Three-dimensional vortices in Abelian gauge theories

In this paper we consider an Abelian Gauge Theory in R⁴${\mathbb{R}}^4$ equipped with the Minkowski metric. This theory leads to a system of equations, the Klein–Gordon–Maxwell equations, which provide models for the interaction between the electromagnetic field and matter. A three-dimensional vortex is a finite energy solution of these equations in which the magnetic field looks like the field created by a finite solenoid. Under suitable assumptions, we prove the existence of vortex solutions.     

Electrically charged solitons in gauge field theory

Monopoles and vortices are well known magnetically charged soliton solutions of gauge field equations. Extending the idea of Dirac on monopoles, Schwinger pioneered the concept of solitons carrying both electric and magnetic charges, called dyons, which are useful in modeling elementary particles. Mathematically, the existence of dyons presents interesting variational partial differential equation problems, subject to topological constraints. This article is a survey on recent progress in the study of dyons.     

Sudakov scaling and the gauge/string duality

We investigate the anomalous dimensions of (super)conformal Wilson operators in the weak-and strong-coupling regimes using the integrability symmetry on both sides of the gauge/string correspondence. We study the origin of the single-logarithmic asymptotic behavior of long operators/strings in the limit of large Lorentz spin. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 2, pp. 204–218, February, 2007.     

On the microscopic origin of integrability in the Seiberg-Witten theory

We discuss the microscopic origin of integrability in the Seiberg-Witten theory. In particular, we discuss the theory in more detail with the simplest higher perturbation in the ultraviolet, where additional explicit results are obtained using bosonization and elliptic uniformization of the spectral curve. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 3, pp. 424–450, March, 2008.     

Integrability in String/Field Theories and Hamiltonian Flows in the Space of Physical Systems

Integrability in string/field theories is known to emerge when considering dynamics in the moduli space of physical theories. This implies that one must study the dynamics with respect to unusual time variables such as coupling constants or other quantities parameterizing the configuration space of physical theories. The dynamics given by variations of coupling constants can be considered as a canonical transformation or, infinitesimally, a Hamiltonian flow in the space of physical systems. We briefly consider an example of integrable mechanical systems. Then any function T generates a one-parameter family of integrable systems in the vicinity of a single system. For an integrable system with several coupling constants, the corresponding Hamiltonians T_i satisfy the Whitham equations and, after quantization (of the original system), become operators satisfying the zero-curvature condition in the coupling-constant space.     

Quantization of the N=2 supersymmetriC KdV hierarchy

We continue the study of the quantization of supersymmetric integrable KdV hierarchies. We consider the N=2 KdV model based on the sl(1)(2|1) affine algebra but with a new algebraic construction for the L-operator, different from the standard Drinfeld-Sokolov reduction. We construct the quantum monodromy matrix satisfying a special version of the reflection equation and show that in the classical limit, this object precisely gives the monodromy matrix of the N=2 supersymmetric KdV system. We also show that at both the classical and the quantum levels, the trace of the monodromy matrix (transfer matrix) is invariant under two supersymmetry transformations and the zero mode of the associated U(1) current. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 303–314, May, 2006.     

一平面可积三次非Hamilton系统的Abel积分

本文讨论一平面可积三次非Hamilton系统在n次多项式扰动下Abel积分零点个数上确界，得到的结论是该Abel积分的零点个数的上确界为n。     

关于AKNS族和一个耦合的MKdV族的规范等价可积系统与r-矩阵

得到了一个新的耦合的MKdV族．通过规范变换，首次从AKNS族中得到耦合的MKdV族的Lax表示、可积系与约束流；利用Lax表示，构造了耦合MKdV族的约束流的γ-矩阵，同时也给出了该方程族约束流的第二组守恒积分与对合性．     

Theory of submanifolds,associativity equations in 2D topological quantum field theories,and Frobenius manifolds

We prove that the associativity equations of two-dimensional topological quantum field theories are very natural reductions of the fundamental nonlinear equations of the theory of submanifolds in pseudo-Euclidean spaces and give a natural class of flat torsionless potential submanifolds. We show that all flat torsionless potential submanifolds in pseudo-Euclidean spaces bear natural structures of Frobenius algebras on their tangent spaces. These Frobenius structures are generated by the corresponding flat first fundamental form and the set of the second fundamental forms of the submanifolds (in fact, the structural constants are given by the set of the Weingarten operators of the submanifolds). We prove that each N-dimensional Frobenius manifold can be locally represented as a flat torsionless potential submanifold in a 2N-dimensional pseudo-Euclidean space. By our construction, this submanifold is uniquely determined up to motions. Moreover, we consider a nonlinear system that is a natural generalization of the associativity equations, namely, the system describing all flat torsionless submanifolds in pseudo-Euclidean spaces, and prove that this system is integrable by the inverse scattering method. To the memory of my wonderful mother Maya Nikolayevna Mokhova (4 May 1926–12 September 2006) Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 368–376, August, 2007.     

