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.

This post imported from StackExchange Physics at 2014-03-31 16:00 (UCT), posted by SE-user Heterotic