The Mathematical Basis Gauge Theories of The Yang-Mills type and Dirac Monopoles
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Date
1982-06
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Addis Ababa University
Abstract
l\fter? g90metric and intuitive introduction to
~~n-trivial fibe~ bundleJ~ a precise presentation of
principal fiber bundles ~ith Lie group structure is given.
The methods of differential geometry are set up. It is
, 'I ,,' I
shown how a gauge potent~alcan be regarded as a connection
in some fiber' bundle" an1 the corresponding gauge field
as,the associated curvat~re. l\s an example of a physical
case which l,,'ads to the 'Jon[;ide~ati.on of non-trivi.al' bundles I
th~magne~i~ monoPol~ is! introduced in its Dirac form together
'I '
\,;;lth some notiol;ts of du~tity symmetry (self dual fields).
'I '
t1onopoles 'in t~e recent \~u-Yang formulation I and
electrodynamicswit~ mohbpoles identified with non-trivial
, i
U(l) bundles are studiedl The yang-~lills action functional
, ,
:is, def ined. : I
I
'of the
-'.-;'
of the
of the
" ,The
i
Y~ng-Millsl field equation are derived in terms
. I
,fielo (curvature) i. , , i The dual synunetric counterparts
" I
field e~uations a~e
, I
established as a consequence
~ianchi identitY~f differential geometry. It is
,
S'hO\'ll) hovi the global asp'ects of, the theory (f ini teness of
the action and the boundary conditions on the gauge
p~tentials) ~re;encoded lin the structure of a bundle. The
. ' I
asymptotically vanishing; gauge potentials (or pure gauges)
~;e classified usi,ng th I notions of instantoI,l number and
homotopy
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Keywords
The Mathematical Basis Gauge Theories