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 P: 360 Hi Tom, You're after a unified description of scalar, fermion and gauge fields… very ambitious. But don't forget the gravitational spin connection and frame. Let $$A$$ be a 1-form gauge field valued in a Lie algebra, say spin(10) if you like GUTs, and $$\omega$$ be the gravitational spin connection 1-form valued in spin(1,3), and $$e$$ be the gravitational frame 1-form valued in the 4 vector representation space of spin(1,3), and let $$\phi$$ be a scalar Higgs field valued in, say, the 10 vector representation space of spin(10). Then, avoiding Coleman-Mandula's assumptions by allowing e to be arbitrary, possibly zero, we can construct a unified connection valued in spin(11,3): $$H = {\scriptsize \frac{1}{2}} \omega + \frac{1}{4} e \phi + A$$ and compute its curvature 2-form as $$F = d H + \frac{1}{2} [H,H] = \frac{1}{2}(R - \frac{1}{8}ee\phi\phi) + \frac{1}{4} (T \phi - e D \phi) + F_A$$ in which $$R$$ is the Riemann curvature 2-form, $$T$$ is torsion, $$D \phi$$ is the gauge covariant 1-form derivative of the Higgs, and $$F_A$$ is the gauge 2-form curvature -- all the pieces we need for building a nice action as a perturbed $$BF$$ theory. To include a generation of fermions, let $$\Psi$$ be an anti-commuting (Grassmann) field valued in the positive real 64 spin representation space of spin(11,3), and consider the "superconnection": $$A_S = H + \Psi$$ The "supercurvature" of this, $$F_S = d A_S + A_S A_S = F + D \Psi + \Psi \Psi$$ includes the covariant Dirac derivative of the fermions in curved spacetime, including a nice interaction with the Higgs, $$D \Psi = (d + \frac{1}{2} \omega + \frac{1}{4} e \phi + A) \Psi$$ We can then build actions, including Dirac, as a perturbed $$B_S F_S$$ theory. Once you see how all this works, the kicker is that this entire algebraic structure, including spin(11,3) + 64, fits inside the E8 Lie algebra.