Anti Self-dual Yang-Mills Research Articles

Antiself-dual Yang–Mills, modified Faddeevs–Jackiw formalism and hidden BRS invariance

We analyze the constraints for a system of antiself-dual Yang–Mills (ASDYM) equations by means of the modified Faddeev–Jackiw method in [Formula: see text] and [Formula: see text] gauges à la Yang. We also establish the Hamiltonian flow for ASDYM system through the hidden BRS invariance in both the gauges. Finally, we remark on the bi-Hamiltonian nature of ASDYM and the compatibility of the symplectic structures therein.

Bianchi Permutability for the Anti-Self-Dual Yang-Mills Equations

The anti-self-dual Yang-Mills equations are known to have reductions to many integrable differential equations. A general Backlund transformation (BT) for the anti-self-dual Yang-Mills (ASDYM) equations generated by a Darboux matrix with an affine dependence on the spectral parameter is obtained, together with its Bianchi permutability equation. We give examples in which we obtain BTs of symmetry reductions of the ASDYM equations by reducing this ASDYM BT. Some discrete integrable systems are obtained directly from reductions of the ASDYM Bianchi system.

Aspects of C3-symmetric calorons from numerical Nahm transform

Calorons are finite action solutions to the anti-selfdual Yang-Mills equations on R3 × S1. They are generally constructed by the so called Nahm construction. We perform the numerical Nahm transform for the Nahm data of 3-calorons with C3-symmetry, which do not have the monopole limits. Dissimilar to the cases of having monopole limits, we can trace the zero-circumference limit of S1. It is found that the action density of the calorons tends to fade away as S1 shrinks.

INTEGRABLE ASPECTS OF NONCOMMUTATIVE ANTI-SELF-DUAL YANG-MILLS EQUATIONS

We discuss extension of soliton theory and integrable systems to non-commutative (NC) spaces, focusing on integrable aspects of NC Anti-Self-Dual Yang-Mills (ASDYM) equations. We give exact soliton solutions (with both finite- and infinite-action solutions) by means of Bäcklund transformations. In the construction of NC soliton solutions, one kind of NC determinants, quasideterminants, play crucial roles. This is partially based on collaboration with C. R. Gilson and J. J. C. Nimmo (Glasgow).

Dirac constraint analysis and symplectic structure of anti-self-dual Yang-Mills equations

We present the explicit form of the symplectic structure of anti-self-dual Yang-Mills (ASDYM) equations in Yang’s J- and K-gauges in order to establish the bi-Hamiltonian structure of this completely integrable system. Dirac’s theory of constraints is applied to the degenerate Lagrangians that yield the ASDYM equations. The constraints are second class as in the case of all completely integrable systems which stands in sharp contrast to the situation in full Yang-Mills theory. We construct the Dirac brackets and the symplectic 2-forms for both J- and K-gauges. The covariant symplectic structure of ASDYM equations is obtained using the Witten-Zuckerman formalism. We show that the appropriate component of the Witten-Zuckerman closed and conserved 2-form vector density reduces to the symplectic 2-form obtained from Dirac’s theory. Finally, we present the Backlund transformation between the J- and K-gauges in order to apply Magri’s theorem to the respective two Hamiltonian structures.

Global anti-self-dual Yang-Mills fields in split signature and their scattering

This article concerns solutions to the anti-self-dual Yang Mills (ASDYM) equations in split signature that are global on the double cover of the appropriate conformally compactified Minkowski space = S2 × S2. Ward's ASDYM twistor construction is adapted to this geometry using a correspondence between points of and holomorphic discs in , twistor space, with boundary on the real slice ℝℙ3. Smooth global U(n) solutions to the ASDYM equations on are shown to be in 1 : 1 correspondence with pairs consisting of an arbitrary holomorphic vector bundle E over together with a positive definite Hermitian metric H on E|ℝℙ3. There are no topological or other restrictions on the bundle E. In ultrahyperbolic signature solutions are generically non-analytic or only finitely differentiable and such solutions arise from a corresponding choice of regularity for H. When E is trivial, the twistor data consists of the Hermitian matrix function H on ℝℙ3 up to constants and the correspondence provides a nonlinear generalisation of the X-ray transform. In general it provides a higher-dimensional analogue of the (inverse) scattering transform in which H plays the role of the reflection coefficient and E the algebraic data.

Integrability aspects of a Schwarzian PDE

We present a reduction of the anti-self-dual Yang–Mills (ASDYM) equations to a system of partial differential equations (PDEs) introduced recently by Nijhoff et al. (Phys. Lett. A 267 (2000) 147). An auto-Bäcklund transformation of the reduced system is also presented. The system under consideration is related to a fourth-order nonlinear PDE of the Schwarzian type. The symmetry group of the latter equation is calculated and similarity reductions to the Schwarzian equation and the full Painlevé III, V and VI are presented.

