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I am a mathematician and reading a physics paper about the holonomy group of Calabi-Yau 3-folds.

In that paper, a Calabi-Yau 3-fold $X$ is defined as a compact 3-dimensional complex manifold with Kahler metric such that the holonomy group $G \subset SU(3)$ but not contained in any $SU(2)$ subgroup of $SU(3)$.

They remark "the condition that $G$ is not contained in $SU(2)$ is a really serious condition for physics since otherwise it would change the supersymmetry".

Could anyone kindly explain this sentence in more detail? I think that if $G\subset SU(2)$ the physics derived from the Calabi-Yau 3-fold has more supersymmetry (because less restriction), but what is wrong about it? One possibility is that too symmetric theory is trivial.

I would appreciate it if someone could kindly explain the physics behind it to a mathematician.

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Let me elaborate on Ryan's correct comments.

The flat background makes all components of the spinors covariantly constant; so the geometry is compatible with all of SUSY.

A generic curved 6-real-dimensional manifold has an $O(6)$ holonomy or $SO(6)\sim SU(4)$ if it is orientable. The $SU(3)$ subgroup preserves 1/4 of the original supercharges – it is the single charge among $4$ in $SU(4)$ that is not included in $3$ of $SU(3)$ and therefore "not participating in the mixing" that destroys the covariant constancy. If the holonomy is $SU(2)$, then 2/4 of the original spinor components i.e. 1/2 of the supersymmetry is preserved.

In reality, the $SU(3)$ holonomy manifolds are the the usual generic Calabi-Yau three-folds. Starting from 16 supercharges i.e. $N=4$ of heterotic string theory, for example, they produce the realistic $N=1$. However, $SU(2)$ holonomy would produce $N=2$ in four dimensions which is too much. $N=2$ SUSY is too large for realistic models – at least for quarks and leptons – because it guarantees too large multiplets, left-right symmetry of spacetime (no chirality), and other strong constraints on the spectrum and the strength of various interactions that would disagree with observations.

The manifolds with the $SU(2)$ holonomy are pretty much just Calabi-Yaus of the form $K3\times T^2$ and perhaps some orbifolds of this manifold. So two of the six dimensions remain flat and decoupled from the other, curved four.

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Say we have a supercharge $Q$ in $\mathbb{R}^{10}$. To turn this into a supercharge on the $\mathbb{R}^4$ effective theory obtained by compactifying on $X$, we need to contract $Q$ with a covariantly constant spinor on $X$. The reason why we want it to be covariantly constant is because we want to take the size of $X$ to zero.

Covariant constant spinors are obtained by taking a spinor at a point and parallel transporting it all over the manifold. We get a well-defined global spinor iff the spinor we started with was invariant under the action of the holonomy group. Thus, we get more of these if the holonomy representation of $X$ is restricted. The simplest it can be (trivial) occurs when $X$ is a flat torus, and we will get different numbers of supercharges arising from $Q$ when the holonomy of $X$ is $SU(2)$ or $SU(3)$.

There's a nice set of lecture notes talking about the relationship between special holonomy and covariantly constant spinors here: http://empg.maths.ed.ac.uk/Activities/Spin/Lecture8.pdf .

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    $\begingroup$ Thanks for the answer. Do you mean that there are more supercharges if the holonomy group is smaller? $\endgroup$ Dec 16 '13 at 6:34
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    $\begingroup$ Yes, since this means we have more covariantly constant spinor fields. $\endgroup$ Dec 16 '13 at 9:17

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