Quaternion-Octonion Analyticity for Abelian and Non-Abelian Gauge Theories of Dyons

Einstein-Schrödinger (ES) non-symmetric theory has been extended to accommodate the Abelian and non-Abelian gauge theories of dyons in terms of the quaternion-octonion metric realization. Corresponding covariant derivatives for complex, quaternion and octonion spaces in internal gauge groups are shown to describe the consistent field equations and generalized Dirac equation of dyons. It is also shown that quaternion and octonion representations extend the so-called unified theory of gravitation and electromagnetism to the Yang-Mill’s fields leading to two SU(2) gauge theories of internal spaces due to the presence of electric and magnetic charges on dyons.     

Supersymmetrization of Quaternion Dirac Equation for Generalized Fields of Dyons

The quaternion Dirac equation in presence of generalized electromagnetic field has been discussed in terms of two gauge potentials of dyons. Accordingly, the supersymmetry has been established consistently and thereafter the one, two and component Dirac Spinors of generalized quaternion Dirac equation of dyons for various energy and spin values are obtained for different cases in order to understand the duality invariance between the electric and magnetic constituents of dyons.     

Bound States of Monopoles and Dyons with Spin-1/2 Particles in SU(2) Gauge Theory Under Moduli Space Approximation

Energy spectrum and degeneracy associated with bound states of monopoles and dyons in non-Abelian gauge theory has been investigated and it is shown that energy levels expand due to the presence of additional degeneracies. Splitting of energy level of dyonium in presence of external magnetic and electric field has also been analyzed confirming the presence of additional degeneracy levels of the system. In addition, the study of behaviour of a fermion moving in the field of non-Abelian dyon in moduli space under SU(2) gauge potential has been undertaken and energy eigen values for the system are carried out. Detailed analysis of relativistic correction in fermion-dyon system in moduli space is presented and angular momentum operators of the system are derived, which demonstrates that besides the contribution of Higgs field, the interaction of spin and orbital angular momentum of moving fermion also contributes to the energy operator.     

Non-Abelian Tensor Multiplet Equations from Twistor Space

We establish a Penrose–Ward transform yielding a bijection between holomorphic principal 2-bundles over a twistor space and non-Abelian self-dual tensor fields on six-dimensional flat space-time. Extending the twistor space to supertwistor space, we derive sets of manifestly ${\mathcal{N} = (1, 0)}$ and ${\mathcal{N} = (2, 0)}$ supersymmetric non-Abelian constraint equations containing the tensor multiplet. We also demonstrate how this construction leads to constraint equations for non-Abelian supersymmetric self-dual strings.     

Classes on Compactifications of the Moduli Space of Curves Through Solutions to the Quantum Master Equation

In this paper we describe a construction which produces classes in compactifications of the moduli space of curves. This construction extends a construction of Kontsevich which produces classes in the open moduli space from the initial data of a cyclic A _∞-algebra. The initial data for our construction are what we call a ‘quantum A _∞-algebra’, which arises as a type of deformation of a cyclic A _∞-algebra. The deformation theory for these structures is described explicitly. We construct a family of examples of quantum A _∞-algebras which extend a family of cyclic A _∞-algebras, introduced by Kontsevich, which are known to produce all the kappa classes using his construction.      

Zero Modes for the Magnetic Pauli Operator in Even-Dimensional Euclidean Space

We study the ground state of the Pauli Hamiltonian with a magnetic field in ${\mathbb{R}^{2d}}$ , d > 1. We consider the case where a scalar potential W is present and the magnetic field B is given by ${B=2i\partial\bar{\partial} W}$ . The main result is that there are no zero modes if the magnetic field decays faster than quadratically at infinity. If the magnetic field decays quadratically then zero modes may appear, and we give a lower bound for the number of them. The results in this paper partly correct a mistake in a paper from 1993.     

Moduli Space of BPS Walls in Supersymmetric Gauge Theories

Existence and uniqueness of the solution are proved for the ‘master equation’ derived from the BPS equation for the vector multiplet scalar in the U(1) gauge theory with N _F charged matter hypermultiplets with eight supercharges. This proof establishes that the solutions of the BPS equations are completely characterized by the moduli matrices divided by the V-equivalence relation for the gauge theory at finite gauge couplings. Therefore the moduli space at finite gauge couplings is topologically the same manifold as that at infinite gauge coupling, where the gauged linear sigma model reduces to a nonlinear sigma model. The proof is extended to the U(N _C) gauge theory with N _F hypermultiplets in the fundamental representation, provided the moduli matrix of the domain wall solution is U(1)-factorizable. Thus the dimension of the moduli space of U(N _C) gauge theory is bounded from below by the dimension of the U(1)-factorizable part of the moduli space. We also obtain sharp estimates of the asymptotic exponential decay which depend on both the gauge coupling and the hypermultiplet mass differences.     

