Space-time locality in theSp(2)-symmetric Lagrangian formalism

The existence of a local solution to the Sp(2) master equation for the gauge field theory is proved in the perturbation theory under standard regularity assumptions for the action. The arbitrariness of solutions to the Sp(2) master equation is described, provided they are proper. The effective action can be chosen to be Sp(2) invariant and (under the additional assumption that the gauge transformation generators are Lorentz tensors) Lorentz invariant. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 115. No. 3, pp. 373–388, June, 1998.     

The holomorphic effective action inN=2D=4 supergauge theories with various gauge groups

We consider the theory of hypermultiplets in arbitrary representations of arbitrary semisimple gauge groups coupled to gauge superfields. Using the N=2 harmonic superspace formulation of these models, we find the general structure of the holomorphic effective action depending on the gauge superfield with values in the Cartan subalgebra of the gauge algebra. We find explicit expressions for the effective actions in the cases where the hypermultiplets are in the fundamental and adjoint representations of SU(n), SO(n), and Sp(2n). Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 3, pp. 444–455, April, 2000.     

Kaluza-Klein or fiber bundles?

Two basic geometric approaches to the modern theory of gauge fields are analyzed and compared. The first approach is an extension of the Kaluza-Klein unified theory of gravity and electromagnetism (1921). The second approach generalizes Cartan's formulation (1925) of Riemannian geometry and GR which now is transformed to fiber bundle theory. The goal of this paper is to show that the above-mentioned geometric approaches to the classical gauge field theory are nonequivalent and lead to different forms of quantum gauge field theory. Bibliography:12 titles. Dedicated to the memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 215–224. Translated by N. P. Konopleva.     

Canonical formulation of the embedded theory of gravity equivalent to Einstein’s general relativity

We study the approach in which independent variables describing gravity are functions of the space-time embedding into a flat space of higher dimension. We formulate a canonical formalism for such a theory in a form that requires imposing additional constraints, which are a part of Einstein’s equations. As a result, we obtain a theory with an eight-parameter gauge symmetry. This theory becomes equivalent to Einstein’s general relativity either after partial gauge fixing or after rewriting the metric in the form that is invariant under the additional gauge transformations. We write the action for such a theory. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 2, pp. 271–288, November, 2007.     

Applications of gauge duality in robust principal component analysis and semidefinite programming

Gauge duality theory was originated by Freund (1987), and was recently further investigated by Friedlander et al. (2014). When solving some matrix optimization problems via gauge dual, one is usually able to avoid full matrix decompositions such as singular value and/or eigenvalue decompositions. In such an approach, a gauge dual problem is solved in the first stage, and then an optimal solution to the primal problem can be recovered from the dual optimal solution obtained in the first stage. Recently, this theory has been applied to a class of semidefinite programming (SDP) problems with promising numerical results by Friedlander and Macêdo (2016). We establish some theoretical results on applying the gauge duality theory to robust principal component analysis (PCA) and general SDP. For each problem, we present its gauge dual problem, characterize the optimality conditions for the primal-dual gauge pair, and validate a way to recover a primal optimal solution from a dual one. These results are extensions of Friedlander and Macêdo (2016) from nuclear norm regularization to robust PCA and from a special class of SDP which requires the coefficient matrix in the linear objective to be positive definite to SDP problems without this restriction. Our results provide further understanding in the potential advantages and disadvantages of the gauge duality theory.     

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.     

Dual gauge programs,with applications to quadratic programming and the minimum-norm problem

A gauge functionf(·) is a nonnegative convex function that is positively homogeneous and satisfiesf(0)=0. Norms and pseudonorms are specific instances of a gauge function. This paper presents a gauge duality theory for a gauge program, which is the problem of minimizing the value of a gauge functionf(·) over a convex set. The gauge dual program is also a gauge program, unlike the standard Lagrange dual. We present sufficient conditions onf(·) that ensure the existence of optimal solutions to the gauge program and its dual, with no duality gap. These sufficient conditions are relatively weak and are easy to verify, and are independent of any qualifications on the constraints. The theory is applied to a class of convex quadratic programs, and to the minimuml _p norm problem. The gauge dual program is shown to provide a smaller duality than the standard dual, in a certain sense discussed in the text.     

Quantum string theory in the space of states in an indefinite metric

The quantum string theory in the indefinite (pseudo-Hilbert) Fock representation is considered. It is shown that all anomalies in the four-dimensional theory of an open bosonic string are canceled for a particular choice of the vacuum state. Then the algebra of the reparametrization symmetry (in the covariant approach) and the algebra of rotations (in the light-cone gauge) become closed. The indefiniteness of the space of states leads to some nonphysical properties of the theory. Modifications of the model under consideration, which are necessary to construct a more realistic theory, are discussed.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 2, pp. 213–228, May, 1996.Translated by L. O. Chekhov.     

Non-Abelian tensor gauge fields

A recently proposed extension of Yang-Mills theory contains non-Abelian tensor gauge fields. The Lagrangian has quadratic kinetic terms, as well as cubic and quartic terms describing nonlinear interaction of tensor gauge fields with the dimensionless coupling constant. We analyze the particle content of non-Abelian tensor gauge fields. In four-dimensional space-time the rank-2 gauge field describes propagating modes of helicity 2 and 0. We introduce interaction of the non-Abelian tensor gauge field with fermions and demonstrate that the free equation of motion for the spinor-vector field correctly describes the propagation of massless modes of helicity 3/2. We have found a new metric-independent gauge invariant density which is a four-dimensional analog of the Chern-Simons density. The Lagrangian augmented by this Chern-Simons-like invariant describes the massive Yang-Mills boson, providing a gauge invariant mass gap for a four-dimensional gauge field theory.     

