Self-dual Yang-Mills Equations Research Articles

Analytic loops and gauge fields

In this paper the linear representations of analytic Moufang loops are investigated. We prove that every representation of semisimple analytic Moufang loop is completely reducible and find all nonassociative irreducible representations. We show that such representations are closely associated with the (anti-)self-dual Yang-Mills equations in ${\bf R}^8$

A bilinear approach to a Pfaffian self-dual Yang-Mills equation

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. By using the bilinear technique of soliton theory, a pfaffian version of the SU(2) self-dual Yang-Mills equation and its solution is constructed.

Instanton-meron hybrid in the background of gravitational instantons

When it comes to the topological aspects, gravity may have profound effects even at the level of particle physics despite its negligibly small relative strength well below the Planck scale. In spite of this intriguing possibility, relatively little attempt has been made toward the exhibition of this phenomenon in relevant physical systems. In the present work, perhaps the simplest and the most straightforward new algorithm for generating solutions to (anti) self-dual Yang-Mills (YM) equation in the typical gravitational instanton backgrounds is proposed and then applied to find the solutions practically in all the gravitational instantons known. Solutions thus obtained turn out to be some kind of instanton-meron hybrids possessing mixed features of both. Namely, they are rather exotic type of configurations obeying first order (anti) self-dual YM equation which are everywhere non-singular and have finite Euclidean YM actions on one hand while exhibiting meron-like large distance behavior and carrying generally fractional topological charge values on the other. Close inspection, however, reveals that the solutions are more like instantons rather than merons in their generic natures.

HIDDEN SYMMETRIES OF THE OPEN N=2 STRING

It is known for ten years that self-dual Yang–Mills theory is the effective field theory of the open N=2 string in (2+2)-dimensional space–time. We uncover an infinite set of Abelian rigid string symmetries, corresponding to the symmetries and integrable hierarchy of the self-dual Yang–Mills equations. The twistor description of the latter naturally connects with the BRST approach to string quantization, providing an interpretation of the picture phenomenon in terms of the moduli space of string backgrounds.

Yang–Mills instantons in the gravitational instanton backgrounds

The simplest and the most straightforward new algorithm for generating solutions to (anti) self-dual Yang-Mills (YM) equation in the typical gravitational instanton backgrounds is proposed. When applied to the Taub-NUT and the Eguchi-Hanson metrics, the two best-known gravitational instantons, the solutions turn out to be the rather exotic type of instanton configurations carrying finite YM action but generally fractional topological charge values.

Formula omitted] geometry from Fedosov's deformation quantization

A geometric derivation of W ∞ gravity based on Fedosov's deformation quantization of symplectic manifolds is presented. To lowest order in Planck's constant it agrees with Hull's geometric formulation of classical non-chiral W ∞ gravity. The fundamental object is a W -valued connection one form belonging to the exterior algebra of the Weyl algebra bundle associated with the symplectic manifold. The W -valued analogs of the self-dual Yang-Mills equations, obtained from a zero curvature condition, naturally lead to the Moyal Plebanski equations, furnishing Moyal deformations of self-dual gravitational backgrounds associated with the complexified cotangent space of a two-dimensional Riemann surface. Deformation quantization of W ∞ gravity is retrieved upon the inclusion of all the ħ terms appearing in the Moyal bracket. Brief comments on non commutative geometry and M(atrix) theory are made.

Bi-differential calculi and integrable models

The existence of an infinite set of conserved currents in completely integrable classical models, including chiral and Toda models as well as the KP and self-dual Yang-Mills equations, is traced back to a simple construction of an infinite chain of closed (respectively, covariantly constant) 1-forms in a (gauged) bi-differential calculus. The latter consists of a differential algebra on which two differential maps act. In a gauged bi-differential calculus these maps are extended to flat covariant derivatives.

Applications of Darboux transformations to the self-dual Yang-Mills equations

The linear problem associated with the self-dual Yang-Mills equations is covariant with respect to Darboux and binary Darboux transformations of almost classical type. This technique is used to construct solutions of the problem in the form of Wronskian-like and Gramm-like determinants. The self-dual conditions can be properly realized for only the latter type of solutions.

SELF-DUAL YANG–MILLS: SYMMETRIES AND MODULI SPACE

Geometry of the solution space of the self-dual Yang–Mills (SDYM) equations in Euclidean four-dimensional space is studied. Combining the twistor and group-theoretic approaches, we describe the full infinite-dimensional symmetry group of the SDYM equations and its action on the space of local solutions to the field equations. It is argued that owing to the relation to a holomorphic analogue of the Chern–Simons theory, the SDYM theory may be as solvable as 2D rational conformal field theories, and successful nonperturbative quantization may be developed. An algebra acting on the space of self-dual conformal structures on a 4-space (an analogue of the Virasoro algebra) and an algebra acting on the space of self-dual connections (an analogue of affine Lie algebras) are described. Relations to problems of topological and N=2 strings are briefly discussed.

Reduction of the self-dual Yang-Mills equations by Bessel and Legendre functions

Two integrable nonlinear equations are derived by dimensional reductions of the SU(2) self-dual Yang-Mills equations. To derive the equations, we use Hirota's method and some properties of special functions, namely those of the Bessel and Legendre functions. Solutions of these equations are represented in terms of Toeplitz determinants whose elements are a superposition of the special functions.

