Going from one notation to another in Yang-Mills

Question

In many books on Yang-Mills theories, written from a physicist's point of view, the curvature tensor is written as:

$$F_{\mu \nu}^a = \partial_\mu A_\nu^a-\partial_\nu A_\mu^a+gf^{abc}A_\mu^bA_\nu^c$$

with the covariant derivative as:

$$D_\mu=I\partial_\mu-igT^aA^a_\mu$$

where $I$ is unit matrix and $T$ are generators of the group and

$$ \mathcal{L} = -\frac{1}{2}\operatorname{Tr}(F^2)=- \frac{1}{4}F^{a\mu \nu} F_{\mu \nu}^a$$

this has been basically taken from Wikipedia

But in the book I have been using, with a more mathematical perspective (Baez & Muniain's Gauge Fields, Knots and Gravity) the vector 1-form $A$ is an $\text{End}(E)$ valued $1$-form described as

$$A = A_{\mu i}^j e^i \otimes e_j \otimes dx^\mu $$

and the covariant derivative and curvature are

$$D_\mu = \partial_\mu + A_\mu,\quad F_{jk} = \partial_j A_k - \partial_k A_j + [A_j, A_k].$$

How can I go from one notation to the other? I can't do it, I think I'm confused by the superscipts $a$ in the physicist's notation.

And what is $F_{\mu\nu}$ for a non-abelian gauge theory? A matrix?

Answer 1

The superscript $a$ in these equations is the group index, that is, it runs over the a set of elements of the representation (I believe the adjoint representation) of the group. The commutator of two quantities $X^b$^ and $Y^c$ becomes $f^{abc}X^bY^c$ where $f^{abc}$ are the (fully antisymmetric structure constants for the group. For example, if the YM group is $SU(2)$ which is isomorphic to $SO(3)$ then $f^{abc}$ is the usual Levi-Civita symbol $\varepsilon_{abc}$, which makes the expression into the familiar commutator.

In the B&M book the group index is suppressed; the implication is that $A_\mu$ is formed from group members and is not a group invariant. I think in the expression for $F_{jk}$ in that book makes the further assumption that the normalization of $A$ is such that the "coupling constant" ($g$ in the physics text) is one.

Answer 2

This is an answer to your comment on Mark's answer, but a bit long for a comment so here it goes.

Presumably, in BM $A$ takes value in a Lie subalgebra of $\mathfrak{gl}(\mathbb{R},n)$ and they are taking $\{e^i\otimes e_j\}$ as a basis for it (think of $e^i\otimes e_j$ as the matrix with 1 in the position $(i,j)$ and $0$ elsewhere). So $A$ and $F$ will both carry two "group" indices (components with respect to a basis of the Lie algebra really), and the appropriate number of Greek cotangent space indices.

In other books, especially when dealing with specific Lie algebras, you label a basis of the Lie algebra with a single index, e.g. for $\mathfrak{su}(2)$ you could take $t_k = i\ \sigma_k$ with $\{\sigma_k\}$ the Pauli matrices. You then expand $A=A^i_\mu\ \ t_i\otimes dx ^\mu$ which is the notation used in Wikipedia.