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

  1. Sep 12, 2007 #1
    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. [tex]\eta[/tex] is covariantly constant on K (comes from SUSY constraints). Thus need subgroup of SO(6), H, which has, for any U in H, [tex]U\eta = \eta[/tex]. Okay so far.

    GS&W then point out that [tex]\mathcal{L}(SO(6)) \equiv \mathcal{L}(SU(4))[/tex]. That I understand. Spinors of definite chirality are then in the [tex]\mathbf{4}[/tex] or [tex]\mathbf{\bar{4}}[/tex] of SU(4). Okay with this. However, I don't see why this applies to [tex]\eta[/tex] since, from my understanding, [tex]\eta[/tex] 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 [tex]\eta[/tex].

    Am I missing something? I can see SO(6) having a [tex]\mathbf{4}[/tex], which splits into a [tex]\mathbf{3}[/tex] and a [tex]\mathbf{1}[/tex] and then the holonomy preserving the singlet (and SU(3) works on the [tex]\mathbf{3}[/tex]) 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 [tex]\eta[/tex] 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.
  2. jcsd
  3. Sep 15, 2007 #2


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    Maybe im 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)
  4. Oct 7, 2007 #3
    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 :)
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