Geometry of W-algebras from the affine Lie algebra point of view

To classify the classical field theories with W-symmetry one has to classify the symplectic leaves of the corresponding W-algebra, which are the intersection of the defining constraint and the coadjoint orbit of the affine Lie algebra if the W-algebra in question is obtained by reducing a WZNW model. The fields that survive the reduction will obey non-linear Poisson bracket (or commutator) relations in general. For example the Toda models are well-known theories which possess such a non-linear W-symmetry and many features of these models can only be understood if one investigates the reduction procedure. In this paper we analyze the SL(n,R) case from which the so-called W_n-algebras can be obtained. One advantage of the reduction viewpoint is that it gives a constructive way to classify the symplectic leaves of the W-algebra which we had done in the n=2 case which will correspond to the coadjoint orbits of the Virasoro algebra and for n=3 which case gives rise to the Zamolodchikov algebra. Our method in principle is capable of constructing explicit representatives on each leaf. Another attractive feature of this approach is the fact that the global nature of the W-transformations can be explicitly described. The reduction method also enables one to determine the ``classical highest weight (h. w.) states'' which are the stable minima of the energy on a W-leaf. These are important as only to those leaves can a highest weight representation space of the W-algebra be associated which contains a ``classical h. w. state''.Comment: 17 pages, LaTeX, revised 1. and 7. chapter

Возможности диагностики и лечения больных ХОБЛ в рамках реальной клинической практики. Подходы к терапии пациентов с различными фенотипами по GOLD (2019): материалы Совета экспертов Сибирского федерального округа, Читы и Бурятии от 15.03.19

Chronic obstructive pulmonary disease (COPD) is a global problem in modern medicine. In recent years, the medical community’s understanding of COPD has changed significantly, which is primarily due to the emergence of a new classification and the identification of various phenotypes of the disease. These changes could not affect the tactics of COPD treatment. The article discusses not only the debatable issues of treating COPD; it provides an overview of changes in international (Global Initiative for Chronic Obstructive Lung Disease, 2018) and national (2019) recommendations, but also the significance and benefits of triple therapy in terms of evidence-based medicine as well as the benefits of extra-fine drugs in the treatment of bronchial obstructive syndrome.Хроническая обструктивная болезнь легких (ХОБЛ) представляет собой глобальную проблему современной медицины. За последние годы представление медицинского сообщества о ХОБЛ существенно изменилось, что связано в первую очередь с появлением новой классификации и выделением различных фенотипов заболевания. Эти изменения не могли не повлиять на тактику лечения ХОБЛ. В статье рассматриваются не только дискуссионные вопросы лечения ХОБЛ, представлен обзор изменений в международных (Гло - бальная инициатива по диагностике, лечению и профилактике ХОБЛ (Global Initiative for Chronic Obstructive Lung Disease, 2018)) и национальных (2019) рекомендациях, но и значение и преимущества тройной терапии с точки зрения доказательной медицины, а также преимущества экстрамелкодисперсных препаратов при лечении бронхообструктивного синдрома

Yang–Mills equations on conformally connected torsion-free 4-manifolds with different signatures

In this paper we study spaces of conformal torsion-free connection of dimension 4 whose connection matrix satisfies the Yang–Mills equations. Here we generalize and strengthen the results obtained by us in previous articles, where the angular metric of these spaces had Minkowski signature. The generalization is that here we investigate the spaces of all possible metric signatures, and the enhancement is due to the fact that additional attention is paid to calculating the curvature matrix and establishing the properties of its components. It is shown that the Yang–Mills equations on 4-manifolds of conformal torsion-free connection for an arbitrary signature of the angular metric are reduced to Einstein's equations, Maxwell's equations and the equality of the Bach tensor of the angular metric and the energy-momentum tensor of the skew-symmetric charge tensor. It is proved that if the Weyl tensor is zero, the Yang–Mills equations have only self-dual or anti-self-dual solutions, i.e the curvature matrix of a conformal connection consists of self-dual or anti-self-dual external 2-forms. With the Minkowski signature (anti)self-dual external 2-forms can only be zero. The components of the curvature matrix are calculated in the case when the angular metric of an arbitrary signature is Einstein, and the connection satisfies the Yang–Mills equations. In the Euclidean and pseudo-Euclidean 4-spaces we give some particular self-dual and anti-self-dual solutions of the Maxwell equations, to which all the Yang–Mills equations are reduced in this case

Gauge-invariant Tensors of 4-Manifold with Conformal Torsion-free Connection and their Applications for Modeling of Space-time

We calculated basic gauge-invariant tensors algebraically expressed through the matrix of conformal curvature. In particular, decomposition of the main tensor into gauge-invariant irreducible summands consists of 4 terms, one of which is determined by only one scalar. First, this scalar enters the Einstein's equations with cosmological term as a cosmological scalar. Second, metric being multiplied by this scalar becomes gauge invariant. Third, the geometric point, which is not gauge-invariant, after multiplying by the square root of this scalar becomes gauge-invariant object — a material point. Fourth, the equations of motion of the material point are exactly the same as in the general relativity, which allows us to identify the square root of this scalar with mass. Thus, we obtained an unexpected result: the cosmological scalar coincides with the square of the mass. Fifth, the cosmological scalar allows us to introduce the gauge-invariant 4-measure on the manifold. Using this measure, we introduce a new variational principle for the Einstein equations with cosmological term. The matrix of conformal curvature except the components of the main tensor contains other components. We found all basic gauge-invariant tensors, expressed in terms of these components. They are 1- or 3-valent. Einstein's equations are equivalent to the gauge invariance of one of these covectors. Therefore the conformal connection manifold, where Einstein's equations are satisfied, can be divided into 4 types according to the type of this covector: timelike, spacelike, light-like or zero