Help with obtaining EoM for Dynamical Chern-Simons

[Alejandro Aguilar]

Hi everyone,

I'm having trouble with obtaining the equations of motion for DCS theory.  According to https://arxiv.org/pdf/0907.2562 , the term *RR is equal to the divergence of the Chern-Simons Current K^\mu. I'm trying to show that the equations of motion in both forms of the action, with *RR and K are equivalent but I seem to be struggling with it. Firstly, because I can't seem to obtain the correct form of the C-Tensor. I tried following one example shown in https://github.com/xAct-contrib/examples and applied the rules defined on it but I'm still missing something.

The second problem I'm having is that since the K^\mu is defined with Christoffel Symbols the ToCannonical command is having problems.

I'm attaching the attempt I've done for my notebook.

I would appreciate any help or suggestions.

Cheers,

Alejandro