# What is the trace in the Chern-Simons action

I have been looking at the Chern-Simons Lagrangian in $$(2+1)$$-dimensional spacetime $$M$$ in terms of a gauge field $$A$$:

$$S[A] = \frac{k}{4 \pi}\int_M \text{Tr}(A \wedge \text{d}A+ \frac{2}{3}A \wedge A \wedge A).$$

The Chern-Simons theory allows $$A$$ to be a "Lie algebra valued $$n$$-form". According to Wikipedia, a Lie algebra $$n$$-form is defined as an object $$A \in (\mathfrak{g} \times M) \otimes \wedge^k T^*(M)$$, where $$\mathfrak{g}$$ is a Lie algebra and $$T^*(M)$$ is the cotangent bundle. I believe this means that I can take my gauge field $$A$$ and decompose it as

$$A = \tilde{A} \otimes \omega = A_{\mu}^a T_a \otimes \text{d}x^\mu$$

where $$\tilde{A}$$ is a Lie algebra valued field on spacetime, $$\omega$$ is an n-form and $$\{ T_a \}$$ and $$\{ \text{d}x^\mu\}$$ are the bases for $$\mathfrak{g}$$ and $$T^*(M)$$ respectively.

From studying Yang-Mills theory, I am aware that I can produce objects which are scalar with respect to gauge transformations by using the Killing form

$$K(X,Y) = \text{Tr}(\text{ad}X \text{ad}Y)$$

where ad is the adjoint representation of the Lie algebra, however the trace notation "Tr" in the Chern-Simons action has always bothered me - is it really a trace in the usual matrix sense? Some literature seems to suggest that the "trace" is in fact an invariant bilinear form on our Lie algebra, i.e. a Killing form. It is at this point that I get confused.

## My question

If Tr is to be interpreted as an inner product, what does $$\text{Tr}(A \wedge \text{d}A)$$ and $$\text{Tr}(A \wedge A \wedge A)$$ mean? An inner product should take two arguments so what are the arguments in each case and how do I explicitly evaluate this?

Generally speaking, for a compact connected Lie group $$G$$ of the form,

$$G = U(1) \times \dots \times U(1) \times G_1\times\dots\times G_s$$

where $$G_i$$ are compact, simple Lie groups, $$\langle \cdot,\cdot\rangle_{\mathfrak g}$$ is an $$\mathrm{Ad}_G$$-invariant, positive definite scalar product on $$\mathfrak g$$ which can be constructed as the direct sum of:

1. a positive definite scalar product on $$\mathfrak{u}(1)\times\dots\times \mathfrak{u}(1)$$;
2. $$\mathrm{Ad}_{G_i}$$-invariant positive definite scalar products on $$\mathfrak g_i$$, which may always be constructed by, for example, the negative of the Killing form.

Now suppose we have a Lie algebra-valued one form, $$A$$, by the notation $$\langle A, A\rangle$$ we mean taking the wedge product and the inner product, above defined.

Now the notation $$\mathrm{Tr}$$ is misleading because to construct the Lagrangian, we take wedge products, and then evaluate an inner product, which happens to involve a trace, but I find it more proper to note we are taking an inner product. Thus by,

$$\mathrm{Tr}(A\wedge A \wedge A)$$

we mean $$\langle [A,A],A\rangle$$ (with wedge products implied), up to some constant depending on the normalization chosen for the structure constants. Two useful references are:

• Mathematical Gauge Theory by Mark J.D. Hamilton
• Differential Geometry, Gauge Theory and Gravity by M. Göckeler and T. Schücker