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

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.

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I question some of the tags. I think that the first and third are appropriate, but not the others. If my answer is indeed correct, the division-algebras might also be appropriate. –  José Figueroa-O'Farrill Nov 3 '10 at 1:37
retagged. ${ }$ –  userN Nov 3 '10 at 2:40
Thank you for that. –  José Figueroa-O'Farrill Nov 3 '10 at 12:06
@Osiris: Would it be possible to add some words of context? Where do you meet/need this identity, if different from supersymmetric Yang-Mills? –  José Figueroa-O'Farrill Nov 3 '10 at 12:07
@José: Thanks a lot. Yes, that's what I really want. It is for SUSY Yang-Mills. –  Osiris Nov 3 '10 at 22:50

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.

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@José Would it be possible to see some more details of why this works? It looks like it might be similar (but not the same) as the discussions in the papers Division Algebras and Supersymmetry I and II discussed at the n-Cafe golem.ph.utexas.edu/category/2009/09/… –  Simon Nov 3 '10 at 2:07
Yes, it is basically the same identity as in those papers. Of course, the result is much older than that. I think it goes back at least to the early 80s and a paper of Kugo and Townsend (inspirebeta.net/record/181889) if not earlier. –  José Figueroa-O'Farrill Nov 3 '10 at 2:18
The susy Yang-Mills identity can be found in Appendix 4.A of Green, Schwarz and Witten, Superstring Theory where the proof for 10 dimensions is worked out in detail. –  Jeff Harvey Nov 3 '10 at 2:21