Some of the heterotic string models have an $E_6\otimes E_8$ symmetry. Examples include some orbifold models, some free fermionic models and Gepner models. We can break the gauge symmetry by including Wilson lines. Does anyone know if it is possible to break $E_6$ to $SO(10)$ while maintaining spacetime SUSY? I have seen examples in Gepner models where $E_6$ is broken to a smaller group than $SO(10)$ while keeping SUSY but none where $E_6$ is broken to $SO(10)$. Is this pure coincidence or is there a reason behind it? Any insights are appreciated!
I thought I might write an update on this in case anyone else is interested...
The answer is yes, we can break $E_6$ (to anything) while maintaining spacetime SUSY. Models with $E_6$ are known as $(2,2)$ and models without $E_6$ as $(0,2)$. $(0,2)$ means N=0 Superconformal symmetry on the the left-moving sector on the worldsheet and N=2 Superconformal symmetry on the right-moving sector on the worldsheet. There is a big literature on $(0,2)$ models but the important information is that they do indeed have spacetime SUSY. In fact, a theorem in this paper states that $N=1$ spacetime SUSY requires (at least) $(0,2)$ worldsheet SUSY.
The take home message is that $E_6$ gauge symmetry on the bosonic sector corresponds to N=2 Superconformal symmetry on the worldsheet.