On Certain Kähler Quotients of Quaternionic Kähler Manifolds

We prove that, given a certain isometric action of a two-dimensional Abelian group A on a quaternionic Kähler manifold M which preserves a submanifold N ? M, the quotient M′ = N/A has a natural Kähler structure. We verify that the assumptions on the group action and on the submanifold N ? M are satisfied for a large class of examples obtained from the supergravity c-map. In particular, we find that all quaternionic Kähler manifolds M in the image of the c-map admit an integrable complex structure compatible with the quaternionic structure, such that N ? M is a complex submanifold. Finally, we discuss how the existence of the Kähler structure on M′ is required by the consistency of spontaneous ${\mathcal{N} = 2}$ to ${\mathcal{N} = 1}$ supersymmetry breaking.     

Hyperkähler metrics and supersymmetry

We describe two constructions of hyperkähler manifolds, one based on a Legendre transform, and one on a sympletic quotient. These constructions arose in the context of supersymmetric nonlinear -models, but can be described entirely geometrically. In this general setting, we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry, along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience.Supported by the Swedish Natural Science Research CouncilResearch supported by the National Science Foundation under Contract No. PHY 81 09110 A-01 and PHY 85-07627     

Linear Perturbations of Hyperkähler Metrics

We study general linear perturbations of a class of 4d real-dimensional hyperkähler 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 space. The deformed Kähler potential follows from these data by a Penrose-type transform. As an illustration of our general framework, we determine the leading exponential deviation of the Atiyah–Hitchin manifold away from its negative mass Taub-NUT limit.     

Contact Spheres and Hyperkähler Geometry

A taut contact sphere on a 3-manifold is a linear 2-sphere of contact forms, all defining the same volume form. In the present paper we completely determine the moduli of taut contact spheres on compact left-quotients of SU(2) (the only closed manifolds admitting such structures). We also show that the moduli space of taut contact spheres embeds into the moduli space of taut contact circles.This moduli problem leads to a new viewpoint on the Gibbons-Hawking ansatz in hyperkähler geometry. The classification of taut contact spheres on closed 3-manifolds includes the known classification of 3-Sasakian 3-manifolds, but the local Riemannian geometry of contact spheres is much richer. We construct two examples of taut contact spheres on open subsets of \({\mathbb{R}^3}\) with nontrivial local geometry; one from the Helmholtz equation on the 2-sphere, and one from the Gibbons-Hawking ansatz. We address the Bernstein problem whether such examples can give rise to complete metrics.     

L 2-Cohomology of Hyperkähler Quotients

Using an argument of Jost and Zuo, we give a criterion which implies that the L ² harmonic forms on a complete noncompact hyperk?hler manifold lie in the middle dimension and are invariant under the isometry group. This is applied to various examples, and in particular gives a verification of some of the predictions of Sen on monopole moduli spaces. Received: 20 September 1999 / Accepted: 2 November 1999     

Hyperkähler Implosion and Nahm’s Equations

We use Nahm data to describe candidates for the universal hyperkähler implosion with respect to a compact Lie group.     

Quaternionic Kähler metrics associated with special Kähler manifolds

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.     

Generalized Kähler Geometry

Generalized Kähler geometry is the natural analogue of Kähler geometry, in the context of generalized complex geometry. Just as we may require a complex structure to be compatible with a Riemannian metric in a way which gives rise to a symplectic form, we may require a generalized complex structure to be compatible with a metric so that it defines a second generalized complex structure. We prove that generalized Kähler geometry is equivalent to the bi-Hermitian geometry on the target of a 2-dimensional sigma model with (2, 2) supersymmetry. We also prove the existence of natural holomorphic Courant algebroids for each of the underlying complex structures, and that these split into a sum of transverse holomorphic Dirac structures. Finally, we explore the analogy between pre-quantum line bundles and gerbes in the context of generalized Kähler geometry.     

Pseudo-Hyperkähler Geometry and Generalized Kähler Geometry

We discuss the conditions for additional supersymmetry and twisted super-symmetry in N = (2, 2) supersymmetric nonlinear sigma models described by one left and one right semi-chiral superfield and carrying a pair of non-commuting complex structures. Focus is on linear non-manifest transformations of these fields that have an algebra that closes off-shell. We find that additional linear supersymmetry has no interesting solution, whereas additional linear twisted supersymmetry has solutions with interesting geometrical properties. We solve the conditions for invariance of the action and show that these solutions correspond to a bi-hermitian metric of signature (2, 2) and a pseudo-hyperkähler geometry of the target space.     

