Cohomological structure of the conformal anomaly

A cohomological classification of conformal anomalies in the dimension D=6 is given. Different anomaly classes have a common origin from the cohomological standpoint, being equivalent to the Euler density E₆. The descent equation technique is developed for conformal anomalies by analogy with gauge theories. All highest cocycles of the Weyl group are investigated. The general technique for constructing all conformal anomalies from the Weyl density E_2n in arbitrary space-time dimensions is presented. The principal difference between structures of these anomalies in dimensions D=4 and D=6 is demonstrated. A conformally invariant operator (constructed from Riemann and Ricci tensors, scalar curvature, and covariant derivatives) acting on a scalar with zero conformal weight is absent in D=6. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 117, No. 3, pp. 351–363, December, 1998.     

Constrained superpotentials in harmonic gauge theories with eight supercharges

We consider D-dimensional supersymmetric gauge theories with eight supercharges (D≤6, N=8) in the framework of harmonic superspaces. The effective Abelian low-energy action for D=5 contains the free and Chern-Simons terms. Effective N=8 superfield actions for D≤4 can be written in terms of superpotentials satisfying the superfield constraints and (6-D)-dimensional Laplace equations. The role of alternative harmonic structures is discussed. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 2, pp. 324–331, August, 1999.     

Geodesic equations for a charged particle in the unified theory of gravitational and electromagnetic interactions

A new model of gravitational and electromagnetic interactions is constructed as a version of the classical Kaluza-Klein theory based on a five-dimensional manifold as the physical space-time. The velocity space of moving particles in the model remains four-dimensional as in the standard relativity theory. The spaces of particle velocities constitute a four-dimensional distribution over a smooth five-dimensional manifold. This distribution depends only on the electromagnetic field and is independent of the metric tensor field. We prove that the equations for the geodesics whose velocity vectors always belong to this distribution are the same as the charged particle equations of motion in the general relativity theory. The gauge transformations are interpreted in geometric terms as a particular form of coordinate transformations on the five-dimensional manifold. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 3, pp. 517–528, June, 1999.     

Solution of the Kemmer and Breit equations for a plane-polarized wave

All solutions of the Kemmer and Breit equations for a plane-polarized wave are presented. In the limit as the particle mass tends to zero, some of the solutions coincide with the electromagnetic field of the wave acting on the particle. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 2, pp. 282–294, May, 1999.     

On integrable systems and supersymmetric gauge theories

We discuss the properties ofN=2 supersymmetric gauge theories underlying the Seiberg-Witten hypothesis. We consider the main points of the theory that describes the finite-gap solutions to integrable equations in terms of complex curves and generating differentials. We clarify the invariant meaning of these definitions. This formalism is applied to the exact nonperturbative solutions found recently in theN=2 supersymmetric non-Abelian gauge theories. In the known cases, we compare this formalism with the results that can be obtained using standard quantum field-theory methods. The paper was written at the request of the Editorial Board. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 3–46, July, 1997.     

Instantons and gravitation theory

Some problems related to using nonperturbative quantization methods in theories of gauge fields and gravitation are studied. The unification of interactions is considered in the context of the geometric theory of gauge fields. The notion of vacuum in the unified interaction theory and the role of instantons in the vacuum structure are considered. The relation between the definitions of instantons and the energymomentum tensor of a gauge field and also the role played by the vacuum solutions to the Einstein equations in the definition of vacuum for gauge fields are demonstrated. The Schwarzschild solution, as well as the entire class of vacuum solutions to the Einstein equations, is a gravitational instanton even though the signature of the space-time metric is hyperbolic. Gravitation, oncluding the Einstein version, is considered a special case of an interaction described by a non-Abelian gauge field. Translated from Teoreticheskaya i Matematicheskaya. Fizika. Vol. 115, No. 2, pp. 312–320, May. 1998.     

Fermi-Walker transport and the Weyl connection

We derive equations relating the Fermi-Walker and the congruent Weyl transports. Using these equations, we show that a non-Abelian gauge field can result in the Thomas precession of a gyroscope. We find solutions to the equations for such a non-Abelian gauge field. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 136–141, April, 1999.     

A quantum model of a real scalar field

A quantum model of a real scalar field with local operator gauge symmetry is discussed. In the localized theory, in order to keep the local operator gauge symmetry, an operator gauge potential BB _μ, is needed. By combining the constraint of operator gauge potentialB _μ, and the microscopic causality theorem, the usual canonical quantization condition of a real scalar field is obtained. Therefore, a quantum model of a real scalar field without the usual procedure of quantizing a related classical model can be directly constructed. Project supported in part by T.D. Lee’s NNSF Grant, National Natural Science Foundation of China, Foundation of Ph. D. Directing Programme of Chinese Universities and the Chinese Academy of Sciences.     

Integrable many-body systems and gauge theories

We review the study of the relation between integrable many-body systems and gauge theories. We show that the degrees of freedom of integrable systems are related to the topological degrees of freedom of gauge theories. We also describe the relation between families of integrable systems and N=2 supersymmetric gauge theories. We show that the degrees of freedom of many-body systems can be identified with the collective coordinates of string theory solitons, theD-branes. This article was written at the request of the Editorial Board. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 1, pp. 3–56, October, 2000.     

