arXiv research

We study complex 4-manifolds with holomorphic self-dual conformal structures, and we obtain an interpretation of the Weyl tensor of such a manifold as the projective curvature of a field of cones on the ambitwistor space. In particular, its vanishing is implied by the existence of some compact, simply-connected, null-g…

Develops higher gauge theory for categorified spaces, solving tensor field equations.

We establish a Penrose-Ward transform yielding a bijection between holomorphic principal 2-bundles over a twistor space and non-Abelian self-dual tensor fields on six-dimensional flat space-time. Extending the twistor space to supertwistor space, we derive sets of manifestly N=(1,0) and N=(2,0) supersymmetric non-Abeli…

Study on type-D Ricci-flat metrics with Killing spinors and Killing vectors.

Study rigidifies geometry of electrostatic systems with specific tensor properties.

Classifies self-dual almost-Kähler 4-manifolds, proving uniqueness up to rescaling.

Study on type-D metrics aligned with Einstein-Maxwell equations, deriving solutions.

We construct manifestly superconformal field theories in six dimensions which contain a non-Abelian tensor multiplet. In particular, we show how principal 3-bundles over a suitable twistor space encode solutions to these self-dual tensor field theories via a Penrose-Ward transform. The resulting higher or categorified …

We find necessary and sufficient conditions for a Riemannian four-dimensional manifold $(M, g)$ with anti-self-dual Weyl tensor to be locally conformal to a Ricci--flat manifold. These conditions are expressed as the vanishing of scalar and tensor conformal invariants. The invariants obstruct the existence of parallel …

Simply connected 4-manifolds with specific Weyl tensor are geodesic balls in space forms.

We establish a $C^{1,α}$ compactness theorem for the metrics with bounded self - dual Weyl tensor and Scalar curvature. The key step is to estimate the $C^{1,α}$ harmonic radius, where we use the blow up analysis as in \cite{Anderson90}. The result is motivated by, and may be applied to the Calabi flow on complex surfa…

A characterization of the Kerr-NUT-(A)de Sitter metric among four dimensional Λ-vacuum spacetimes admitting a Killing vector is obtained in terms of the proportionality of the self-dual Weyl tensor and a natural self-dual double two-form constructed from the Killing vector. This result recovers and extends a previous c…

Classification of toric self-dual Einstein gravitational instantons with negative cosmological constant

The abstract conjectures and proves conditions for scalar-flat Kähler surfaces with specific tensor properties.

Proves Kähler-Einstein property for certain Einstein 4-manifolds.

The Goldberg-Sachs theorem is generalized for all four-dimensional manifolds endowed with torsion-free connection compatible with the metric, the treatment includes all signatures as well as complex manifolds. It is shown that when the Weyl tensor is algebraically special severe geometric restrictions are imposed. In p…

Study classifies Einstein 4-manifolds with specific curvature properties.

The paper examines four-dimensional manifolds with specific curvature constraints.

Utilizing a number of results of Dittmann, we investigate the nature of the Yang-Mills field over the eight-dimensional convex set, endowed with the Bures metric, of three-level quantum systems. Adopting a numerical strategy, we first decompose the field into self-dual and anti-self-dual components, by implementing the…

Researchers find non-diagonal Einstein metrics in various signatures.

We study the spectral geometry of the conformal Jacobi operator on a 4-dimensional Riemannian manifold (M,g). We show that (M,g) is conformally Osserman if and only if (M,g) is self-dual or anti self-dual. Equivalently, this means that the curvature tensor of (M,g) is given by a quaternionic structure, at least pointwi…

We show the existence of a modified Cliff(1,1) structure compatible with an Osserman 0-model of signature (2,2). We then apply this algebraic result to certain classes of pseudo-Riemannian manifolds of signature (2,2). We obtain a new characterization of the Weyl curvature tensor of an (anti-)self-dual manifold and we …

