Ward's conjecture

In mathematics, Ward's conjecture is the conjecture made by Ward (1985, p. 451) that "many (and perhaps all?) of the ordinary and partial equations that are regarded as being integrable or solvable may be obtained from the anti-self-dual gauge field equations (or its generalizations) by reduction".

Example

$F=-\star F$

in the (2,2) signature with gauge group SU(2) can be reduced along one timelike and one null direction to give an equation equivalent to the nonlinear Schroedinger equation for the gauge field after suitable gauge transformation choices.

Penrose-Ward transform

Via the Penrose-Ward transform these solutions give the holomorphic vector bundles often seen in the context of algebraic integrable systems.

References

• Ward, R. S. (1985), "Integrable and solvable systems, and relations among them", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 315 (1533): 451–457, Bibcode:1985RSPTA.315..451W, doi:10.1098/rsta.1985.0051, ISSN 0080-4614, MR 0836745