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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?

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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:

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