Progress in classically solving ten-dimensional supersymmetric reduced Yang-Mills theories

It is shown that there exists an on-shell light cone gauge where half of the fermionic components of the super vector potential vanish, so that part of the superspace flatness conditions becomes linear. After reduction to (1 + 1) space-time dimensions, the general solution of this subset of equations is derived. The remaining non-linear equations are written in a form which is analogous to Yang equations, albeit with superderivatives involving sixteen fermionic coordinates. It is shown that this non-linear part may, nevertheless, be solved by methods similar to powerful techniques previously developed for the (purely bosonic) self-dual Yang Mills equations in four dimensions.

Exact Solutions for Self-dual Yang–Mills Equations

The (constrained) canonical reduction of four dimensional self-dual Yang–Millstheory to 2, (2+1) dimensional sine-Gordon theory and 2 dimensional Liouvilles theory areconsidered. The Bäcklund transformations (BTs) areimplemented to obtain new classes of exact solutions for the reduced 2 dimensional sine-Gordonand Liouville models. Another transformation is developed and used to obtain exact solution forthe 2+1 and the original 3+1 sine-Gordon models.

Lax equations in 10-dimensional supersymmetric classical Yang-Mills theories

In a recent paper (hep-th/9811108), Saveliev and the author showed that there exits an on-shell light cone gauge where the non-linear part of the field equations reduces to a (super) version of Yang's equations which may be solved by methods inspired by the ones previously developed for self-dual Yang-Mills equations in four dimensions. Here, the analogy between these latter theories and the present ones is pushed further by writing down a set of super partial linear differential equations whose consistency conditions may be derived from the SUSY Y-M equations in ten dimensions, and which are the analogues of the Lax pair of Belavin and Zakharov. On the simplest example of the two pole ansatz, it is shown that the same solution-generating techniques are at work, as for the derivation of the celebrated multi-instanton solutions carried out in the late seventies. The present Lax representation, however, is only a consequence of (instead of being equivalent to) the field equations, in contrast with the Belavin Zakharov Lax pair.

Yang-Mills-Higgs soliton dynamics in 2+1 dimensions

Dimensional reduction of the self-dual Yang-Mills equation in 2+2 dimensions produces an integrable Yang-Mills-Higgs-Bogomolnyi equation in 2+1 dimensions. For theSU(1,1) gauge group, a t'Hooft-like ansatz is used to construct a monopole-like solution and an N-soliton-type solution, which describes both the static deformed monopoles and the exotic monopole dynamics including a transmutation. How the monopole solution results from the twistor formalism is shown. Multimonopole solutions are commented on.

General solution of 7D octonionic top equation

The general solution of a 7D analogue of the 3D Euler top equation is shown to be given by an integration over a Riemann surface with genus 9. The 7D model is derived from the 8D Spin(7) invariant self-dual Yang-Mills equation depending only upon one variable and is regarded as a model describing self-dual membrane instantons. Several integrable reductions of the 7D top to lower target space dimensions are discussed and one of them gives 6, 5, 4D descendants and the 3D Euler top associated with Riemann surfaces with genus 6, 5, 2 and 1, respectively.

Null Killing vectors and reductions of the self-duality equations

We find all null conformal Killing vectors in with the metric . We classify one and two-dimensional totally null algebras generated by such vectors. Reductions of the self-dual Yang-Mills equations are obtained for gauge fields invariant with respect to these algebras. PACS Numbers: 0210, 1130

The reduced self-dual Yang–Mills equation, binary and infinitesimal Darboux transformations

The self-dual Yang–Mills equation in the (2+1)-dimensional space–time is considered. Binary Darboux transformation of a new kind is applied to obtain the infinite hierarchy of solutions expressed through ones of corresponding Lax pairs on an initial solution of the equation studied. Sufficient conditions are given for the solutions built to take values in the class of unimodular positive-definite matrices. A new infinitesimal Darboux transformation is introduced and a particular solution of the linearization of the self-dual Yang–Mills equation is given. Two hierarchies of infinitesimal symmetries depending on a solution of the nonlinear equation are extracted.

Bilinear Form Approach to the Self-Dual Yang-Mills Equations and Integrable Systems in (2+1)-Dimension

We apply Hirota's bilinear form approach to the analysis of the SU(2) self-dual Yang-Mills equations. Solutions of the bilinear forms are represented by using the Toda-molecule type determinants with new linear dispersion relations. We also discuss dimensional reductions of the bilinear forms and the (2+1)-dimensional integrable equations derived from them. These integrable equations are higher dimensional extension of the nonlinear Schroinger equation and the derivative nonlinear Schrodinger equation.

On infinite-dimensional algebras of symmetries of the self-dual Yang–Mills equations

Infinite-dimensional algebras of hidden symmetries of the self-dual Yang–Mills equations are considered. A current-type algebra of symmetries and an affine extension of conformal symmetries introduced recently are discussed using the twistor picture. It is shown that the extended conformal symmetries of the self-dual Yang–Mills equations have a simple description in terms of Ward’s twistor construction.

Super-Matrix KdV and Super-Generalized NS Equations from Self-Dual Yang–Mills Systems with Supergauge Groups

Super-matrix KdV and super-generalized nonlinear Schrodinger equations are shown to arise from a symmetry reduction of ordinary self-dual Yang–Mills equations with supergauge groups.