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I know that there are few Q&A here regarding implementing summation convention and also other packages like xAct to do such things efficiently. But I was hoping for a very well cut-out technique for my problem at hand.

I want to compute (Einstein's summation implied)

$g^{ab}g^{cd}g^{ef}g^{ij}g^{kl}\epsilon_{bdfjl}\partial_{[c}A_{e]}\partial_{[i}A_{k]}$.

As some of you can guess that such terms appear in trying to handle Chern-Simons theories in curved spacetimes.

Anyway, you can assume here that all the indices take five coordinate values. Metric $g^{ab}$'s are known for all components. Levi civita symbol $\epsilon$ is known in some convention and the 5-component vector fields $A_a$'s are also some function of some coordinates (I am keeping things general. E.g. I can choose time component $A_t$ to be a function of radial coordinate only and so on).

My question is how do I implement this summed over terms in mathematica by first defining the vector fields (function of some coordinates) and metric. Note that one component is free, so basically I have 5 terms.

Thanks! Any help is much appreciated!

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