This is my latest fuzzy thought... Woit's theory ("Euclidean twistor unification") is a form of chiral graviweak unification. In such theories, a four-dimensional rotational symmetry is factored into left-handed and right-handed components, which become the gauge symmetries of the weak force and the gravitational force respectively. Woit goes further, considering a particular quotient bundle in twistor space, in order to obtain the hypercharge and color gauge symmetries as well.
Some things are lacking in Woit's theory - like fermions, and three generations. But let's suppose he does have a way to obtain gravity and the standard model gauge fields, by looking at the twistor space of a 4d manifold in a particular way... If we switch back to Weinstein's Geometric Unity, it embeds a 4d manifold into a 14d manifold which is actually the metric bundle of the 4d manifold. Then we are to have a 14d Yang-Mills field coupled to the 4d manifold by a "shiab operator" in such a way as to somehow give us 4d gravity, and a 14d Dirac field that in 4d will give us two of the three fermion generations (with the heavy third generation to come as a piece of a 14d Rarita-Schwinger field).
What interests me is whether Woit's construction can be obtained as a subset of Weinstein's. One reason to consider this, is that Weinstein also talks about obtaining gravity from an SL(2,C) subgroup of his complexified 14d gauge group - a group which also features in some chiral approaches to gravity. Unfortunately it is unclear to me how this related to the shiab operator, which is to mimic, for the 14d gauge field, the Ricci decomposition of the 4d metric.
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