Matrix Models and Matrix Integrals

We discuss the relations between arbitrary solutions of the loop equations describing the Hermitean one-matrix model and particular (multicut) solutions corresponding to concrete matrix integrals. These latter have a series of specific properties and, in particular, are described in terms of the Seiberg-Witten-Whitham theory. We consider the simplest example of an ordinary integral in detail. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 77–89, January, 2006.     

Symplectic Topology of Integrable Hamiltonian Systems, II: Topological Classification

The main purpose of this paper is to give a topological and symplectic classification of completely integrable Hamiltonian systems in terms of characteristic classes and other local and global invariants.     

Multipoint correlation functions in Liouville field theory and minimal Liouville gravity

We study (n+3)-point correlation functions of exponential fields in the Liouville field theory with n degenerate and three arbitrary fields and derive an analytic expression for these correlation functions in terms of Coulomb integrals. We consider the application of these results to the minimal Liouville gravity. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 3, pp. 536–556, March, 2008.     

Semiclassical geometry and integrability of the AdS/CFT correspondence

We discuss the semiclassical geometry and integrable systems related to the gauge—string duality. We analyze semiclassical solutions of the Bethe ansatz equations arising in the context of the AdS/CFT correspondence, comparing them to stationary phase equations for the matrix integrals. We demonstrate how the underlying geometry is related to the integrable sigma models of the dual string theory and investigate some details of this correspondence.Translated from Teoreticheskaya Matematicheskaya Fizika, Vol. 142, No. 2, pp. 265–283, February, 2005.     

On the measure with maximal entropy for the Teichmüller flow on the moduli space of Abelian differentials

The Teichmüller flow g _t on the moduli space of Abelian differentials with zeros of given orders on a Riemann surface of a given genus is considered. This flow is known to preserve a finite absolutely continuous measure and is ergodic on every connected component ? of the moduli space. The main result of the paper is that µ/µ(?) is the unique measure with maximal entropy for the restriction of g _t to ?. The proof is based on the symbolic representation of g _t .     

Concave gauge functions and applications

Many problems of optimization involve the minimization of an objective function on a convex cone. In this respect we define a concave gauge function which will be used in interior point methods.Application are given in particular on the space of real symmetric matrices.     

Linear estimate of the number of zeros of Abelian integrals for quadratic centers having almost all their orbits formed by cubics

We study the number of zeros of Abelian integrals for the quadratic centers having almost all their orbits formed by cubics, when we perturb such systems inside the class of all polynomial systems of degreen     

Vacuum solutions in the Neveu-Schwarz field theory of a fermionic string and the zero-curvature representation in graded spaces

We demonstrate that vacuum solutions in the Neveu-Schwarz field theory of a fermionic string incorporating the GSO(−) sector are naturally related to the zero-curvature representation in a graded space of special form. We use the same representation to describe the equivalence of the cubic and nonpolynomial theories with the GSO(−) sector also taken into account. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 159, No. 1, pp. 131–141, April, 2009.     

一类可积非Hamilton系统的Abel积分的零点估计

讨论了一类含参可积非Hamilton系统在一般二次多项式扰动下的Abel积分的零点,得出了不同参数范围下的Abel积分的零点数目的估计.     

Strings, susy gauge theories, and integrable systems

We consider supersymmetric Yang-Mills theories in the general framework of string theory (M-theory) and describe particular compactifications leading to the Seiberg-Witten (SW) exact effective theories. We show strong parallels between SW and integrable system theories. This paper was written at the request of the Editorial Board. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 2, pp. 179–243. November, 1999.     

Abelian Chern-Simons vortices and holomorphic Burgers hierarchy

We consider the Abelian Chern-Simons gauge field theory in 2+1 dimensions and its relation to the holomorphic Burgers hierarchy. We show that the relation between the complex potential and the complex gauge field as in incompressible and irrotational hydrodynamics has the meaning of the analytic Cole-Hopf transformation, linearizing the Burgers hierarchy and transforming it into the holomorphic Schrödinger hierarchy. The motion of planar vortices in Chern-Simons theory, which appear as pole singularities of the gauge field, then corresponds to the motion of zeros of the hierarchy. We use boost transformations of the complex Galilei group of the hierarchy to construct a rich set of exact solutions describing the integrable dynamics of planar vortices and vortex lattices in terms of generalized Kampe de Feriet and Hermite polynomials. We apply the results to the holomorphic reduction of the Ishimori model and the corresponding hierarchy, describing the dynamics of magnetic vortices and the corresponding lattices in terms of complexified Calogero-Moser models. We find corrections (in terms of Airy functions) to the two-vortex dynamics from the Moyal space-time noncommutativity.