How Does Topological Action Simplify with Levi-Civita Tensor Contractions?

[jinbaw]

I'm trying to simplify an action that has the term: levicivita_[a,b,c,d]*levicivita^[mu,nu,rho,sigma]*R^[a,b]_[mu,nu]*R^[c,d]_[rho,sigma]

where a,b,c, and d are flat indices and mu nu rho sigma are curved indices

I got the term: 4*e^mu_a*e^nu_b*e^rho_c*e^sigma_d*R^a,b_mu,nu*R^c,d_rho,sigma

My question is if i have for example a=c levicivita_[a,b,c,d] is 0. however if i have a =c and mu=rho in the answer i got... i won't get a zero. is there some wrong in my computations? thank you

 

[arkajad]

I think the magic is that, for instance, $$4{R^{12}}_{12}{R^{12}}_{12}$$ indeed appears in your term, but it will also appear in those five other terms that you call "permutations" - and they will cancel out. The point of your expansion is to get nice contractions in the formulas, but it is not computationally optimal in the sense that there will be many cancellations of terms. In other words: you are adding and subtracting the same terms in order to get certain nice expressions like square of the scalar curvature etc.

Does it make sense?