Clifford vector valued 1-form and gravity

[Octonion]

I am currently trying to read through Garret Lisi's paper, An Exceptionally Simple Theory of Everything, and am having trouble understanding what it means for the gravitational fields to be described by a spin connection that is a Clifford bivector valued 1-form \begin{equation} \omega \in so(3,1) = Cl^2(3,1). \end{equation} I understand how the electroweak and strong are described by the special unitary group but I'm not sure what to make of gravity.

Additionally, how exactly does the frame defined by a Clifford vector valued 1-form \begin{equation}e \in Cl^1(3,1)\end{equation} combine with "a multiplet of Higgs scalar fields" in order to give fermions masses.

 

[Ben Niehoff]

I haven't read Garrett's paper in detail, but I can guess what I think he means here. Typically the spin connection is $\mathfrak{so}(3,1)$-valued and is written

$$\omega^a{}_b$$

where {a,b} are tangent space indices. The metric-compatibility condition requires that the spin connection be $\eta$-antisymmetric; that is

$$\eta_{ac} \omega^c{}_b = - \eta_{bc} \omega^c{}_a$$

To get a Clifford-bivector-valued form, just contract with some gamma matrices:

$$\eta_{ac} \omega^c{}_b \gamma^a \gamma^b = \eta_{ac} \omega^c{}_b \gamma^{[a} \gamma^{b]} = \eta_{ac} \omega^c{}_b \gamma^{ab}$$

due to antisymmetry. It's the same idea with the frame fields; contract their tangent-space index with a gamma matrix. As for how they describe gravity, one can rewrite the Einstein-Hilbert action in terms of $\{e^a, \omega^a{}_b\}$ as fundamental variables. The result is called the Palatini action.

As for how the frame interacts with the Higgs scalars to give the fermions masses, I can't help you there. However, Garrett Lisi does read this board and tends to show up when people ask about his paper.