Narrow Results

A SU(2) Gauge Theory with spherical symmetry over the Minkowski space-time is considered. The self-duality equation of the gauge fields are written and their solutions are obtained. Two exact solutions, one of which is statical and another of dynamical type are given. All the calculations are performed using an analytical program written in GRTensor computer algebra package, which runs on the MapleV platform.

A model of Poincaré gauge self-dual theory over the Minkowski space–time, endowed with spherical symmetry, is considered. The self-duality (S-D) conditions are imposed and two sets of the self-duality equations for the gauge fields are obtained: one for the torsion tensor and another for the curvature tensor. An analytical solution of the gauge field equations is also obtained. The Yang–Mills (Y–M) equations of the gauge fields are also derived. It is shown that these Y–M equations can be obtained from the S-D equations, a result which is generally true for any S-D gauge theory. Most of all calculations are performed using the GRTensorII package, running on the MapleV platform.

A pair of Mond–Weir type nondifferentiable second-order symmetric primal and dual problems in mathematical programming is formulated. Weak duality, strong duality, and converse duality theorems are established under η-pseudobonvexity assumptions. Symmetric minimax mixed integer primal and dual problems are also investigated. Moreover, the self duality theorem is also discussed.

The non-Abelian self-dual action in three dimensions is derived using the self-interaction mechanism.

It is well known that Einstein gravity can be formulated as a gauge theory of Lorentz group where spin connections play a role of gauge fields and Riemann curvature tensors correspond to their field strengths. One can then pose an interesting question: What is the Einstein equation from the gauge theory point of view? Or equivalently, what is the gauge theory object corresponding to Einstein manifolds? We show that the Einstein equations in four dimensions are precisely self-duality equations in Yang–Mills gauge theory and so Einstein manifolds correspond to Yang–Mills instantons in SO(4) = SU(2)_L × SU(2)_R gauge theory. Specifically, we prove that any Einstein manifold with or without a cosmological constant always arises as the sum of SU(2)_L instantons and SU(2)_R anti-instantons. This result explains why an Einstein manifold must be stable because two kinds of instantons belong to different gauge groups, instantons in SU(2)_L and anti-instantons in SU(2)_R, and so they cannot decay into a vacuum. We further illuminate the stability of Einstein manifolds by showing that they carry nontrivial topological invariants.

Partially massless theory in three dimensions is revisited and its relationship with the self-dual massive gravity is considered. The only mode of the partially massless theory is shown explicitly through an action for a scalar field on (A)dS background. This fact can be generalized to higher dimensions. This degree of freedom is altered when a triadic Chern–Simons is introduced, giving rise to the self-dual massive gravity on (A)dS background. We present another physical system with partially massless symmetry and its connection with topologically massive gravity is discussed.

We discuss three mathematical structures which arise in topologically massive Abelian gauge theory. First, the Euclidean topologically massive Abelian gauge theory defines a contact structure on a manifold. We briefly discuss three solutions and the related contact structures on the flat 3-torus, the AdS space, the 3-sphere which respectively correspond to Bianchi type I, VIII, IX spaces. We also present solutions on Bianchi type II, VI and VII spaces. Secondly, we discuss a family of complex (anti-)self-dual solutions of the Euclidean theory in Cartesian coordinates on which are given by (anti)holomorpic functions. The orthogonality relation of contact structures which are determined by the real parts of these complex solutions separates them into two classes: the self-dual and the anti-self-dual solutions. Thirdly, we apply the curl transformation to this theory. An arbitrary solution is given by a vector tangent to a sphere whose radius is determined by the topological mass in transform space. Meanwhile a gauge transformation corresponds to a vector normal to this sphere. We discuss the quantization of topological mass in an example.

We present a unified analysis of the self-dual, second order, topologically massive and the recently introduced fourth-order models of massive gravity in 3D. We show that there is a family of first-order actions which interpolate between these different single excitation models. We show how the master actions are related by duality transformation. We construct by the same method the master action which relates the fourth-order new massive model with two excitations and the usual second-order model with Fierz–Pauli mass. We show that the more general model obtained by adding a Chern–Simons term to the new massive model is equivalent off-shell to the second-order spontaneously broken linearized massive gravity.

We examine a generalization of the usual self-duality equations for Yang–Mills theory when the color space admits a nontrivial involution. This involution allows us to construct a nontrivial twist which may be combined with the Hodge star to form a twisted self-dual curvature. We will construct a simple example of twisted self-duality for $su(2)\oplus su(2)$ gauge theory along with its explicit solutions, both in Euclidean and Minkowski backgrounds, and then dimensionally reduce from four dimensions to obtain families of nontrivial nonlinear equations in lower dimensions. This twisted self-duality constraint will be shown to arise in $E_7$ exceptional field theory through a Scherk–Schwarz reduction and we will show how an Eguchi–Hanson gravitational instanton also obeys the twisted self-duality condition.

We define a duality of solutions in coalitional games that we call a twisted duality. This twisted duality extends the rule of self-duality in bankruptcy problems. After showing that the prekernel and prenucleolus exhibit twisted duality, we define a twisted reduced game property and characterize the prekernel and prenucleolus by axioms including this property. We note that the Shapley value satisfies twisted duality, and we define another twisted reduced game property by and characterize the Shapley value by axioms including this property.

We define Harada rings of a component type. The class of them contains indecomposable serial rings. We consider the structure of them and show that they have weakly symmetric self-dualities.

In this paper, an intrinsic description of some families of linear codes with symmetries is given, showing that they can be described more generally as quasi-group codes, that is, as linear codes allowing a group of permutation automorphisms which acts freely on the set of coordinates. An algebraic description, including the concatenated structure, of such codes is presented. This allows to construct quasi-group codes from codes over rings, and vice versa. The last part of the paper is dedicated to the investigation of self-duality of quasi-group codes.

In this paper, we construct a self-dual complete resolution of a module defined by a pair of embedded complete intersection ideals in a local ring. Our construction is based on a gluing construction of Herzog and Martsinkovsky and exploits the structure of Koszul homology in the embedded complete intersection case. As a consequence of our construction, we produce an isomorphism between certain stable homology and cohomology modules.

Seiberg–Witten equations on 8-manifolds with Spin(7)-holonomy are introduced in [A. H. Bilge, T. Dereli and S. Koçak, Monopole equations on 8-manifolds with Spin(7) holonomy, Commun. Math. Phys.203(1) (1999) 21–30] and it is given a local solution for these equations. In this work, we write down similar Seiberg–Witten equations on 8-dimensional manifolds with SU(4)-structure and give a global solution for these equations.

Bianchi-IX four metrics are SU(2) invariant solutions of vacuum Einstein equation, for which the connection-wise self-dual case describes the Euler top, while the curvature-wise self-dual case yields the Ricci flat classical Darboux–Halphen system. It is possible to see such a solution exhibiting Ricci flow. The classical Darboux–Halphen system is a special case of the generalized one that arises from a reduction of the self-dual Yang–Mills equation and the solutions to the related homogeneous quadratic differential equations provide the desired metric. A few integrable and near-integrable dynamical systems related to the Darboux–Halphen system and occurring in the study of Bianchi-IX gravitational instanton have been listed as well. We explore in details whether self-duality implies integrability.

We review recent advances in the full counting statistics (FCS) of transport of charge through an impurity in a 1D quantum wire. The model also applies to a coherent conductor in a resistive environment and to a resistively shunted Josephson device. Starting out from the path integral Coulomb gas representation both the weak and strong tunneling series representations for the cumulant generating function can be found in analytic form in particular regions of the parameter space. The zero temperature case is discussed in some detail.