## Wednesday, 7 December 2011

### 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.