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:
