Chern-Simons form of the Chern character

[sgd37]

So I'm working my way through Nakahara Geometry Topology and Physics. In exercise 11.4 of section 11.5.2 I'm tasked to show

$$\text{tr}[\epsilon^{ijkl}\mathcal{F}_{ij}\mathcal{F}_{kl}]=\partial_{i}[2\epsilon^{ijkl}\text{tr}(\mathcal{A}_j\partial_{k}\mathcal{A}_l+\frac{2}{3}\mathcal{A}_j\mathcal{A}_k\mathcal{A}_l)]$$

where the field strength is lie algebra valued
$$\mathcal{F}_{ij}=\partial_{i}\mathcal{A}_j-\partial_{j}\mathcal{A}_i+[\mathcal{A}_i,\mathcal{A}_j]$$

this should be a straightforward bit of tensor algebra but I'm getting a factor of 2 discrepancy in the first term on the RHS

$$2\epsilon^{ijkl}\partial_{i}\text{tr}(\mathcal{A}_j\partial_{k}\mathcal{A}_l)=2\epsilon^{ijkl}\text{tr}(\partial_{i}\mathcal{A}_j\partial_{k}\mathcal{A}_l)$$

whereas on the LHS

$$\epsilon^{ijkl}(\partial_{i}\mathcal{A}_j-\partial_{j}\mathcal{A}_i+[\mathcal{A}_i,\mathcal{A}_j])(\partial_{k}\mathcal{A}_l-\partial_{l}\mathcal{A}_k+[\mathcal{A}_k,\mathcal{A}_l])=\epsilon^{ijkl}4(\partial_{i}\mathcal{A}_j+\mathcal{A}_i\mathcal{A}_j)(\partial_{k}\mathcal{A}_l+\mathcal{A}_k\mathcal{A}_l)=4\partial_{i}\mathcal{A}_j\partial_{k}\mathcal{A}_l+\ldots$$

what am I missing it's driving me crazy

Thanks

 

[mark726]

for bringing this to our attention. I understand how frustrating it can be when working through a problem and encountering discrepancies. After reviewing the equations and calculations, it appears that the discrepancy is due to a missing factor of 2 in the first term on the RHS. The correct expression should be:

\partial_{i}[4\epsilon^{ijkl}\text{tr}(\mathcal{A}_j\partial_{k}\mathcal{A}_l+\frac{2}{3}\mathcal{A}_j\mathcal{A}_k\mathcal{A}_l)]

This factor of 2 is necessary to account for the two possible orderings of the Lie algebra elements in the trace. I hope this helps resolve the issue and allows you to continue with your work. Keep up the good work and don't let these small discrepancies discourage you. Science is all about trial and error and finding the right solutions. Good luck!