# 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

## Abstract

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.