We introduce a general notion of twistorial map and classify twistorial harmonic morphisms with one-dimensional fibres from self-dual four-manifolds. Such maps can be characterised as those which pull back Abelian monopoles to self-dual connections. In fact, the constructions involve solving a generalised monopole equa…

In the context of D-dimensional Euclidean gravity, we define the natural generalisation to D-dimensions of the self-dual Yang-Mills equations, as duality conditions on the curvature 2-form of a Riemannian manifold. Solutions to these self-duality equations are provided by manifolds of SU(2), SU(3), G_2 and Spin(7) holo…

Study generalizes Yang-Mills equations for special complex surfaces.

We consider self-dual metrics on 3CP^2 of positive scalar curvature admitting a non-trivial Killing field, but which is not conformally isometric to LeBrun's metrics. Firstly, we determine defining equations of the twistor spaces of such self-dual metrics. Next we prove that conversely, the complex threefolds defined b…

The study examines four-dimensional gradient Ricci solitons and their properties.

Study on Yang-Mills fields on $S^4$ proving self-duality constraints.

The twistor space of self-dual positive Einstein manifolds naturally admits two 1-parameter families of Riemannian metrics, one is the family of canonical deformation metrics and the other is the family introduced by B. Chow and D. Yang in 1989. The purpose of this paper is to compare these two families. In particular …

This paper derives new identities for the Weyl tensor on a gradient Ricci soliton, particularly in dimension four. First, we prove a Bochner-Weitzenböck type formula for the norm of the self-dual Weyl tensor and discuss its applications, including connections between geometry and topology. In the second part, we are co…

Researchers compute and describe differential invariants for self-dual conformal structures.

Study integrability of conformal geodesics on gravitational instantons.

In this paper we announce a gluing theorem for conformal structures with anti-self-dual (ASD) Weyl tensor that applies in geometrical situations that are more general than those considered by previous authors. By adapting a method proposed by Floer, sufficient conditions are given for the existence of ASD conformal str…

A new formalism simplifies SO(3) Yang-Mills theory connections.

Refined asymptotics of scalar-flat ALE four-manifolds

The article provides conditions for unobstructedness of ASD manifolds.

Study finds obstacles to solutions for specific equations on compact surfaces.

New method solves 'googly problem' for Schwarzschild black holes.

We derive a formula for the global gravitational anomaly of the self-dual field theory on an arbitrary compact oriented Riemannian manifold. Along the way, we uncover interesting links between the theory of determinant line bundles of Dirac operators, Siegel theta functions and a functor constructed by Hopkins and Sing…

Let M be a closed oriented 4-manifold, with Riemannian metric g, and a spin^C structure induced by an almost-complex structure ω. Each connection A on the determinant line bundle induces a unique connection \nabla^A, and Dirac operator \D^A on spinor fields. Let σ: W^+ --> Λ^+ be the natural squaring map, taking self-d…

The Weyl functional is analyzed on 4-manifolds with positive scalar curvature.

We generalize duality in Abelian gauge theory on globally hyperbolic spacetimes.

New integrable matrix PDEs derived from Frölicher-Nijenhuis brackets.

A family of new twistor string theories is constructed and shown to be free from world-sheet anomalies. The spectra in space-time are calculated and shown to give Einstein supergravities with second order field equations instead of the higher derivative conformal supergravities that arose from earlier twistor strings. …

Derives equations for non-abelian self-dual strings and finds a categorified monopole solution.

Classifies weakly Einstein curvature tensors in 4D Euclidean space.

Tensor fields depending on other tensor fields are considered. The concept of extended tensor fields is introduced and the theory of differentiation for such fields is developed.

We show that, on a 4-manifold M endowed with a spin^c structure induced by an almost-complex structure, a self-dual (= positive) spinor field φ\in Γ(W^+) is the same as a bundle morphism φ: TM \to TM acting on the fiber by self-dual conformal transformations, such that the Clifford multiplication is just the evaluation…