Self-duality in Gauge Theory and Integrable Systems

Jeremy Schiff

Thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences, Columbia University. Final version, deposited July 1991


Three papers are presented. In "Hyperbolic Vortices and Some Non-Self-Dual Classical Solutions of SU(3) gauge theory", a proposal of Burzlaff [Phys.Rev.D 24 (1981) 546] is followed to obtain a series of non-self-dual classical solutions of four-dimensional SU(3) gauge theory; this is done by finding solutions of the classical equations of motion of an abelian Higgs model on hyperbolic space. The lowest value of the Yang-Mills action for these solutions is roughly 3.3 times the standard instanton action.

In "Kahler-Chern-Simons Theory and Symmetries of Anti-Self-Dual Gauge Fields", Kahler-Chern-Simons theory, which was proposed as a generalization of ordinary Chern-Simons theory, is explored in more detail. The theory describes anti-self-dual instantons on a four-dimensional Kahler manifold. The phase space is the space of gauge potentials, whose symplectic reduction by the constraints of anti-self-duality leads to the moduli space of instantons. Infinitesimal Backlund transformations, previously related to "hidden symmetries" of instantons, are canonical transformations generated by the anti-self-duality constraints. The quantum wave functions naturally lead to a generalized Wess-Zumino-Witten action, which in turn has associated chiral current algebras. The dimensional reduction of the anti-self-duality equations leading to integrable two-dimensional theories is briefly discussed in this framework.

In "The Self-Dual Yang-Mills Equations as a Master Integrable System" a systematic method of dimensional reduction of the self-dual Yang-Mills equations to obtain two-dimensional integrable systems, and simple three dimensional extensions thereof, is examined. This unifies existing knowledge about such reductions. The method produces the recursion operators of various two-dimensional integrable systems; for gauge group SL(2,C) the recursion operators of the KdV, MKdV, Gardner KdV and NLS hierarchies appear, and for SL(3,C) the recursion operators of the Boussinesq and fractional KdV hierarchies. We also obtain the Sine-Gordon and Liouville equations. The different possible reductions for SL(N,C) are classified, giving a conjecture on the existence of large numbers of new integrable systems, and possibly even a scheme for classification of integrable systems.