# An Identity in SUSY sigma model

1. Jun 25, 2008

### Claret

I am struggleing in an identity, i.e. $$\nabla_m R_{ikjl}(\overline{\epsilon}\psi^m)(\overline{\psi^i}\psi^j)(\overline{\psi^k}\psi^l)=0$$ ,

where $$i,j,k,l,m$$ are dummy indices, $$\nabla_m$$ is covariant derivative, $$R_{ikjl}$$ is Riemann-Christoffel curvature tensor, and it is known that, for any two arbitary spinors $$\psi^i,\psi^j$$, $$\overline{\psi^i}\psi^j=\overline{\psi^j}\psi^i$$.

I think one could use Bianchi Identity to prove this, but I failed.....who can do me a favor? Thanks!