SelfDuality in Gauge Theory and Integrable Systems.
Abstract
Three papers are presented. In "Hyperbolic Vortices and Some NonSelfDual Classical Solutions of SU(3) gauge theory", a proposal of Burzlaff (Phys.Rev.D 24 (1981) 546) is followed to obtain a series of nonselfdual classical solutions of fourdimensional 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 YangMills action for these solutions is roughly 3.3 times the standard instanton action. In "KahlerChernSimons Theory and Symmetries of AntiSelfDual Gauge Fields", KahlerChernSimons theory, which was proposed as a generalization of ordinary Chern Simons theory, is explored in more detail. The theory describes antiselfdual instantons on a fourdimensional Kahler manifold. The phase space is the space of gauge potentials, whose symplectic reduction by the constraints of antiselfduality leads to the moduli space of instantons. Infinitesimal Backlund transformations, previously related to "hidden symmetries" of instantons, are canonical transformations generated by the antiselfduality constraints. The quantum wave functions naturally lead to a generalized WessZumino Witten action, which in turn has associated chiral current algebras. The dimensional reduction of the antiselfduality equations to integrable twodimensional theories is briefly discussed in this framework. In "The SelfDual YangMills Equations as a Master Integrable System" a systematic method of dimensional reduction of the selfdual YangMills equations to obtain twodimensional 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 twodimensional 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 SineGordon 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.
 Publication:

Ph.D. Thesis
 Pub Date:
 1991
 Bibcode:
 1991PhDT.......266S
 Keywords:

 Physics: Elementary Particles and High Energy; Mathematics