Chern-Simons action for non-abelian brane worldvolume and Tsetlyn's symmetric trace prescription I am trying to reproduce the results of the (famous) Myer's paper "Dielectric Branes" https://arxiv.org/abs/hep-th/9910053. I am struggling a bit to obtain the numerical factors in the equations. When he explains the D0-D2 bound state in terms of a stack of D0 branes, thus the non-abelian prescription, he decomposes the Chern-Simons term as follows in his equation (67a),
$$
{\rm Tr}\left(\Phi^j\Phi^i[\Phi^k~\partial_k C_{ijt}^{(3)}(t)+
C_{ijk}^{(3)}(t)~D_t\Phi^k]\right) \tag{67a}
$$
I am assuming that the $Tr(...)$ symbol here is actually Tsetlyn's symmetric trace description,
$$
\text{STr}(A_1...A_n)=\frac1{n!}(A_1...A_n+\text{all possible permutations}),
$$
which is supposed to capture the $\alpha'$ corrections in the non-abelian D$p$-brane action up to order four or so.
My problem is the following: Equation (67) is supposed to be equal to
$$
\frac{1}{3} {\rm Tr}(\Phi^i\Phi^j\Phi^k)~ F_{tijk}^{(4)}(t), \tag{67b}
$$
To obtain (67b), you have to realize that $\partial_k C_{ijk}(t)=0$ since $k$ are spatial coordinates and $C^{(3)}$ only depends on time, and then you do a partial integration over the other term, which (says me) leads to
$$
 \text{Tr}(\Phi^i\Phi^j\Phi^k)F_{tijk}^{(4)}.
$$
So, then I say that this $1/3$ probably comes from the symmetrized trace prescription, but I do not see how, since I believe that
$$
\text{STr}[\Phi^i\Phi^j\Phi^k F_{tijk}^{(4)}]=0,
$$
because of the indexes $F_{tijk}$ is antisymmetric in $i,j,k$. Maybe this factor $1/3$ is coming from somewhere else?  
 A: Tsetlyn's symmetric trace prescription is not used here. We basically only need to use (i) that a form field $C^{(3)}$ is totally antisymmetric in its indices, and (ii) that the trace ${\rm Tr}$ over the $N\times N$ matrix-valued scalars $\Phi^{\ell}$ is cyclic: 
$$\begin{align}
&~{\rm Tr}\left(\Phi^j\Phi^i[\Phi^k~\partial_k C_{ijt}^{(3)}(t)+
C_{ijk}^{(3)}(t)~D_t\Phi^k]\right) \tag{67a} \cr
~=~~&-{\rm Tr}\left(\Phi^i\Phi^j[\Phi^k~\partial_k C_{ijt}^{(3)}(t)+
D_t\Phi^k~C_{ijk}^{(3)}(t)]\right) \cr
~=~~&-{\rm Tr}\left(\Phi^i\Phi^j\Phi^k~\partial_i C_{jkt}^{(3)}(t)+\frac{1}{3}D_t[\Phi^i\Phi^j\Phi^k]~C_{ijk}^{(3)}(t)\right)\cr
~\stackrel{\begin{array}{c}\text{int. by}\cr\text{parts}\end{array} }{\sim}& \frac{1}{3} {\rm Tr}(\Phi^i\Phi^j\Phi^k)~[-3\partial_i C_{jkt}^{(3)}(t)+ \partial_tC_{ijk}^{(3)}(t)] \cr
~=~~&\frac{1}{3} {\rm Tr}(\Phi^i\Phi^j\Phi^k)~[-3\partial_{[i} C_{jk]t}^{(3)}(t)+ \partial_tC_{[ijk]}^{(3)}(t)]\cr
~=~~&\frac{1}{3} {\rm Tr}(\Phi^i\Phi^j\Phi^k)~4 \partial_{[t}C_{ijk]}^{(3)}(t)\cr
~\stackrel{(B.4.3)}{=}&\frac{1}{3} {\rm Tr}(\Phi^i\Phi^j\Phi^k)~ (\mathrm{d}C^{(3)}(t))_{tijk}\cr
~=~~&\frac{1}{3} {\rm Tr}(\Phi^i\Phi^j\Phi^k)~ F_{tijk}^{(4)}(t). \tag{67b}
\end{align}$$
References:


*

*J. Polchinski, String Theory Vol. 2, 1998; eq. (B.4.3).

