More on Spinors and Clifford

After John Baez wrote about Clifford bundles in TWF211 and Peter Woit commented to which Lubos commeted let me reiterate my favourite property of Clifford algebras: Fierz identities.

Remember that given a vector space V with a non-degenerate bilinear form you can define a Clifford C(V) algebra by saying that there is a linear map gamma from V to C(V) such that
{gamma(v),gamma(w)}=2 b(v,w) 1 and such that any other such map factors. Physicists usually denote the generators of C(V) by gamma(i) where i labels an orthonormal basis of V.

The important property is that as a vector space, there is a bijection between C(V) and the exteriour algebra over V. That is if V is the tangent vector space over some point of a manifold, you can view any element of C(V) as a differential form. Fierz identities tell you how this map looks like concretely.

If you work in higher dimensional theories and care about fermions you probably spend a significant amount of your time working out the details of this in specific cases.

Now, what is this relation of Clifford algebras and susy that Peter was refering to? Lubos guessed that this was just the fact that the supercharge is a spinor and thus a module over a clifford algebra. But there is a much more interesting relation and that involves division algebras.

Remember, there are four division algebras: The reals, the complex number, quaternions and octonions. They have a number of defining properties, and amongst them is alternativity: Not all of them are associative and you can define

t(a,b,c) = a(bc)-(ab)c

Alternativity tells you that t is anti-symmetric under swapping two of a,b,c. Note that this is quadratic in the structure constants of the division algebra as there are two products envolved.

Furthermore, you should know that there is a nice representation of SO(9,1) gamma matrices (generators of the above Clifford algebra): For gamma^+ and gamma^- you use the usual form for Weyl spinors and the remaining eight are block offdiagional. The blocks are given by

c^i_jk = c_ijk if all 1<=i,j,k<=7
c^8_jk = delta_jk if 1<=jk<=7 and cyclic
and 0 if there are two 8's.

where the c_ijk are the structure constants of the octonions that is summarized by the triangle diagram. There are similar expressions for the gammas for SO(5,1) and SO(3,1) and SO(2,1) related to the other division algebras.

Now you open GSW 1 in the appendix to chapter 4 where super YM is discussed. There, it is argued that you would guess that the susy variation of the gauge field is proportional to the spinor and that the susy variation of the spinor is proportional to the field strength (using gamma matrices to soak up the indices). Then they guess an action of the form

L= F^2 + psi D psi

where D is the covariant Dirac operator. Its relatively easy to adjust the relative factor so the variations of the two terms transform into each other. But there is one term that is special: If you vary the gauge field within D you get a term with three psi's and that is the only such term. So it has to vanish by itself. It also comes with two gamma matrices. Here, the above mentioned Fierz identities come to help: They tell you that exactly in 3,4,6 and 10 dimensions this combination of gamma's vanishes. This is why there is sYM in those dimensions.

But now you can plug in the above representation of gamma matrices. The candidate expression is quadratic in gamma's. Guess what the Fierz identity corresponds to in terms of the division algebras!

So, the working of susy theories rely on Fierz identities and those are deeply encoded in the structure of the various Clifford algebras.

NB that the little group in 10D is SO(8) which has triality (the Dynkin diagram is the star): The vector and the two spinor representations are equivalent. This was also needed in the above representation (i didn't use different types for the three different indices of the gamma's). And this 8 is of course related to the 8 in Bott periodicity. But this is just because with Clifford algebras (and other "special" algebraic structures like division algebras and exceptional Lie groups), everything is related. Fnord.