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.     

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.     

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.     

Symmetries in Generalized Kähler Geometry

We define the notion of a moment map and reduction in both generalized complex geometry and generalized Kähler geometry. As an application, we give very simple explicit constructions of bi-Hermitian structures on $\mathbb{C}\mathbb{P}^{N}We define the notion of a moment map and reduction in both generalized complex geometry and generalized K?hler geometry. As an application, we give very simple explicit constructions of bi-Hermitian structures on , Hirzebruch surfaces, the blow up of at arbitrarily many points, and other toric varieties, as well as complex Grassmannians.     

Instantons, Poisson Structures and Generalized Kähler Geometry

Using the idea of a generalized Kähler structure, we construct bihermitian metrics on CP² and CP¹×CP¹, and show that any such structure on a compact 4-manifold M defines one on the moduli space of anti-self-dual connections on a fixed principal bundle over M. We highlight the role of holomorphic Poisson structures in all these constructions.     

Generalized Kähler Geometry, Gerbes, and all that

We review recent advances in generalized Kähler geometry while stressing the use of Poisson and symplectic geometry. The derivation of a generalized Kähler potential is sketched and relevant global issues are discussed.     

Generating Techniques and Analytically Extended Solutions of the Einstein–Maxwell Equations

The correspondence of arbitrary parameters inexact axisymmetric solutions of the Einstein-Maxwellequations constructed with the aid of differentgenerating methods to the analytically extendedparameter sets is discussed and examples of the extendedsolutions are given.     

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.     

Generalized Kähler Geometry from Supersymmetric Sigma Models

We give a physical derivation of generalized Kähler geometry. Starting from a supersymmetric nonlinear sigma model, we rederive and explain the results of Gualtieri (Generalized complex geometry, DPhil thesis, Oxford University, 2004) regarding the equivalence between generalized Kähler geometry and the bi-hermitean geometry of Gates et al. (Nucl Phys B248:157, 1984). When cast in the language of supersymmetric sigma models, this relation maps precisely to that between the Lagrangian and the Hamiltonian formalisms. We also discuss topological twist in this context.     

The Sasaki join and admissible Kähler constructions

We give a survey of our recent work (Boyer and Tønnesen-Friedman (2013) [50], [51], [30], Boyer and Tønnesen-Friedman (2014) [33], [29], [36]) describing a method which combines the Sasaki join construction of Boyer et al. (2007) [31] with the admissible Kähler construction of Apostolov et al. (2006, 2004, 2008) [26], [27], [14], [25] to obtain new extremal and new constant scalar curvature Sasaki metrics, including Sasaki–Einstein metrics. The constant scalar curvature Sasaki metrics also provide explicit solutions to the CR Yamabe problem. In this regard we give examples of the lack of uniqueness when the Yamabe invariant $\lambda \left(M\right)$ is positive.     

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.     

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.     

The extended conformal Einstein field equations with matter: The Einstein–Maxwell field

A discussion is given of the conformal Einstein field equations coupled with matter whose energy–momentum tensor is trace-free. These resulting equations are expressed in terms of a generic Weyl connection. The article shows how in the presence of matter it is possible to construct a conformal gauge which allows to know a priori the location of the conformal boundary. In vacuum this gauge reduces to the so-called conformal Gaussian gauge. These ideas are applied to obtain (i) a new proof of the stability of Einstein–Maxwell de Sitter-like spacetimes; (ii) a proof of the semi-global stability of purely radiative Einstein–Maxwell spacetimes.     

The Maxwell–Einstein system,Ward identities and the Vilkovisky construction

The gauge fixing dependence of the one-loop effective action of quantum gravity in the proper-time representation is investigated for a space of arbitrary curvature, and the investigation is extended to Maxwell–Einstein theory. The construction of Vilkovisky and DeWitt for removal of this dependence is then considered in general gauges, and it is shown that nontrivial criteria arising from a Ward identity of the theory must be obeyed by the regularization scheme, if the construction is to remove the gauge dependence of quadratic and quartic divergences. The results apply also to non-Abelian gauge theories; they are used to address the question of gauge dependence of asymptotic freedom arising through internal graviton lines at one-loop order as suggested by Robinson and Wilczek.     

Extremal Kähler metrics and Bach–Merkulov equations

In this paper, we study a coupled system of equations on oriented compact 4-manifolds which we call the Bach–Merkulov equations. These equations can be thought of as the conformally invariant version of the classical Einstein–Maxwell equations. Inspired by the work of C. LeBrun on Einstein–Maxwell equations on compact Kähler surfaces, we give a variational characterization of solutions to Bach–Merkulov equations as critical points of the Weyl functional. We also show that extremal Kähler metrics are solutions to these equations, although, contrary to the Einstein–Maxwell analogue, they are not necessarily minimizers of the Weyl functional. We illustrate this phenomenon by studying the Calabi action on Hirzebruch surfaces.