We consider strings with an N=2 local superconformal symmetry on the worldsheet. The critical dimension for this theory is four (two complex dimensions) with the signature (2, 2). A Kahler function giving rise to self-dual gravity is the only physical degree of freedom of this theory. Some miraculous symmetries are observed corresponding to the exchange of worldsheet and target moduli. The open and heterotic versions of this string theory correspond to self-dual Yang-Mills fields coupled to self-dual gravity in four dimensions.

Self-Duality and N = 2 String Magic

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 paraKahler 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 1 4 . A locally conformally flat hyper-paraKahler (hypersymplectic) structure with parallel Lee form on Kodaira–Thurston surfaces is obtained. Anti-self-dual non-Weyl flat neutral metric on Inoe surfaces modeled on Sol 1 4 is presented. An example of anti-self-dual neutral metric which is not locally conformally hyper-paraKahler is constructed.

ParaHermitian and paraquaternionic manifolds

* Basic Algebraic Notions* Basic Geometrical Notions* Walker Structures* Three-Dimensional Lorentzian Walker Manifolds* Four-Dimensional Walker Manifolds* The Spectral Geometry of the Curvature Tensor* Hermitian Geometry* Special Walker Manifolds

The Geometry of Walker Manifolds

A para-Kahler manifold can be defined as a pseudo-Riemannian manifold with a parallel skew-symmetric para-complex structure , that is, a parallel field of skew-symmetric endomorphisms with or, equivalently, as a symplectic manifold with a bi-Lagrangian structure , that is, two complementary integrable Lagrangian distributions. A homogeneous manifold of a semisimple Lie group admits an invariant para-Kahler structure if and only if it is a covering of the adjoint orbit of a semisimple element . A description is given of all invariant para-Kahler structures on such a homogeneous manifold. With the use of a para-complex analogue of basic formulae of Kahler geometry it is proved that any invariant para-complex structure on defines a unique para-Kahler Einstein structure with given non-zero scalar curvature. An explicit formula for the Einstein metric is given. A survey of recent results on para-complex geometry is included.

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Homogeneous para-Kähler Einstein manifolds

Suppose that we havethe unit Euclidean ball in Rn and construct new bodies using three op- erations — linear transformations, closure in the radial metric, and multiplicative summation defined by kxkK+0L = p kxkKkxkL. We prove that in dimension 3 this procedure gives all origin-symmetric convex bodies, while this is no longer true in dimensions 4 and higher. We introduce the concept of embedding of a normed space in L0 that naturally extends the corresponding properties of Lp-spaces with p 6 0, and show that the procedure described above gives exactly the unit balls of subspaces of L0 in everydimension. We provideFourier analytic and geometric characterizations of spaces embedding in L0, and prove several facts confirming the place of L0 in the scale of Lp-spaces.

The geometry of $L_0$

The twistor space Z of an oriented Riemannian 4-manifold M admits a natural 1-parameter family of Riemannian metrics ht compatible with the almost-complex structures J, and J2 introduced, respectively, by Atiyah, Hitchin and Singer, and Eells and Salamon. In the present note we describe the (real-analytic) manifolds M for which the Ricci tensor of (Z , ht) is ./"-Hermitian, n = 1 or 2. This is used to supply examples giving a negative answer to the Blair and Ianus question of whether a compact almost-Kahler manifold with Hermitian Ricci tensor is Kahlerian.

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Twistor spaces with hermitian ricci tensor

On the Riemannian curvature of a twistor space

In this paper, we obtain a classification (up to conformal equivalence) of the compact self-dual Hermitian surfaces. As an application, we prove that every compact Hermitian surface of pointwise constant holomorphic sectional curvature with respect to either the Riemannian or the Hermitian connection is Kahler.

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Compact self-dual Hermitian surfaces

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Hyperbolic Twistor Spaces

CONTENTSForewordChapter I. Introduction §1. Instanton-type solutions in the σ-model theory §2. Harmonic maps. Basic notions §3. Chern's theorem and Gaussian maps §4. Twistors and bundles of almost complex structuresChapter II. Twistor spaces of Riemannian manifolds §1. The bundle of almost Hermitian structures §2. Reduction of a twistor space §3. Twistor manifolds of symmetric spaces §4. Loop spacesChapter III. Harmonic maps §1. Twistor constructions of harmonic maps §2. Harmonic maps into complex projective spaces §3. Harmonic maps into complex Grassmann manifolds §4. Harmonic maps into groups and symmetric spaces §5. Harmonic maps and loop spacesReferences

https://iopscience.iop.org/article/10.1070/RM1993v048n03ABEH001031/pdf

Johann Davidov

Papers

Twistor spaces with hermitian ricci tensor

On the Riemannian curvature of a twistor space

Compact self-dual Hermitian surfaces

Hyperbolic Twistor Spaces

Twistor spaces and harmonic maps