), 723; ibid. vol.113 (2005), 481]. This implies that we can develop the higher-order …
Metric Reconstruction and Second Order Perturbation for Generic Spherically Symmetric spacetime
School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS（杭州高等研究院基础物理与数学学院，UCAS）；University of Chinese Academy of Sciences（中国科学院大学）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）
An outline of a proof of the decomposition of linear metric perturbations into gauge-invariant and gauge-variant parts on an arbitrary background spacetime which admits ADM decomposition is briefly discussed. We explicitly construct the gauge-invariant and gauge-variant parts of the linear metric perturbations based on some assumptions. This implies that we can develop the higher-order gauge-invariant perturbation theory on an arbitrary background spacetime.
We study gravitational creation of light fermions in the presence of classical scalar metric perturbations about a flat Friedmann-Lemaitre- Robertson-Walker (FLRW) background. These perturbations can be large during preheating, breaking the conformal flatness of the background spacetime. We compute numerically the total number of particles generated by the modes of the metric perturbations which have grown sufficiently to become classical. In the absence of inhomogeneities massless fermions are not gravitationally produced, and then this effect may be relevant for abundance estimates of light …
Construction of the gauge-invariant variables for the linear metric perturbation, which was proposed in the paper [K. Nakamura, arXiv:1101.1147], is discussed through an alternative approach. Our starting point of the construction of the gauge-invariant variables is an non-trivial non-local decomposition of the linear metric perturbation. Assuming the existence of some Green functions, we reproduce results in the above paper. This supports the consistency of the result and implies that one can develop the general-relativistic higher-order gauge-invariant perturbation theory on general backgrou…
Gauge-invariant treatments of general-relativistic higher-order perturbations on generic background spacetime is proposed. We show the fact that the linear-order metric perturbation is decomposed into gauge-invariant and gauge-variant parts, which was the important premise of this general framework. This means that the development the higher-order gauge-invariant perturbation theory on generic background spacetime is possible.
An outline of a proof of the decomposition of the linear metric perturbation into gauge-invariant and gauge-variant parts on an arbitrary background spacetime is discussed through an exlicit construction of gauge-invariant and gauge-variant parts. Although this outline is incomplete, yet, due to our assumptions, we propose a conjecture which states that the linear metric perturbation is always decomposed into its gauge-invariant and gageu-variant parts. If this conjecture is true, we can develop the higher-order gauge-invariant perturbation theory on an arbitrary background spacetime.
The procedure to find gauge invariant variables for two-parameter nonlinear perturbations in general relativity is considered. For each order metric perturbation, we define the variable which is defined by the appropriate combination with lower order metric perturbations. Under the gauge transformation, this variable is transformed in the manner similar to the gauge transformation of the linear order metric perturbation. We confirm this up to third order. This implies that gauge invariant variables for higher order metric perturbations can be found by using a procedure similar to that for line…
