Anti Self Research Articles

Dolbeault harmonic (1,1)-forms on 4-dimensional compact quotients of Lie groups with a left invariant almost Hermitian structure

We study Dolbeault harmonic (1,1)-forms on compact quotients M=Γ﹨G of 4-dimensional Lie groups G admitting a left invariant almost Hermitian structure (J,ω). In this case, we prove that the space of Dolbeault harmonic (1,1)-forms on (M,J,ω) has dimension b−+1 if and only if there exists a left invariant anti self dual (1,1)-form γ on (G,J) satisfying idcγ=dω. Otherwise, its dimension is b−. In this way, we answer to a question by Zhang.

Conformal geodesics on gravitational instantons

AbstractWe study the integrability of the conformal geodesic flow (also known as the conformal circle flow) on the SO(3)–invariant gravitational instantons. On a hyper–Kähler four–manifold the conformal geodesic equations reduce to geodesic equations of a charged particle moving in a constant self–dual magnetic field. In the case of the anti–self–dual Taub NUT instanton we integrate these equations completely by separating the Hamilton–Jacobi equations, and finding a commuting set of first integrals. This gives the first example of an integrable conformal geodesic flow on a four–manifold which is not a symmetric space. In the case of the Eguchi–Hanson we find all conformal geodesics which lie on the three–dimensional orbits of the isometry group. In the non–hyper–Kähler case of the Fubini–Study metric on $\mathbb{CP}^2$ we use the first integrals arising from the conformal Killing–Yano tensors to recover the known complete integrability of conformal geodesics.

Wave equations on anti self dual (ASD) manifolds

In this paper, we study and perform analyses of the wave equation on some manifolds with non diagonal metric \(g_{ij}\) which are of neutral signatures. These include the invariance properties, variational symmetries and conservation laws. In the recent past, wave equations on the standard (space time) Lorentzian manifolds have been performed but not on the manifolds from metrics of neutral signatures.

Some (Anti-) Self Duality Solutions on Six Manifolds

The bundle of the 2-forms on 6-manifold decomposes into three subbundles such that Λ2 (ℝ6) = Λ21 ⊕ Λ26 ⊕ Λ28 with dimensions 1, 6 and 8, respectively. The self and anti self duality solutions of the 2-forms, called ϕ-duality are handled and these solutions show that the anti self dual gauge fields live on the subbundles Λ21 and Λ26 while the self ones equations on Λ28. Also the solution on Λ21 presents a flat connection. In addition, the curvatures of the connections on Λ26 and Λ28 have , and so the topological invariants determined by the Chern classes, i.e. topological charge, consist only on the second Chern class. In the result of this case, the anti self and self ϕ-dual gauge invariant Lagrangians of defined on both subbundles are bounded by the same topological charge. Also, one gives a quantization case to be relating to the instanton number.

Twistor Geometry of a Pair of Second Order ODEs

We discuss the twistor correspondence between path geometries in three dimensions with vanishing Wilczynski invariants and anti-self-dual conformal structures of signature $(2, 2)$. We show how to reconstruct a system of ODEs with vanishing invariants for a given conformal structure, highlighting the Ricci-flat case in particular. Using this framework, we give a new derivation of the Wilczynski invariants for a system of ODEs whose solution space is endowed with a conformal structure. We explain how to reconstruct the conformal structure directly from the integral curves, and present new examples of systems of ODEs with point symmetry algebra of dimension four and greater which give rise to anti--self--dual structures with conformal symmetry algebra of the same dimension. Some of these examples are $(2, 2)$ analogues of plane wave space--times in General Relativity. Finally we discuss a variational principle for twistor curves arising from the Finsler structures with scalar flag curvature.

Classification of self dual quadratic bent functions

We classify all self dual and anti self dual quadratic bent functions in 2n variables under the action of the orthogonal group $${{O}(2n,\mathbb F_2)}$$ . This is done through a classification of all 2n × 2n involutory alternating matrices over $${\mathbb F_2}$$ under the action of the orthogonal group. The sizes of the $${{O}(2n,\mathbb F_2)}$$ -orbits of self dual and anti self dual quadratic bent functions are determined explicitly.

Four–dimensional metrics conformal to Kähler

AbstractWe derive some necessary conditions on a Riemannian metric (M,g) in four dimensions for it to be locally conformal to Kähler. If the conformal curvature is non anti–self–dual, the self–dual Weyl spinor must be of algebraic typeDand satisfy a simple first order conformally invariant condition which is necessary and sufficient for the existence of a Kähler metric in the conformal class. In the anti–self–dual case we establish a one to one correspondence between Kähler metrics in the conformal class and non–zero parallel sections of a certain connection on a natural rank ten vector bundle overM. We use this characterisation to provide examples of ASD metrics which are not conformal to Kähler.We establish a link between the ‘conformal to Kähler condition’ in dimension four and the metrisability of projective structures in dimension two. A projective structure on a surfaceUis metrisable if and only if the induced (2, 2) conformal structure onM=TUadmits a Kähler metric or a para–Kähler metric.