Anti-self-dual Yang–Mills equations on noncommutative space–time

By replacing the ordinary product with the so-called ☆-product, one can construct an analog of the anti-self-dual Yang–Mills (ASDYM) equations on the noncommutative R 4 . Many properties of the ordinary ASDYM equations turn out to be inherited by the ☆-product ASDYM equation. In particular, the twistorial interpretation of the ordinary ASDYM equations can be extended to the noncommutative R 4 , from which one can also derive the fundamental structures for integrability such as a zero-curvature representation, an associated linear system, the Riemann–Hilbert problem, etc. These properties are further preserved under dimensional reduction to the principal chiral field model and Hitchin’s Higgs pair equations. However, some structures relying on finite dimensional linear algebra break down in the ☆-product analogs.

The generalized Moyal–Nahm and continuous Moyal–Toda equations

We present in detail a class of solutions to the 4D SU(∞) Moyal–anti-self-dual Yang–Mills (ASDYM) equations (an effective 6D theory) that are related to reductions of the generalized Moyal–Nahm equations using the Ivanova–Popov ansatz. The former yields solutions to the ASDYM/SDYM equations for arbitrary gauge groups in four dimensions. A further dimensional reduction of the above effective 6D equations yields solutions to the Moyal–anti-self-dual gravitational equations in four dimensions. The self-dual Yang–Mills/self-dual gravity case requires a separate study. The SU(2) Toda lattice and SU(∞) (continuous) Moyal–Toda lattice equations are derived from the Moyal–Nahm equations. An explicit map taking the Moyal heavenly form (after a rotational Killing symmetry reduction of the Moyal heavenly equations) into the SU(2) Toda lattice field is found. Finally, the generalized Moyal–Nahm equations are conjectured that contain the (continuous) SU(∞) Moyal–Toda lattice equations, after a suitable reduction process. Embeddings of the different types of Moyal–Toda lattice equations into the Moyal–Nahm equations are described.

Twistor correspondences for the soliton hierarchies

In this article we propose a new overview on the theory of integrable systems based on symmetry reduction of the anti-self-dual Yang—Mills equations and its twistor correspondence. First, the non-linear Schrödinger (NS) equations and the Korteweg de Vries (KdV) equations are shown to be symmetry reductions of the anti-self-dual Yang—Mills (ASDYM) equation with real forms of SL (2, C ) as gauge groups. We obtain a twistor correspondence between solutions of the NS and KdV equations and certain holomorphic vector bundles with a symmetry on the total space of the complex line bundle of Chern class two on the Riemann sphere. Remarkably, when the Chern class is increased, the correspondence extends to the NS and KdV hierarchies. If the symmetry condition is dropped we obtain a twistor correspondence for a hierarchy for the Bogomolny equations, which yields the KdV and NS hierarchies when the symmetry is imposed. The inverse scattering transform is shown to be a coordinate realization of the twistor correspondence. Both the pure solitons and the solitonless cases are treated. The k-soliton solutions arise from the kth “Ward ansatze” in an analogous fashion to the monopole solutions.

High symmetry solutions of the anti-self-dual Yang–Mills equations

Beginning with the anti-self-dual Yang–Mills (ASDYM) equations for an arbitrary Lie algebra on Minkowski space, this paper specializes to the case in which the vector potentials are independent of all the space-time coordinates, i.e., are space-time constants. The resulting equations are three algebraic equations on the algebra. These equations are then simplified by using a null basis. Two of the equations can be immediately solved while the third remains, in general, quite difficult to deal with. Two general cases are considered: finite-dimensional Lie groups and the infinite-dimensional diffeomorphism groups on finite-dimensional manifolds. In a few of the special cases, e.g., SL(2,C) and the Virasoro algebra, the solutions can easily be found. The study of the the diffeomorphism groups leads unexpectedly to the Monge–Ampère equation. In particular, the four-dimensional volume preserving diffeomorphism group is identical with the vacuum anti-self-dual Einstein equations. In conclusion, the question of the associated Lax pair equations and its relation to the Riemann–Hilbert splitting problem on S2 is examined.

On the density of the Ward ansätze in the space of anti-self-dual Yang Mills solutions

A general patching matrixP for the twistor construction of antiself-dual Yang-Mills fields is approximated by a terminating Laurent series. The approximate patching matrixP(m) is triangularized (so that it becomes one of the Ward ansatze) and the associated Riemann-Hilbert problem is solved, thereby generating an anti-self-dual solution of the Yang-Mills equations. The approximate patching matrices and the associated (exact) anti-self-dual Yang-Mills solutions are then shown to converge onP and its corresponding solution so that the Ward ansatze forms a dense subset in the solution space in the Weierstrass sense.

A simple solution generation method for anti-self-dual Yang-Mills equations

A simple, algebraic technique for generating large classes of solutions of the SL( n, C ) anti-self-dual Yang-Mills (ASDYM) equations on Minkowski space is given, beginning from any given (local) solution.

Bäcklund transformations for the anti-self-dual Yang–Mills equations

Beginning from any given (local) solution of the GL(n,C) anti-self-dual Yang–Mills (ASDYM) equations on Minkowski space, a simple technique for the generation of large classes of solutions (perhaps in some sense all) is given. The origin of this technique is described in terms of two versions of the Ward construction. The resulting description of Bäcklund transformations is sufficiently simple that it is then possible to identify the group generated by the collection of all such Bäcklund transformations and the space on which it acts in terms of concrete functions.