Pseudoclassical Foldy-Wouthuysen transformation and canonical quantization of (D=2n)-dimensional relativistic particle with spin in an external electromagnetic field

The canonical quantization of a (D=2n)-dimensional Dirac particle with spin in an arbitrary external electromagnetic field is performed in a gauge that makes it possible to describe simultaneously particles and antiparticles (both massive and massiess) already at the classical level. A pseudoclassical Foldy-Wouthuysen transformation is used to find the canonical (Newton-Wigner) coordinates. The connection between this quantization scheme and Blount's picture describing the behavior of a Dirac particle in an external electromagnetic field is discussed.Erevan Physics Institute. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 3, pp. 378–383, March, 1995.     

Yang—mills fields with external current

A procedure for quantizing a non-Abelian gauge theory with Lagrangian $$L = - \frac{1}{4}F_{\mu v}^a F_a^{\mu v} + J_a^\mu V_\mu ^a$$ near a nontrivial classical solution is considered. The theories are classified with respect to the external current. The gluon propagator in a model spherically symmetric non-Abelian field is constructed and investigated.     

Semilinear subelliptic problems with critical growth on Carnot groups

We prove existence and multiplicity of solutions for the semilinear subelliptic problem with critical growth in Ω, u = 0 on ∂Ω, where is a sublaplacian on a Carnot group , 2* = 2Q/(Q − 2) is the critical Sobolev exponent for and Ω is a bounded domain of .     

Dirac-wave maps

Motivated by supersymmetric field theories, we couple the nonlinear sigma model on 1 + 1-dimensional Minkowski space with a spinor field. The corresponding Lagrangian is . The classical solutions of this model are called Dirac-wave maps. We prove that there exists a unique global solution for given Cauchy data.Received: 28 May 2004, Accepted: 9 June 2004, Published online: 16 July 2004     

A Lie-theoretic setting for the classical interpolation theories

The three classical interpolation theories — Newton-Lagrange, Thiele and Pick-Nevanlinna — are developed within a common Lie-theoretic framework. They essentially involve a recursive process, each step geometrically providing an analytic map from a Riemann surface to a Grassmann manifold. The operation which passes from the (n−1)st to the nth involves the action of what the physicists call a group of gauge transformations. There is also a first-order difference operator which maps the set of solutions of the nth order interpolation to the (n−1)st: This difference operator is, in each case, covariant with respect to the action of the Lie groups involved. For Newton-Lagrange interpolation, this Lie group is the group of affine transformations of the complex plane; for Thiele interpolation the group SL(2, C) of projective transformations; and for Pick-Nevanlinna interpolation the subgroup SU(1, 1) of SL(2, C) which leaves invariant the disk in the complex plane. National Research Council Senior Research Associate at the Ames Research Center (NASA)}.     

Time periodic solutions of smooth convergent dissipative ε-approximations of the modified Navier-Stikes equations

In this paper, we prove the global existence of time periodic classical solutions v' of dissipative ε-approximations (4)–(6) for the three-dimensional modified Navier-Stokes Eqs. (1)–(3) that satisfy the first boundary condition. We also study the convergence for ε → 0 of solutions {v'} to time periodic classical solutions v of Eqs. (1)–(3). Bibliography: 21 titles. Dedicated to V. A. Solonnikov on his sixtieth anniversary Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 213, 1994, pp. 116–130.     

KZ equation,G-opers,quantum Drinfeld–Sokolov reduction,and quantum Cayley–Hamilton identity

The Lax operator of Gaudin-type models is a 1-form at the classical level. In virtue of the quantization scheme proposed by D. Talalaev, it is natural to treat the quantum Lax operator as a connection; this connection is a partcular case of the Knizhnik–Zamolodchikov connection. In this paper, we find a gauge trasformation that produces the “second normal form,” or the “Drinfeld–Sokolov” form. Moreover, the differential operator nurally corresponding to this form is given precisely by the quantum characteristic polynomial of the Lax operator (this operator is called the G-oper or Baxter operator). This observation allows us to relate solutions of the KZ and Baxter equations in an obvious way, and to prove that the immanent KZ equation has only meromorphic solutions. As a corollary, we obtain the quantum Cayley–Hamilton identity for Gaudin-type Lax operators (including the general case). The presented construction sheds a new light on the geometric Langlands correspondence. We also discuss the relation with the Harish-Chandra homomorphism. Bibliography: 19 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 246–259.     

On some nonlinear evolution equations with the strong dissipation,II

In this paper we continue the existence theories of classical solutions of nonlinear evolution equations with the strong dissipation studied in a previous paper [5]. In particular, we give sufficient conditions under which some of the equations have global solutions and at the same time we find steady state solutions of these equations which are exponentially stable as t → ∞. In the application, we improve the existence results to the equations which describe a local statement of balance of momentum for materials for which the stress is related to strain and strain rate through some constitutive equation (cf. Greenberg et al. [6], Greenberg [7], Davis [2], Clements [1], etc.).     

Spectra of radiation for spinless and spin relativistic particles in strong magnetic fields

Synchro-curvature radiation put forward by Zhang and Cheng is a new and universal radiation mechanism, it describes in detail the radiation properties of a relativistic charged particle moving in a curved magnetic field. This new radiation generalizes all the classical results of ordinary synchrotron and curvature radiations and reveals inherent linkage and unification between them. Additionly, a general, simple and unitary formula is provided for discussing the radiation problem in research of pulsars. However, the magnitude of the magnetic field of a pulsar is so strong (10⁷-10⁹T) that the quantum effects cannot be neglected. The GFWW method developed recently by Lieu and Axford is applied to generalize the results of Zhang and Cheng. The quantum limited synchro-curvature radiation spectra for spinless K-G particles and spin-1/2 Dirac particles are presented, respectively. Their radiation properties are also discussed. Project supported by the National Natural Science Foundation of China and the National “Climbing Project”.