arXiv research

Study pinches Weyl curvature on 4-manifolds, proving anti-self-duality.

We explore the geometry of the Nahm-Schmid equations, a version of Nahm's equations in split signature. Our discussion ties up different aspects of their integrable nature: dimensional reduction from the Yang--Mills anti-self-duality equations, explicit solutions, Lax-pair formulation, conservation laws and spectral cu…

We investigate the integrability of almost complex structures on the twistor space of an almost quaternionic manifold constructed with the help of a quaternionic connection. We show that if there is an integrable structure it is independent on the quaternionic connection. In dimension four, we express the anti-self-dua…

We demonstrate how the complex integral formula for the Airy functions arises from Penrose's twistor contour integral formula. We then use the Lax formulation of the isomonodromy problem with one irregular singularity of order four to show that the Airy equation arises from the anti-self-duality equations for conformal…

We investigate the integrability of natural almost complex structures on the twistor space of an almost para-quaternionic manifold as well as the integrability of natural almost paracomplex structures on the reflector space of an almost para-quaternionic manifold constructed with the help of a para-quaternionic connect…

Study of mixed equation combining gauge theory and symplectic geometry.

The existence of K-instantons on a cylinder M^7 = R_tau x K/H over a homogeneous nearly K"ahler 6-manifold K/H requires a conformally parallel or a cocalibrated G_2-structure on M^7. The generalized anti-self-duality on M^7 implies a Chern-Simons flow on K/H which runs between instantons on the coset. For K-equivariant…

We construct a Fourier--Mukai transform for smooth complex vector bundles $E$ over a torus bundle $π:M \to B,$ the vector bundles being endowed with various structures of increasing complexity. At a minimum, we consider vector bundles $E$ with a flat partial unitary connection, that is families or deformations of flat …

Study of Yang-Mills fields on 4-manifolds using modified Lévy Laplacians.

We present a novel formulation of the instanton equations in 8-dimensional Yang-Mills theory. This formulation reveals these equations as the last member of a series of gauge-theoretical equations associated with the real division algebras, including flatness in dimension 2 and (anti-)self-duality in 4. Using this form…

We review aspects of twistor theory, its aims and achievements spanning thelast five decades. In the twistor approach, space--time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex three--fold -- the twistor space. After giving an elementary construction of this s…

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

We briefly review the hierarchy for the hyper-Kähler equations and define a notion of symmetry for solutions of this hierarchy. A four-dimensional hyper-Kähler metric admits a hidden symmetry if it embeds into a hierarchy with a symmetry. It is shown that a hyper-Kähler metric admits a hidden symmetry if it admits a ce…

Study critical metrics on Riemannian manifolds, finding new minimizers and rigidity results.