Conification of Kähler and Hyper-Kähler Manifolds

Given a Kähler manifold M endowed with a Hamiltonian Killing vector field Z, we construct a conical Kähler manifold ${\hat{M}}$ such that M is recovered as a Kähler quotient of ${\hat{M}}$ . Similarly, given a hyper-Kähler manifold (M, g, J ₁, J ₂, J ₃) endowed with a Killing vector field Z, Hamiltonian with respect to the Kähler form of J ₁ and satisfying ${\mathcal{L}_ZJ_2 = -2J_3}$ , we construct a hyper-Kähler cone ${\hat{M}}$ such that M is a certain hyper-Kähler quotient of ${\hat{M}}$ . In this way, we recover a theorem by Haydys. Our work is motivated by the problem of relating the supergravity c-map to the rigid c-map. We show that any hyper-Kähler manifold in the image of the c-map admits a Killing vector field with the above properties. Therefore, it gives rise to a hyper-Kähler cone, which in turn defines a quaternionic Kähler manifold. Our results for the signature of the metric and the sign of the scalar curvature are consistent with what we know about the supergravity c-map.     

Special Kähler Manifolds

We give an intrinsic definition of the special geometry which arises in global N= 2 supersymmetry in four dimensions. The base of an algebraic integrable system exhibits this geometry, and with an integrality hypothesis any special K?hler manifold is so related to an integrable system. The cotangent bundle of a special K?hler manifold carries a hyperk?hler metric. We also define special geometry in supergravity in terms of the special geometry in global supersymmetry. Received: 5 December 1997 / Accepted: 16 November 1998     

Kähler spinning particles

We construct the U(N) spinning particle theories, which describe particles moving on Kähler spaces. These particles have the same relation to the N = 2 string as usual spinning particles have to the NSR string. We find the restrictions on the target space of the theories coming from supersymmetry and from global anomalies. Finally, we show that the partition functions of the theories agree with what is expected from their spectra, unlike that of the N = 2 string in which there is an anomalous dependence on the proper time.     

On a characteristic of the first eigenvalue of the Dirac operator on compact spin symmetric spaces with a Kähler or Quaternion-Kähler structure

It is shown that on a compact spin symmetric space with a Kähler or Quaternion-Kähler structure, the first eigenvalue of the Dirac operator is linked to a “lowest” action of the holonomy, given by the fiberwise action on spinors of the canonical forms characterized by this holonomy. The result is also verified for the symmetric space $F_4/{\mathrm{Spin}}_9$, proving that it is valid for all the “possible” holonomies in Berger’s list occurring in that context. The proof is based on a characterization of the first eigenvalue of the Dirac operator given in Milhorat (2005) and Milhorat (2006). By the way, we review an incorrect statement in the proof of the first lemma in Milhorat (2005).     

Hyperkähler metric and GMN ansatz on focus–focus fibrations

In this paper, we study the hyperkähler metric and practise GMN’s construction of hyperkähler metric on focus–focus fibrations. We explicitly compute the action–angle coordinates on the local model of focus–focus fibrations, and show its semi-global invariant should be harmonic to admit a compatible holomorphic 2-form. Then we study the canonical semi-flat metric on it. After the instanton correction, finally we are able to get a reconstruction of the generalized Ooguri–Vafa metric.     

Instantons and Kähler manifolds

It is shown that for two-dimensional Euclidean chiral models of the field theory with values in arbitrary Kähler manifold duality equations reduce to the Cauchy-Riemann equations on this manifold. A class of models is described possessing such type solutions, the so called instanton solutions.     

Mass in Kähler Geometry

We prove a simple, explicit formula for the mass of any asymptotically locally Euclidean (ALE) Kähler manifold, assuming only the sort of weak fall-off conditions required for the mass to actually be well-defined. For ALE scalar-flat Kähler manifolds, the mass turns out to be a topological invariant, depending only on the underlying smooth manifold, the first Chern class of the complex structure, and the Kähler class of the metric. When the metric is actually AE (asymptotically Euclidean), our formula not only implies a positive mass theorem for Kähler metrics, but also yields a Penrose-type inequality for the mass.     

Generalized Kähler Potentials from Supergravity

We consider supersymmetric \({\mathcal{N} = 2}\) solutions with non–vanishing NS three–form. Building on worldsheet results, we reduce the problem to a single generalized Monge–Ampère equation on the generalized Kähler potential K recently interpreted geometrically by Lindström, Ro?ek, Von Unge and Zabzine. One input in the procedure is a holomorphic function w that can be thought of as the effective superpotential for a D3 brane probe. The procedure is hence likely to be useful for finding gravity duals to field theories with non–vanishing abelian superpotential, such as Leigh–Strassler theories. We indeed show that a purely NS precursor of the Lunin–Maldacena dual to the β–deformed \({\mathcal{N} = 4}\) super–Yang–Mills falls in our class.     

Eingenvalues of the Dirac operator on compact Kähler manifolds

Kählerian twistor operators are introduced to get lower bounds for the eigenvalues of the Dirac operator on compact spin Kähler manifolds. In odd complex dimensions, manifolds with the smallest eigenvalues are characterized by an over determined system of differential equations similar to the Riemannian case. In these dimensions, we show the existence of a unique natural Kählerian twistor operator. It is also proved that, on a Kähler manifold with nonzero scalar curvature, the space of Riemannian twistor-spinors is trivial.This work has been partially supported by the EEC programme GADGET Contract Nr. SC1-0105