Dyons in topological field theories

We examine a class of topological field theories defined by Lagrangians that under certain conditions can be written as the sum of two characteristic numbers or winding numbers. Therefore, the action or the energy is a topological invariant and stable under perturbations. The sufficient conditions required for stability take the form of first-order field equations, analogous to the self-duality and Bogomol'nyi equations in Yang-Mills(-Higgs) theory. Solutions to the first-order equations automatically satisfy the full field equations. We show the existence of nontrivial, nonsingular, minimum energy spherically symmetric dyon solutions and that they are stable. We also discuss evidence for a dual field theory to Yang-Mills-Higgs in topological field theory. The existence of dual field theories and electric monopoles is predicted by Montonen and Olive.     

Moduli Spaces of Instantons on the Taub-NUT Space

We present ADHM-Nahm data for instantons on the Taub-NUT space and encode these data in terms of Bow Diagrams. We study the moduli spaces of the instantons and present these spaces as finite hyperkähler quotients. As an example, we find an explicit expression for the metric on the moduli space of one SU(2) instanton.We motivate our construction by identifying a corresponding string theory brane configuration. By following string theory dualities we are led to supersymmetric gauge theories with impurities.     

Sedenionic Formulation for Generalized Fields of Dyons

In this paper, the conic sedenionic formulation is presented for the unification of generalized field equations of dyons (electromagnetic theory) and gravito-dyons (linear gravity). Sedenionic definitions of generalized potential and current equations for dyons are discussed in consistent manner. The sixteen dimensional field equations related to linear gravity and electromagnetism have been written as one sedenionic equation in elegant and compact form.     

Gauge Formulation for Two Potential Theory of Dyons

Dual electrodynamics and corresponding Maxwell’s equations (in the presence of monopole only) are revisited from the symmetry of duality and gauge invariance. Accordingly, the manifestly covariant, dual symmetric and gauge invariant two potential theory of generalized electromagnetic fields of dyons has been developed consistently from U(1)×U(1) gauge symmetry. Corresponding field equations and equation of motion are derived from Lagrangian formulation adopted for U(1)×U(1) gauge symmetry for the justification of two four potentials of dyons.     

Spontaneous Edge Currents for the Dirac Equation in Two Space Dimensions

Spontaneous edge currents are known to occur in systems of two space dimensions in a strong magnetic field. The latter creates chirality and determines the direction of the currents. Here we show that an analogous effect occurs in a field-free situation when time reversal symmetry is broken by the mass term of the Dirac equation in two space dimensions. On a half plane, one sees explicitly that the strength of the edge current is proportional to the difference between the chemical potentials at the edge and in the bulk, so that the effect is analogous to the Hall effect, but with an internal potential. The edge conductivity differs from the bulk (Hall) conductivity on the whole plane. This results from the dependence of the edge conductivity on the choice of a selfadjoint extension of the Dirac Hamiltonian. The invariance of the edge conductivity with respect to small perturbations is studied in this example by topological techniques Mathematics Subject Classification (2000). 81Q10, 58J32     

Dirac Equation in Space–Time with Torsion

The Dirac equation in a curved space–time endowed with compatible affine connection is reconsidered. After a detailed decomposition of the total action, the equation is obtained by varying with respect to the Dirac spinor and the torsion field. The result is a known Dirac-like equation with constraints that can be interpreted as the equation of a self-interacting spin 1/2 particle in curved space–time. The scheme is then translated into the language of the 2-spinor formalism of curved space–time based on the choice of a null tetrad frame. The spinorial equation so obtained coincides with the standard one in case of no torsion, while in general it remains a nonlinear equation describing a self-interacting spin 1/2 particle. The nonlinearity is produced by the interaction of the particle with its own current that remains conserved as in the free torsion case.     

Phase Space Compression in One-Dimensional Complex Ginzburg-Landau Equation

The transition from stationary to oscillatory states in dynamical systems under phase space compression is investigated. By considering the model for the spatially one-dimensional complex Ginzburg-Landau equation, we find that defect turbulence can be substituted with stationary and oscillatory signals by applying system perturbation and confining variable into various ranges. The transition procedure described by the oscillatory frequency is studied via numerical simulations in detail.     

Split Octonion Reformulation for Electromagnetic Chiral Media of Massive Dyons

In an explicit, unified, and covariant formulation of an octonion algebra, we study and generalize the electromagnetic chiral fields equations of massive dyons with the split octonionic representation. Starting with 2 × 2 Zorn's vector matrix realization of split-octonion and its dual Euclidean spaces, we represent the unified structure of split octonionic electric and magnetic induction vectors for chiral media. As such, in present paper, we describe the chiral parameter and pairing constants in terms of split octonionic matrix representation of Drude-Born-Fedorov constitutive relations. We have expressed a split octonionic electromagnetic field vector for chiral media, which exhibits the unified field structure of electric and magnetic chiral fields of dyons. The beauty of split octonionic representation of Zorn vector matrix realization is that, the every scalar and vector components have its own meaning in the generalized chiral electromagnetism of dyons. Correspondingly, we obtained the alternative form of generalized Proca–Maxwell's equations of massive dyons in chiral media. Furthermore, the continuity equations, Poynting theorem and wave propagation for generalized electromagnetic fields of chiral media of massive dyons are established by split octonionic form of Zorn vector matrix algebra.