Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

112 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
1. INTRODUCTION
The study of symmetries in physics, has become a subject of most
importance, for example in particle physics, gauge symmetry is the central
idea to understanding the fundamental forces, Lorentz invariance is
crucial in quantum field theory, general coordinate covariance in General
Relativity, etc.
In recent years a new wave of dualities has emerge, as a most important
technique in the understanding of non–perturbative aspects of
quantum field theory, and string theory.
In superstring theory, duality shows the equivalence among different
types of perturbative string theories and gives insight of a new underlying
theory known as M–theory. Specially interesting are T–duality, and S–
duality; the first one gives a new view of space time physics, revealing
the existence of a minimal length.
Duality is an old known type of symmetry which by interchanging
the electric, and magnetic fields leaves invariant the vacuum Maxwell
equations. It was extended by Dirac to include sources, with the well
known price of the predictions of monopoles, which appear as the dual
particles of the electrically charged ones, and whose existence has not
been confirmed. Dirac obtained that the charges (couplings) of the magnetical
and electrical charged particles are inverse to each other, so if the
electrical force is “weak”, then the magnetic force between monopoles
will be “strong”, then one description would be treated perturbatively,
and other one not, but of course we could always use the duality to go
from one theory to the other.
The Strong/weak coupling duality (S–duality) in superstring and supersymmetric
gauge theories in various dimensions has been, in the last
five years, the major tool to study the strong coupling dynamics of these
theories. Much of these results require supersymmetry through the notion
of BPS state. These states describe the physical spectrum and they
are protected of quantum corrections leaving the strong/coupling duality
under control to extract physical information. In particular, the
exact Wilson effective action of supersymmetric gauge theories
has been computed, showing the duality symmetries of these effective
theories. It turns out that the dual description is quite adequate to address
standard non-perturbative problems of Yang–Mills theory. In the
non–supersymmetric case there are no BPS states and the situation is
much more involved. This latter case is an open question and it is still
under current investigation.
In the specific case of non–supersymmetric gauge theories in four dimensions,
the subject has been explored recently in the Abelian as well