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.


2 Answers 2


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.


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 .

  • 1
    $\begingroup$ Thanks for the answer. Do you mean that there are more supercharges if the holonomy group is smaller? $\endgroup$ Dec 16, 2013 at 6:34
  • 2
    $\begingroup$ Yes, since this means we have more covariantly constant spinor fields. $\endgroup$ Dec 16, 2013 at 9:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.