In some forms of string theory, including E 8 × E 8 heterotic string theory, the resultant four-dimensional theory after spontaneous compactification on a six-dimensional Calabi–Yau manifold resembles a GUT based on the group E 6. Notably E 6 is the only exceptional simple Lie group to have any complex representations, a requirement for a theory to contain chiral fermions (namely all weakly-interacting fermions). Hence the other four (G 2, F 4, E 7, and E 8) can't be the gauge group of a GUT.