Self-dual Yang-Mills fields in eight dimensions

Strongly self-dual Yang-Mills fields in even-dimensional spaces are characterised by a set of constraints on the eigenvalues of the Yang-Mills fieldsF . We derive a topological bound on R⁸, , wherep1 is the first Pontryagin class of the SO(n) Yang-Mills bundle, andk is a constant. Strongly self-dual Yang-Mills fields realise the lower bound.     

Self-dual Yang-Mills fields and Nahm's equations

For the gauge fields with values in arbitrary semisimple Lie algebra we introduce the ansatzes which reduce the self-duality equations in the Euclidean spaceR ^4,0 to the well-known Nahm equations.     

On the classical lump of Yang-Mills fields

It is shown that for pure Yang-Mills fields there is no lump phenomena if the total energy is infinite and diverges mildly in a certain sense. This improves the well-known classical result on the absence of a lump for the finite total energy case. Some exact lump solutions are obtained showing that the energy condition cannot be removed. The results are valid for more general fields.This work was supported by the Chinese Natural Science Foundation, the Chinese Fund of Doctor Programs, and the University of Paris VI, and University of Bourgogne.     

Self-dual Yang-Mills fields in d=7,8, octonions and ward equations

The Yang-Mills theories in d=7 and d=8 with the arbitrary gauge group G are considered. Generalized self-duality-type relations for gauge fields are reduced to systems of nonlinear differential equations on functions of one variable (Ward equations). Ward equations may be reduced to equations which follow from Yang-Baxter equations. This permits us to obtain new classes of explicit solutions of the Yang-Mills equations in d=7 and d=8.     

Frame-like action and unfolded formulation for massive higher-spin fields

Unfolded equations of motion for symmetric massive bosonic fields of any spin in Minkowski and (A)dS$(A)\mathit{dS}$ spaces are presented. Manifestly gauge invariant action for a spin s?2$s?2$ massive field in any dimension is constructed in terms of gauge invariant curvatures.     

On the holomorphic factorization for superconformal fields

For a generic value of the central charge, we prove the holomorphic factorization of partition functions for free superconformal fields which are defined on a compact Riemann surface without boundary. The partition functions are viewed as functionals of the Beltrami coefficients and their fermionic partners, which variables parametrize superconformal classes of metrics.     

A monopole-like classical solution to a purely fermionic model of even dimensions

We present a static solution to the classical field equations of a purely spinorial model with SO(2n) internal symmetry in 2n dimensions. The model contains composite vector fields which have solutions of the Wu-Yang monopole type.     

Classical solutions to a quadratically nonlinear gauge theory

We present some classical solutions to a gauge theory based on quadratically nonlinear Lie algebras without a central term. We observe that instanton-like and meron-like solutions force the internal fields to behave like solitons.     

Asymptotic freedom for quantum chromodynamics with top yukawa and Higgs interactions

Asymptotic freedom of QCD is extended to the enlarged system including the top Yukawa and Higgs interactions. Necessary and sufficient conditions for asymptotic freedom are given. By expansion with respect to powers of, all couplings are determined which are compatible with asymptotic freedom. It is found that the Higgs coupling is a function of the top coupling and that both couplings have upper bounds which correspond to the nontrivial case of reduction. The ultraviolet behavior of the coupling is controlled in all orders of the expansion.Work supported in part by the NSF: PHY-91-23780.     

Harmonics on hyperspheres,separation of variables and the bethe ansatz

A class of integrable evolution systems in the spacetimeR ²ⁿ⁺¹ (n 2) based on the generalized self-dual Yang-Mills equations are constructed. It is proved that the Darboux matrix method is applicable to these systems and a lot of explicit solutions are obtained. Starting with the trivial solutions, single soliton solutions and multi-soliton solutions are constructed. They are almost localized and the interaction between solitons is almost elastic.Supported by Chinese national basic research project Nonlinear Science and Japanese Inoue Scientific Foundation.     

On the stability of gauge fields in higher dimensions

The stability of solutions to gauge field equations in higher dimensions is studied. It is found that in the absence of a topological lower bound, these solutions are unstable.     

An example of cancellation of infinities in the star-quantization of fields

Within the *-quantization framework, it is shown how to remove some of the divergences occurring in theø ₂ ⁴ -theory by introducing a-dependent *-product cohomologically equivalent to the normal *-product.     

Functional integral of QED in terms of gauge-invariant quantities

We reduce the functional integral of quantum electrodynamics to an integral containing only local gauge invariant quantities. The set of (bosonic) invariants contains bilinear combinations of the spinor field and a real-valued covector field.     

A noncommutative geometric approach to left-right symmetric weak interactions

By combining the generalized exterior algebra of forms over a noncommutative algebra with the gauging of discrete directions and the associated Higgs fields, we consider the construction of the bosonic sector of left-right symmetric models of the form SU(2) _L SU(2) _R U(1). We see that within this formalism maximal use can be made of the gauge connection associated with the noncommutative graded algebra.     

Spinor electrodynamics in terms of gauge-invariant quantities

We give a formulation of classical spinor electrodynamics in terms of gauge-invariant quantities. The set of invariants consists of bilinear combinations of spinor fields (currents), a real-valued covector field, and a complex scalar field of modulus one. The presented result is a first step towards formulating quantum electrodynamics in terms of gauge-invariant fields.     

On the topological theory of self-dual connections over noncompact Riemann surfaces

A topological action for self-dual connections over noncompact Riemann surfaces is proposed. TheJ formulation and the associated linear system are obtained. A new connection is constructed, depending on a Kac-Moody parameter such that its flatness condition is theJ-equation associated to the self-dual problem. The algebra of infinitesimal Bäcklund transformations depending on this Kac-Moody parameter is constructed.     

Comment on noncommutative deformation of instantons

It was argued in [Y. Maeda, A. Sako, Noncommutative deformation of instantons, J. Geom. Phys. 58 (2008) 1784] that the noncommutative deformation of instantons on a 4-torus T⁴$T^4$ should alter the instanton numbers for arbitrary noncommutativity parameter θ$\theta$. We show that this is not the case for the U(N²)$U\left(N^2\right)$ theory discussed there. And we discuss the instanton numbers in general gauge theories on the noncommutative T⁴$T^4$.     

Yang—Mills instantons over Riemann surfaces

Exact solutions to the self-dual Yang—Mills equations over Riemann surfaces of arbitrary genus are constructed. They are characterized by the conformal class of the Riemann surface. They correspond to U(1) instantonic solutions for an Abelian-Higgs system. A functional action of a genus g Riemann surface is constructed, with minimal points being the two-dimensional self-dual connections. The exact solutions may be interpreted as connecting initial and final nontrivial vacuum states of a conformal theory, in the sense of Segal, with a Feynman functor constructed from the functional integral of the action.     

Are vortex numbers preserved?

We study noncommutative vortex solutions that minimize the action functional of the Abelian Higgs model in 2-dimensional noncommutative Euclidean space. We first consider vortex solutions which are deformed from solutions defined on commutative Euclidean space to the noncommutative one. We construct solutions whose vortex numbers are unchanged under the noncommutative deformation. Another class of noncommutative vortex solutions via a Fock space representation is also studied.