The MSSM (sort of) from the heterotic string

Today I have a new paper out with several collaborators, in which we construct compactifications of the heterotic string which lead to exactly the light field content of the MSSM in four dimensions. In itself, this is not a new achievement; it has been about six years since such models first appeared in the literature. The difference is that these models are obtained by deformations of the so-called 'standard embedding' solution.

Although you probably either know this already, or won't understand it without a lot more background, let me briefly explain what this means. In 1985, a seminal paper (of which Philip Candelas, one of my collaborators on the new work, was a co-author) established that there is a canonical way to compactify the $E_8{\times}E_8$ heterotic string theory on any Calabi-Yau threefold $X$, leading to an $\mathcal{N}=1$ supersymmetric theory in flat four-dimensional spacetime. More specifically, it yields an $E_6$ gauge theory, with matter content determined by the topology of the manifold — there are $h^{2,1}(X)$ ($h^{1,1}(X)$) chiral multiplets in the $\mathbf{27}$ ($\mathbf{\overline{27}}$) representation, where the notation $h^{p,q}(X)$ refers to the Hodge numbers of $X$.

There are lots of problems with the standard embedding as I've described it so far: we know that if $E_6$ is indeed a gauge symmetry in the real world, it must be broken at a very high scale. Furthermore, as well as one generation of standard model fermions, the $\mathbf{27}$ of $E_6$ contains a lot of extra junk, and we also typically get a large number of vector-like pairs $\mathbf{27}\oplus\mathbf{\overline{27}}$, even if we arrange for three chiral generations. But in principle, all these problems can be solved with some fiddling. If $X$ has a non-trivial fundamental group, we can turn on discrete VEVs for Wilson line operators corresponding to homotopically non-trivial loops in the extra dimensions (i.e. we associate $E_6$-valued analogues of Aharanov-Bohm phases with such loops), which breaks $E_6$ to some smaller group. If there are vector-like pairs in the theory, we can also give VEVs to these, to Higgs the remaining gauge group to that of the standard model. This procedure is still somewhat problematic though, because although vector-like pairs correspond to $D$-flat directions, there may be an $F$-term potential preventing them obtaining a VEV, and this is much harder to calculate. The $F$-terms also determine which extra fields obtain mass from this Higgs mechanism, so it also very difficult to determine the resulting light spectrum. Perhaps most worrying though, is that we need to take their VEVs to be up very close to the compactification scale, and the effective field theory is likely to break down.

To avoid the above problems, we can note the following: the charged fields in four dimensions are modes of the ten-dimensional super-Yang-Mills fields, so giving them VEVs corresponds to deforming the gauge bundle over the compact space $X$. In the new paper, we directly constructed the corresponding bundles on a particular Calabi-Yau manifold (discovered by Volker Braun a couple of years ago). This allowed us to check that they solve the appropriate equations (i.e. in four-dimensional language, the deformations are not obstructed by a potential), and directly calculate the light spectrum. The punchline is that eight of the families of bundles we constructed lead to the massless spectrum of the MSSM $mdash; three generations of standard model fermions and their superpartners, as well as a single pair of Higgs doublets.

So does one of these eight models describe the real world? Well, probably not, but you never know. We have not yet calculated the Yukawa couplings in the model, nor dealt with the problems of supersymmetry breaking or moduli stabilisation, mainly because just constructing these models and calculating the spectrum took two years of on-and-off work. These steps would have to be carried out before we know whether any of these models are realistic. But in any case, it is nice to be able to answer a question which has been outstanding for well over twenty years: can any standard embedding solution be modified in such a way as to yield just the light field content of the MSSM? Yes! And in fact, in the last section of the paper we argue that with the current state of knowledge, it seems that our models are unique in this regard.