Strominger–Yau–Zaslow Geometry, Affine Spheres and Painlevé III

We give a gauge invariant characterisation of the elliptic affine sphere equation and the closely related Tzitzeica equation as reductions of real forms of \({SL(3, \mathbb{C})}\) anti–self–dual Yang–Mills equations by two translations, or equivalently as a special case of the Hitchin equation.

Anti-self-dual Conformal Structures with Null Killing Vectors from Projective Structures

Using twistor methods, we explicitly construct all local forms of four–dimensional real analytic neutral signature anti–self–dual conformal structures (M, [g]) with a null conformal Killing vector. We show that M is foliated by anti-self-dual null surfaces, and the two-dimensional leaf space inherits a natural projective structure. The twistor space of this projective structure is the quotient of the twistor space of (M, [g]) by the group action induced by the conformal Killing vector. We obtain a local classification which branches according to whether or not the conformal Killing vector is hyper-surface orthogonal in (M, [g]). We give examples of conformal classes which contain Ricci–flat metrics on compact complex surfaces and discuss other conformal classes with no Ricci–flat metrics.

Calibrated lifts of minimal submanifolds

A canonical real line bundle associated to a minimal Lagrangian submanifold in a Kähler–Einstein manifold X is known to be special Lagrangian when considered as a subset of the canonical line bundle of X with a natural Calabi–Yau structure. We first verify this result by standard moving frame computation, and obtain a uniform lower bound for the mass of compact minimal Lagrangian submanifolds in C P n . Similar correspondence is then proved for integrable G 2 and Spin ( 7 ) structures on the bundle of anti self dual 2-forms and a Spin bundle respectively of a self dual Einstein 4-manifold N constructed by Bryant and Salamon. In this case, analogues of tangent and normal bundles of certain minimal surfaces in N are calibrated, i.e., associative, coassociative, or Cayley.

Trasplante de progenitores hematopoyéticos en la esclerosis múltiple

The idea of treating auto immune diseases which are resistant to conventional immunodepressive treatment by autologous transplantation of haematopoietic stem cells obtained from peripheral blood (TAPH) is based on the hypothesis that immuno ablative treatment would destroy the patients anti self lymphocytes and the reinfusion of stem cells would give rise to lymphocytes which would be tolerant of the antigens responsible for the auto immune response. By July 2000 90 patients with multiple sclerosis (ES) had received a TAPH according to data of the European Group for Blood and Marrow Transplantation/European League against Rheumatism (EBMT/EULAR). In the literature there are reports of the results obtained in 43 patients in whom EM was the sole indication for treatment [30 secondary progressive (SP), 10 primary progressive (PP) and 3 progressive relapsing (PR) ES]. The transplant related mortality (death within the first 100 days) was 4.6% (2/43), in both cases caused by infection. In the study which included 24 patients, the TAPH was considered to have been effective, since 92% of the patients with SP MS had no further progression after 3 years. However, of the patients with PP EM only 40% scored the same or better on the EDSS, as compared to their basal scores, at the end of the study. In any case, this treatment cannot be considered to cure the MS since the probability of patients having no signs of active disease 3 years after transplantation is 12% in SP MS and 0% in PP MS.TAPH remains an experimental treatment. Only in the future will it be clear whether this treatment is really useful for patients with multiple sclerosis.

Physiological Approaches to Neuroprotection: Boosting of Protective Autoimmunity

In glaucoma, as in other degenerative diseases of the central nervous system (CNS), there are some neurons which, although susceptible to degeneration, are amenable to neuroprotection. Until very recently, attempts to attenuate the spread of damage after CNS trauma or in neurodegenerative diseases did not include recruitment of the immune system, because it was assumed that in the CNS any immune activity, particularly autoimmune activity, would be harmful. Using the injured optic nerve of the rat as a model, we recently showed, however, that this ‘secondary degeneration’ can be slowed down by a well-controlled adaptive immune response, which is mediated by T cells against CNS self-antigens, such as myelin basic protein (MBP), and can be achieved either by passive transfer of MBP-activated T cells or by active immunization with MBP. Accordingly, we suggested that autoimmune T cells can be neuroprotective. We further showed that the neuroprotective autoimmunity is a physiological response to the injury, perhaps insufficient in its natural state, but amenable to boosting. The neuroprotective activity of these T cells probably depends on their reactivation by their specific antigen after they are targeted to the injured nerve. As nerve degeneration is initiated and sustained by many factors, it would probably best be counteracted by a comprehensive type of therapy rather than treatment that addresses only some aspects of nerve damage. T cell therapy, being physiological rather than pharmaceutical in nature, may provide a global answer. However, since the neuroprotective immune response is directed against the self, it must be rigorously regulated to avoid inducing an autoimmune disease. We showed that the synthetic copolymer Cop-1 can passively or actively evoke T cell-mediated neuroprotection, probably by cross-reacting with MBP. Safe synthetic peptides that resemble self-antigens and are cross-activated by CNS-associated self antigens may be a useful starting point for the development of anti–self immunity for neuroprotective purposes.