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The field stregth tensor of a Yang-Mills theory is defined as $$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+ie\left[A_\mu,A_\nu\right].$$ In general, the gauge field is in the adjoint representation of the gauge group (we normally say it takes value in the algebra) so it is written as $$A_\mu=A_\mu^aT_a,\quad a=1,2,\ldots dim(G),$$ where $dim(G)$ ...

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To answer your second question, group theory has a lot of application in physics, especially in particle physics. I guess you are unfamiliar with Special unitary group. Basically, if we consider $SU(2) \times U(1)|$ Yang-Mills system along with the Higgs field, we'd describe electromagnetism along with weak force. Also, let us consider $SU(3)|$ Yang-Mills ...

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I can't comment on string theory, but in quantum field theory the U(1), SU(2), etc symmetry groups are local gauge symmetries. They are not a symmetry of the spacetime in which the symmetry is formulated. So whether the spacetime is discrete or not makes no difference to the local gauge symmetry. As far as I know the physical significance of the local gauge ...

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