Tetrad-Gauge Theory of Gravity

We present a tetrad–gauge theory of gravity based on the local Lorentz group in a four-dimensional Riemann–Cartan space–time. Using the tetrad formalism allows avoiding problems connected with the noncompactness of the group and includes the possibility of choosing the local inertial reference frame arbitrarily at any point in the space–time. The initial quantities of the theory are the tetrad and gauge fields in terms of which we express the metric, connection, torsion, and curvature tensor. The gauge fields of the theory are coupled only to the gravitational field described by the tetrad fields. The equations in the theory can be solved both for a many-body system like the Solar System and in the general case of a static centrally symmetric field. The metric thus found coincides with the metric obtained in general relativity using the same approximations, but the interpretation of gravity is quite different. Here, the space–time torsion is responsible for gravity, and there is no curvature because the curvature tensor is a linear combination of the gauge field tensors, which are absent in the case of pure gravity. The gauge fields of the theory, which (together with the tetrad fields) define the structure of space–time, are not directly coupled to ordinary matter and can be interpreted as the fields describing dark energy and dark matter.     

Exact solutions of classical electrodynamics and the Yang-Mills-Wong theory in even-dimensional space-time

Exact solutions of classical gauge theories in even-dimensional (D=2n) space-time are discussed. Common and specific properties of these solutions are analyzed for the particular dimensions D=2, D=4, and D=6. The consistent formulation of classical gauge field theories with pointlike charged or colored particles is proposed for D=6. The particle Lagrangian must then depend on the acceleration. In D=2, radiation is absent and all processes are invertible w.r.t. time. In D=6, the expression for the radiation intensity, as well as the equation of motion of a self-interacting particle, is obtained; trembling always leads to radiation. Non-Abelian solutions are absent for any D≠4, and only Coulomb-like solutions, which correspond to the Abelian limit of the D-dimensional Yang-Mills-Wong theory, are admitted. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 119–135, April, 1999.     

Superfield formulation of the SP(2)-covariant quantization method in gauge theories

A Lagrange quantization scheme for a general gauge theory based on the BRST-and anti-BRST-symmetry principles is proposed in the superspace D=d+2, where d is the space-time dimension. For superfields, the BRST- and anti-BRST-transformations are realized as shifts along the auxiliary (Grassmann) coordinates of the superspace.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 2, pp. 229–237, May, 1996.Translated by L. O. Chekhov.     

Infrared Variables for the SU(3) Yang–Mills Field

We generalize the self-dual parameterization of the SU(2) Yang–Mills field proposed by Niemi and Faddeev for describing the infrared limit of the theory to the case of the gauge group SU(3). We demonstrate that the duality property intrinsic to the SU(2) gauge field cannot be transferred automatically to the higher-rank group case. We interpret the algebraic structures appearing in the Lagrangian for the new compact variables in terms of the group products SU(2)³.     

Construction of a perturbatively correct light-front Hamiltonian for a (2+1)-dimensional gauge theory

We discuss a perturbation theory on the light front regularized by a method analogous to Pauli–Villars regularization for the (2+1)-dimensional SU(N)-symmetric gauge theory. This allows constructing a correct renormalized light-front Hamiltonian.     

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.     

The condensate 〈(A µ 2 )〉 in commutative and noncommutative theories

We show that the gauge invariance of the operator ∫ dx tr(A _μ ² −2/(gξ)x ^υθ_μυ A ^μ) in a noncommutative gauge theory does not lead to the gauge independence of its vacuum condensate. We obtain the generalized Ward identities for Green’s functions containing the operator lim_Ω→∞(1/Ω)∫_Ω dx tr (A _μ ² ) in commutative and noncommutative gauge theories. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 3, pp. 350–356, September, 2006.     

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.     

Pure gauge configurations and tachyon solutions for cubic fermionic string field theory equations of motion

Special perturbative pure gauge solutions parameterized by a pair of wedge states are parts of the nontrivial (not purely gauge) tachyon solutions of the cubic fermionic string field theory describing the non-BPS brane true vacuum. We demonstrate explicitly that for the large parameter of the perturbation expansion, these pure gauge configurations are no longer solutions of the equations of motion. We show that this problem is solved by adding an extra term that is just the term needed for the first Sen conjecture to hold.     

On the canonical quantization of anomalousSu(N) chiral Yang-Mills models

Canonical quantization of the anomalous SU(N) Yang-Mills models is considered. It is shown that the gauge invariance of quantum theory can be preserved in spite of the degeneracy of the Wess-Zumino action.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 2, pp. 315–326, August, 1996.     

SU(2)规范场的恰当形式(欧空间)

应用数学机械化方法研究欧氏空间中SU(2)规范场的正规化问题.首先对Yang-Mills方程的表述进行了讨论,给出了一种具有物理和几何意义的YM-方程,称其为恰当的(exact)YM-方程.对于这种恰当的YM-方程,构造了一类线性微分变换,称之为SU(2)规范场的示性变换.经由示性变换,将非线性的恰当的YM-方程变为一组Laplace方程,实现了SU(2)规范场方程的线性化,即场方程的正规化.从而证明了SU(2)规范场存在3个独立的Yang-Mills规范场.