Hyperkahler Research Articles

Kähler Manifolds with Almost Non-negative Ricci Curvature

Compact Kahler manifolds with semi-positive Ricci curvature have been investigated by various authors. From Peternell’s work, if M is a compact Kahler n-manifold with semi-positive Ricci curvature and finite fundamental group, then the universal cover has a decomposition \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{M} \cong X_{1} \times \cdots \times X_{m} \), where Xj is a Calabi-Yau manifold, or a hyperKahler manifold, or Xj satisfies H0(Xj , Ωp) = 0. The purpose of this paper is to generalize this theorem to almost non-negative Ricci curvature Kahler manifolds by using the Gromov-Hausdorff convergence. Let M be a compact complex n-manifold with non-vanishing Euler number. If for any ∈ > 0, there exists a Kahler structure (J∈, g∈) on M such that the volume \( {\text{Vol}}_{{g_{ \in } }} {\left( M \right)} −∈g∈, where V and Λ are two constants independent of ∈. Then the fundamental group of M is finite, and M is diffeomorphic to a complex manifold X such that the universal covering of X has a decomposition, \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{X} \cong X_{1} \times \cdots \times X_{s} \), where Xi is a Calabi-Yau manifold, or a hyperKahler manifold, or Xi satisfies H0(Xi , Ωp) = {0}, p > 0.

Hyperkähler sigma models on cotangent bundles of Hermitian symmetric spaces using projective superspace

Kahler manifolds have a natural hyperkahler structure associated with (part of) their cotangent bundles. Using projective superspace, we construct four-dimensional N = 2 models on the tangent bundles of some classical Hermitian symmetric spaces (specifically, the four regular series of irreducible compact symmetric Kahler manifolds, and their non-compact versions). A further dualization yields the Kahler potential for the hyperkahler metric on the cotangent bundle.

Local Structure of Hyperkähler Singularities in O'Grady's Examples

We study the local structure of the singularity in the moduli space of sheaves on a K3 surface which has been resolved by K. O’Grady in his construction of new examples of hyperkaehler manifolds. In particular, we identify the singularity with the closure of a certain nilpotent orbit in the coadjoint representation of the group Sp(4). We also prove that the moduli spaces for some other sets of numerical parameters do not admit a smooth symplectic resolution of singularities. 2000 Math. Subj. Class. 14D20.

Deformation Finiteness for Real Hyperkähler Manifolds

We show that the number of equivariant deformation classes of real structures in a given deformation class of compact hyperkahler manifolds is finite.

Bubbling supertubes and foaming black holes

We construct smooth BPS three-charge geometries that resolve the zero-entropy singularity of the U(1) x U(1) invariant black ring. This singularity is resolved by a geometric transition that results in geometries without any branes sources or singularities but with non-trivial topology. These geometries are both ground states of the black ring, and non-trivial microstates of the D1-D5-P system. We also find the form of the geometries that result from the geometric transition of N zero-entropy black rings, and argue that, in general, such geometries give a very large number of smooth bound-state three-charge solutions, parameterized by 6N functions. The generic microstate solution is specified by a four-dimensional hyper-Kahler geometry of a certain signature, and contains a ``foam'' of non-trivial two-spheres. We conjecture that these geometries will account for a significant part of the entropy of the D1-D5-P black hole, and that Mathur's conjecture might reduce to counting certain hyper-Kahler manifolds.

Transgression on hyperkähler manifolds and generalized higher torsion forms

We propose a generalization of the Hodge dd c -lemma to the case of hyperkahler manifolds. As an application we derive a global construction of the fourth order transgression of the Chern character forms of hyperholomorphic bundles over compact hyperkahler manifolds. In Section 3 we consider the fourth order transgression for the infinite-dimensional bundle arising from local families of hyperkahler manifolds. We propose a local construction of the fourth order transgression of the Chern character form. We derive an explicit expression for the arising hypertorsion differential form. Its zero-degree part may be expressed in terms of the Laplace operators defined on the fibres of the local family.

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.

Cohomogeneity-Three HyperKähler Metrics on Nilpotent Orbits

Let \({\cal O}\) be a nilpotent orbit in ℊℂ where G is a compact, simple group and ℊ = Lie(G). It is known that \({\cal O}\) carries a unique G-invariant hyperKahler metric admitting a hyperKahler potential compatible with the Kirillov–Kostant–Souriau symplectic form. In this work, the hyperKahler potential is explicitly calculated when \({\cal O}\) is of cohomogeneity three under the action of G. It is found that such a structure lies on a one-parameter family of hyperKahler metrics with G-invariant Kahler potentials if and only if ℊ is Sp3, su6, So7, So12 or e7 and otherwise is the unique G-invariant hyperKahler metric with G-invariant Kahler potential.

Gravitational Instantons of Type Dk

We use two different methods to obtain Asymptotically Locally Flat hyperkahler metrics of type D_k.

