Quote by garrett
It would be straightforward to formulate a similar theory  including gravity, electroweakish gauge fields, and real fermion fields  using only F4 instead of E8.

Let's give this a shot then! Just to be clear, my objective is to work this through until I can see to my own satisfaction that it
is a welldefined quantum theory. So let me sketch in advance how it looks like things are supposed to go. Your equations 3.7 and 3.8 will still hold, except that things are now f4valued. Gravitational so(3,1) will drop out, and there will be fermions and gauge bosons left over.
First question: which parts of f4 will play the role of [tex]\underset{.}{\Psi}[/tex]? It looks like I can break it down as f4 = so(8)+(8+8+8) or as f4 = so(9)+16  would these lead to distinct "modified BF F4" theories?