To define the Yang-Mills Lagrangian, we need to define the ‘Trace’ of an End(E) valued form. Recall that the Trace of a matrix is the sum of its diagonal enTries. The Trace is independent of the choice of basis – an invariant notion that is independent of the choice of basis. A definition of the Trace that mkes this clear is as follows. Consider – an isomorphism that does not depend on any choice of basis – so the pairing between V and defines a linear map To see that this v is really a Trace, pick of V and let be a dual basis of . Writing as We have which is of course the sum of the diagonal enTries. \newline This implies that if we have a section T of we can define a funciton Tr(T) on the base manifold M whose value at is the Trace of the endomorphism T(p) of the fiber : If and we define

Now we can write down the **Yang-Mills Lagrangian**: If D is a connection on E, this is the n-form given by

where F is the curvature of D. Note that by the defintion of the hodge star operator (also in this collection of notes), we can write this in local co-ordinates as

If we integrate over M we get the **Yang-Mills action**

This needs some elaboration. So let us explain these formulas better. We choose the physics convention where the generators of the Lie algebra are Hermitian.

by another convention (there is a lot of ambiguity of signs in this subject). The first thing to note is that F has vector **and** Lie algebra indices. The Trace is over the Lie algebra, **not** over the vector indices. The vector indices are just those of the field sTrength in QED. In Yang-Mills the curvature form is Lie Algebra valued. \newline In this case where the summation convention is used,and where are the generators of . To be explicit, F has not only tensor components but maTrix components The inner product of F with itself where \textasteriskcentered is the Hodge \textasteriskcentered – operator. Thus we are calculating . It is a standard exercise to find the exterior product of two r-forms. We find Note that the differential forms don’t ‘know’ the Lie algebra. The algebra hasn’t come into the calcuation yet. where e have used the standard normalization convention for the , . (This comes from the fact we want to live in , and the generators of are taken to be where are the Pauli matrices.) Thus, we find which can be written as