Anti-self-dual Yang-Mills Equations Research Articles

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.

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.

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.

Canonical structures on anti-self-dual four-manifolds and the diffeomorphism group

An unusual and attractive system is studied that arises from the anti-self-dual (ASD) Yang–Mills equations with maximal translational symmetry and with gauge group the volume preserving diffeomorphisms of an auxiliary four-manifold ℳ. The resulting equations lead to a system consisting of a volume form together with four independent vector fields on ℳ satisfying three simple Lie bracket relations. This structure is shown to give rise to a two-sphere’s worth of closed simple two-forms which in turn lead to the standard hyper-Kahler structure of an anti-self-dual metric on ℳ. The system determines not only the ASD metric, but also a frame that is proportional to an orthonormal frame. It is shown that the freedom in the choice of frame is related to a pair of solutions of a modified Laplacian and can always be chosen so that the proportionality factor is unity. The Plebanski first and second heavenly forms for general ASD metrics are written out in terms of these structures (the first being the standard description in terms of a complex structure and Kahler scalar). One of the scalars is interpreted as the generating function for the diffeomorphisms (symplectomorphism) in line with the origin of the system as the ASD Yang–Mills equations with the volume preserving diffeomorphisms as gauge group.