Clifford Algebra and Gamma matrices: is this relation generally true for any dimension?

Question

I expect the following relation to be vanishing. But it seems not that obvious.

$\Gamma_{ab}^{\lambda}t^at^b \Gamma_{\lambda c(d)}t^c=0$

where $t^a$ are even ghosts, "$ab$" are indices for matrix element, and $\lambda$ denote different Gamma matrices. The Einstein summation convention is used above, i.e. we will sum over all indices except $d$.

I checked for both 3D and 4D Clifford algebra. The relation above seems to be right. But not sure whether it is generally true.

Does the following equation also vanishes?

$\Gamma_{\lambda a b}t^a t^b C^{\lambda} C^{\alpha}C^{\beta}=0$

where $C^{\lambda}$ are odd ghosts, i.e. $C^{\alpha}C^{\beta}=-C^{\beta}C^{\alpha}$.

The left hand side of the equation above is supposed to be something in $\wedge^2 V$, where $V=\{ C^{\lambda}|\lambda=1,2,\cdots,D \}$. $D$ is the dimension of the space.

Answer 1

It's not clear to me what you mean by "even ghosts". Do you mean perhaps that $t^a t^b = t^b t^a$?

If so, then you will find that the identity is only valid in 3, 4, 6 and 10 dimensions and with lorentzian signature. Indeed, this identity is essentially the condition for the vanishing of a fermionic trilinear which appears in the supersymmetric variation of Yang-Mills coupled minimally to an adjoint fermion, which in turn is the obstruction to the existence of "pure" supersymmetric Yang-Mills.

It is no accident that those dimensions are 2 plus the dimensions of the real division algebras: $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$. In fact, the identities are well-known identities for these algebras. In particular, when the dust clears, the ten-dimensional identities are the celebrated Moufang identities.

Of course, if I got the definition of the even ghosts wrong, then what I say above is probably wrong.