Spinorial content of the standard model, a different look at super-symmetry and fuzzy E-infinity hyper Kähler

Abstract The paper is written partially very informal with one formal section about hyper Kahler manifolds of the E-infinity fuzzy type. We discuss spinors and the particle content of minimally extended super-symmetric standard model.

Vacua of Massive Hyper-Kahler Sigma Models with Non-Abelian Quotient

The Higgs branch of N=2 supersymmetric gauge theories with non-Abelian gauge groups are described by hyper-Kahler (HK) nonlinear sigma models with potential terms. With the non-Abelian HK quotient by U(M) and SU(M) gauge groups, we give the massive HK sigma models that are not toric in the N=1 superfield formalism and the harmonic superspace formalism. The U(M) quotient gives N!/[M! (N-M)!] discrete vacua that may allow various types of domain walls, whereas the SU(M) quotient gives no discrete vacua.

Symplectic Deformations of Kähler Manifolds

Given a compact symplectic manifold $(M,\,\kappa)$, $H^{2}(M,\,{\Bbb{R}})$\, represents, in a natural sense, the tangent space of the moduli space of germs of deformations of the symplectic structure. In the case $(M,\,\kappa,\,J)$ is a compact Kahler manifold, the author provides a complete description of the subset of $H^{2}(M,\,{\Bbb{R}})$ corresponding to Kahler deformations, including the non-generic case, where (at least locally) some hyperkahler manifold factors out from $M$. Several examples are also discussed.

Toric Hyperkahler Manifolds with Quaternionic Kahler Bases and Supergravity Solutions

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 connection with the Pedersen-Poon toric hyperkahler metrics is explained and it is shown that there is a class of solutions of the generalized monopole equation in $\mathbb{R}^2 \otimes Im\mathbb{H}$ related to eigenfunctions of certain linear equation. This hyperkahler examples are lifted to solutions of the D=11 supergravity and type IIA and IIB backgrounds are found by use of dualities. As before, all the description is achieved in terms of a single eigenfunction F. Some explicit F are found, together with the Toda structure corresponding to the trajectories of the Killing vectors of the Calderbank-Pedersen bases.

A survey on Nahm transform

We review the construction known as the Nahm transform in a generalised context, which includes all the examples of this construction already described in the literature. The Nahm transform for translation invariant instantons on R 4 is presented in an uniform manner. We also briefly analyse two new examples, the second of which being the first example involving a four-manifold that is not hyperkähler.

Subvarieties in non-compact hyperkähler manifolds

Let M be a hyperkahler manifold, not necessarily compact, and S =∼C P 1 the set of complex structures induced by the quaternionic action. Trianalytic subvariety of M is a subvariety which is complex analytic with respect to all I ∈ C P 1 . We show that for all I ∈ S outside of a countable set, all compact complex subvarieties Z ⊂ (M,I) are trianalytic. For M compact, this result was proven in [V1] using Hodge theory.

Flowing with eight supersymmetries inM-theory andF-theory

We consider holographic RG flow solutions with eight supersymmetries and study the geometry transverse to the brane. For both M2-branes and for D3-branes in F-theory this leads to an eight-manifold with only a four-form flux. In both settings there is a natural four-dimensional hyper-Kahler slice that appears on the Coulomb branch. In the IIB theory this hyper-Kahler manifold encodes the Seiberg-Witten coupling over the Coulomb branch of a U(1) probe theory. We focus primarily upon a new flow solution in M-theory. This solution is first obtained using gauged supergravity and then lifted to eleven dimensions. In this new solution, the brane probes have an Eguchi-Hanson moduli space with the M2-branes spread over the non-trivial 2-sphere. It is also shown that the new solution is valid for a class of orbifold theories. We discuss how the hyper-Kahler structure on the slice extends to some form of G-structure in the eight-manifold, and describe how this can be computed.

Energy quantization for triholomorphic maps

Let \(\{u_n\}\subset H^1(M, N)\) be weakly convergent stationary triholomorphic maps from a hyperkahler manifold M to another hyperkahler manifold N. We establish an energy quantization for the density function of the defect measure on the concentration set.

Superconformal Symmetry and HyperKähler Manifolds with Torsion

The geometry arising from Michelson & Strominger's study of N=4B supersymmetric quantum mechanics with superconformal D(2,1;alpha)-symmetry is a hyperKaehler manifold with torsion (HKT) together with a special homothety. It is shown that different parameters alpha are related via changes in potentials for the HKT target spaces. For alpha not 0 or -1, we describe how each such HKT 4m-manifold M is derived from a space N of dimension 4m-4 which is quaternionic Kaehler with torsion and carries an Abelian instanton.

HARMONIC KAHLER FORMS ON HYPERKAHLER MANIFOLDS

Let M be a hyperkahler manifold with the hyperkahler structure (g, I, J, K). In [5], D. Huybrechts suggests that it is an open and interesting question whether any Kahler class that stays Kahler in the twister family can actually be represented by an harmonic Kahler form. In this paper we will consider both this problem and the set of all the primitive harmonic Kahler forms on M.

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