Holonomy, SO(6), SU(3) and SU(4)

[AlphaNumeric2]

This springs from section 15.1.3 of Superstring Theory (Vol 2) by GS&W (should anyone have that to hand).

K is a compact 6 dimensional space, thus it's holonomy group is a subgroup of SO(6). Fine. $$\eta$$ is covariantly constant on K (comes from SUSY constraints). Thus need subgroup of SO(6), H, which has, for any U in H, $$U\eta = \eta$$. Okay so far.

GS&W then point out that $$\mathcal{L}(SO(6)) \equiv \mathcal{L}(SU(4))$$. That I understand. Spinors of definite chirality are then in the $$\mathbf{4}$$ or $$\mathbf{\bar{4}}$$ of SU(4). Okay with this. However, I don't see why this applies to $$\eta$$ since, from my understanding, $$\eta$$ would in a complex basis on a complex manifold, be a 3 component complex spinor, yet GS&W then talk about SU(4) matrices acting on a 4 component $$\eta$$.

Am I missing something? I can see SO(6) having a $$\mathbf{4}$$, which splits into a $$\mathbf{3}$$ and a $$\mathbf{1}$$ and then the holonomy preserving the singlet (and SU(3) works on the $$\mathbf{3}$$) so that there's one and one only covariantly constant spinor on K (as is needed by the string constraints), but going into a 4 component complex basis just seems confusing.

Is this just a particular way of represending a spinor on a 6 dimensional manifold? Wouldn't the 4 components give $$\eta$$ too many degrees of freedom? I thought I had my head around the whole Calabi Yau thing and it's construction via supersymmetry breaking but the 4 component spinor has thrown me.

Thanks in advance for any help.

 

[Haelfix]

Maybe I am missing the question, but SU(4) has a maximal subalgebra (spinor) that goes like Sp4 or SU(2) * SU(2) so it makes good sense to work in a 4 component complex basis. If you were looking at the real irreps then yes you would look at the maximal subalgebra that goes like SU(3) *U(1)

 

[AlphaNumeric2]

Sorry for the delay in replying.

Yeah, I was getting mixed up about real and complex reps and the symmetries involved which kept the number of degrees of freedom the same. A lot more reading and thinking has helped.

Thanks :)