# What are the $A_{\mu}{}^a$ fields in Yang-Mills theory?

At some point of the demonstration of Yang Mills theory we assume an ansatz that $$A_{\mu}=t^a A_{\mu}{}^a$$ where $$a=1, \ldots,n^2-1$$ and the $$t^a$$ are the generators of the $$SU(n)$$ symmetry in order to have a hermitian $$n \times n A_{\mu}$$.

I don't understand exactly what are the $$A_{\mu}{}^a$$ are they a constant in that space? Are they a matrix? Like a vector field times the identity? Is the definition of $$A_{\mu}$$ a matrix product?

• The $A^a_\mu(x)$ are the gauge fields. You have 8 of them for SU(3) (QCD), as the index $a=1,\cdots,8$ dictates and that makes sense because we have 8 gluons. The object $A_\mu:=t_aA^a_\mu(x)$ is indeed a matrix. – Thomas Wening Jan 14 at 17:31
• Or, for a hopefully more familiar example, consider the U(1) symmetry of electromagnetism. There you have A_mu, a four-vector. – puppetsock Jan 14 at 17:58

If the structure group of the theory is $$G$$, with Lie algebra $$\mathfrak g$$, then the (local) Yang-Mills fields, are locally defined differential $$1$$-forms that take value in the Lie algebra $$\mathfrak g$$.
If the local gauge neighborhood is $$U$$, and we suppose that it is also a coordinate neighborhood, then we may write $$A(x)=A_\mu^a(x)\mathrm dx^\mu\otimes t_a.$$
If spacetime is $$n$$ dimensional, and $$G$$ is $$N$$ dimensional, then the $$A^a_\mu(x)$$ are $$n\times N$$ local functions on the gauge domain $$U$$ that constitute the component functions of the local Yang-Mills field $$A$$.
For each value of the indices $$\mu$$ and $$a$$, $$A^a_\mu$$ is an ordinary real-valued function. The expression $$A_\mu=A^a_\mu t_a$$ (for a fixed value of $$\mu$$) is a $$\mathfrak g$$-valued local function. Usually in physics, $$G$$ is a matrix group, and $$\mathfrak g$$ is a matrix Lie algebra, in this case the generators $$t_a$$ are matrices, so $$A_\mu^a t_a$$ is (also) a matrix-valued function.