Anand-Ward Solitons

This subcategory was created to investigate the properties of solutions to an integrable Chiral model put forward by Richard Ward.

Very briefly, a Chiral Model is a physical model for fundamental forces in physics. It is a field theory, in that there are no particles in the sense of Newtonian mechanics, but fields taking values in a group. The classical solutions of the model are defined either as critical points of an energy functional, or directly as solutions of a differential equation. The equations are closely related to the harmonic map equations in differential geometry, and since harmonic maps are generalisations of minimal surfaces (i.e., soap films), this provides some insight into possible physical motivations of the theory.

Chiral models are known to admit soliton solutions, which in the physical sense means that the energy densities of solutions look like configurations of particles:

(when graphed as a positive real-valued function of two variables); and they maintain this lumpy appearance as they evolve, with the possible changes in the topological type of the positive-energy subset (i.e., support) interpreted as an interaction of particles.

The model considered here is a 2+1 dimensional, SU(N) model, which means that the field is a map from R³ to the special unitary group, satisfying a differential equation whose symbol (leading order terms) is indefinite. In contrast, classical solutions to a gauge theory can be thought of as maps into a Lie algebra. Ward's equations are

The first three terms constitute the standard Chiral Model, to which Ward has added lower-order (torsion) terms. The lower order terms destroy the Lorentz invariance of the equations, but result in completely integrable equations. Completely integrable systems arose in Hamiltonian mechanics where they were defined to be systems possessing a complete set of integrals. From the point of view of differential geometry, the most practical definition is that the equations can be rewritten by defining a connection (or parametrised family of connections) for which the equation is the curvature of the connection. Solutions of the equations then correspond to flat connections, and the fact that flat connections have covariant constant guages locally gives a natural method of integrating the equations.

These ideas are not new, but applications to new systems are of considerable research interest. It is a matter of current interest to understand better the extent to which the theory of completely integrable systems can be understood within the Twistor Theory developed to study Yang-Mills gauge theory (see [L. Mason and N. Woodhouse, Integrability, Self-duality and Twistor Theory, OUP 1996]).

In this case, the mini-Twistor theory of magnetic monopoles [N. J. Hitchin Monopoles and Geodesics, Commun. Math. Phys. 83 (1982), 579--602] (which is subsumed in the Yang-mills picture) can be used. I will not go into this story in detail. The details are available in the preprint Ward's Solitons with some additional explaination added in the lecture notes (in german). The moral of the story is that after imposing the boundary condition that the solutions extend analytically to RP³ (where we view R³ as an open cell in real projective space) the general solution is of the form

It follows that the space of solutions has infinitely many finite-dimensional components, indexed by k, the size of the constant matrices

appearing in the closed form. Because the static solutions correspond to harmonic spheres in SU(N), it is known that the component number (the size of the matrix alpha) is the energy of the static solutions in this component.

Ward's equations now become the matrix equation and the solutions will be smooth if the data satisfy the nondegeneracy conditions Data satisfying these conditions parametrise the space of solutions uniquely up to the action

of

This is where you come in. By choosing 2x2, 3x3 or 4x4 monad data, from the AW Solitons submenu, you can enter your own parameters for solutions in the first three components of the moduli space of SU(3) solutions. You can then set the time using the Set Parameters dialogue (time is parameter a) and view the energy density of your solution. You can also change the horizontal scale (b) vertical scale (c) and cut off (d) above which value, the energy will be replace with the value (e). The last feature is included because extremely concentrated energy-densities can produce puzzling displays.

All of the usual display controls can be used to view the surface. In particular Morphing can be used to create a moving picture of interacting solitons. (You will probably want to hold all of b,c,d, and e constant and vary a only.)

Of course, you may want to start by viewing the pre-programmed solutions first.

Any questions about the use of the subcategory, this documentation, the geometry behind the construction of the solutions, or interpretations of the parameters in terms of the behaviour of the solutions can be directed to me!