Dynamics of $E_6$ Chiral Gauge Theories

We present exact non-perturbative solutions to chiral gauge theories based on the $E_6$ gauge group and several matter fermions in the fundamental $\mathbf{27}$-dimensional representation. They are obtained when supersymmetric versions are perturbed by small supersymmetry breaking by anomaly mediation. The universality classes obtained are very different from what can be conjectured by the tumbling hypothesis. In particular, the case with three $\mathbf{27}$s may have an unbroken $\text{SU}(3)$ symmetry with massless composite fermions in $\mathbf{10}$ of $\text{SU}(3)$. For this case, we employed numerical techniques to obtain the exact ground state.