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P: 755
 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 well-defined 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 f4-valued. 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 $$\underset{.}{\Psi}$$? 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?