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Apr24-11, 02:08 PM
garrett's Avatar
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 [tex]A[/tex] be a 1-form gauge field valued in a Lie algebra, say spin(10) if you like GUTs, and [tex]\omega[/tex] be the gravitational spin connection 1-form valued in spin(1,3), and [tex]e[/tex] be the gravitational frame 1-form valued in the 4 vector representation space of spin(1,3), and let [tex]\phi[/tex] 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):
[tex]H = {\scriptsize \frac{1}{2}} \omega + \frac{1}{4} e \phi + A[/tex]
and compute its curvature 2-form as
[tex]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 [/tex]
in which [tex]R[/tex] is the Riemann curvature 2-form, [tex]T[/tex] is torsion, [tex]D \phi[/tex] is the gauge covariant 1-form derivative of the Higgs, and [tex]F_A[/tex] is the gauge 2-form curvature -- all the pieces we need for building a nice action as a perturbed [tex]BF[/tex] theory. To include a generation of fermions, let [tex]\Psi[/tex] be an anti-commuting (Grassmann) field valued in the positive real 64 spin representation space of spin(11,3), and consider the "superconnection":
[tex]A_S = H + \Psi[/tex]
The "supercurvature" of this,
[tex]F_S = d A_S + A_S A_S = F + D \Psi + \Psi \Psi[/tex]
includes the covariant Dirac derivative of the fermions in curved spacetime, including a nice interaction with the Higgs,
[tex]D \Psi = (d + \frac{1}{2} \omega + \frac{1}{4} e \phi + A) \Psi[/tex]
We can then build actions, including Dirac, as a perturbed [tex]B_S F_S[/tex] 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.