TITLE: Self-dual gauge theories

ABSTRACT:
 

The self-duality equations are important in gauge theories because they show the connection between gauge models with internal symmetry groups and gauge theory of gravity. They are differential equations of the first order and it is easier to investigate the solutions for different particular configurations of the gauge fields and of space-times.One of the most important property of the self-duality equations is that they imply the Yang-Mills field equations. In this paper we will prove this property for the general case of a gauge theory with compact Lie group of symmetry over a 4-dimensional space-time manifold.

It is important to remark that there are 3m independent self-duality equations (of the first order) while the number of Yang-Mills equations is equal to 4m, where m is the dimension of the gauge group. Both of themhave 4m unknown functions which are the gauge potentials A^a_s(x);a=1,2,....,m; m =0,1,2,3. But, we have, in addition, $m$ gauge conditions for A^a_s(x) (for example Coulomb, Lorentz or axial gauge) which together with the self-duality equation constitute a system of 4m equations. The Bianchi identities for the self-dual stress tensor F^a_mn coincide with the Yang-Mills equations and do not imply therefore suplementary conditions.

We use the axial gauge in order to obtain the self duality equations for a SU(2) gauge theory over a cureved space-time. The compatibility between self-duality and Yang-Mills equations is studied and some classes of solutions are obtained. In fact, we will write the Einstein-Yang-Mills equations and we will analyse only the Yang-Mills sector. The Einstein equations can not be obtained of course from self-duality. They should be obtained if we would consider a gauge theory having PXSU(2) as symmetry group, where P is the Poincare group. More generally, a gauge theory of N-extended supersymmetry can be developed by imposing the self-duality condition.

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