The plan for this blog was to refine my understanding of GU, even to the point of doing calculations. And that's still an agenda. But a few extra things have come up.
First, I need to get a job, and that will eat into my time a lot.
Second, I have noticed an interesting potential crossover with the 1997 paper "Super Yang-Mills in (11,3) Dimensions" by Ergin Sezgin. Let's list the similarities:
1) Works with a Yang-Mills field in 14 dimensions with multi-time signature.
2) The definition of the shiab operator requires a choice of two invariant 1-forms. Sezgin's generalized superalgebra requires a choice of two invariant null vectors.
3) GU contains a special sub-manifold with 3 space dimensions (and 1 time dimension), the base space. Sezgin's model contains a brane with 3 space dimensions (and 3 time dimensions).
4) Sezgin's model is supersymmetric. GU may contain a BRST symmetry of the kind described in "Special Quantum Field Theories in Eight And Other Dimensions" (which cites Eric Weinstein's dissertation).
Here are some further comparisons:
A) Signature and GUT group
GU is usually described in (7,7) signature, factoring into (3,1) signature on the base and (4,6) signature on the fiber, the latter yielding a spin group Spin(6,4) which is then to reduce to Spin(6) x Spin(4) ~ SU(4) x SU(2) x SU(2), i.e. Pati-Salam.
One might suppose that GU in (11,3) signature factors into (1,3) signature on the base and (10,0) signature on the fiber, yielding the well-known GUT group Spin(10). Keeping the conventions of the previous paragraph, that looks like 1 space dimensions and 3 time dimensions on the base. However, I'm hoping that the right choices in defining the Frobenius metric used in GU, will allow us to have (3,1) signature on the base, while still having Spin(10) on the fiber.
B) Base space as brane
As noted, Sezgin's model contains a brane with (3,3) signature. In the context of a GU-Sezgin crossover, one might hope to obtain this by appending the two null directions to the (3,1) base space. However, doing that actually produces a 6-dimensional object with (4,2) signature, a kind of local confornal embedding. One might hope to get (3,3) through a Wick rotation, but then we'd need to understand what that does to the fibers, built as they are on pointwise metric on (3,1) space.
Still, there seems considerable prospect of a convergence between GU and Sezgin here. In GU, the only field native to the base space is the metric of general relativity. SL(4,R) is the local symmetry group in some formulations of general relativity, and its Lie algebra is isomorphic to that of SO(3,3), a local symmetry of (3,3) space-time.
A remaining issue would be the field content of the brane. Sezgin's brane has a field content of scalars from the directions normal to the brane, and spinors from the 14-dimensional gaugino. GU aims to get the standard model on the base space, from the 14-dimensional fields.
C) Other differences
Sezgin is simply working in flat space. But GU requires a nonuniform metric in 14 dimensions. One can presumably add metric dependence to Sezgin's equations, but this introduces the possibility of gravitational anomalies.
Sezgin uses a homogeneous gauge group, but the GU gauge group adds an affine factor of translations in field configuration space (which is combined with the usual gauge group in a semidirect product, in imitation of the Poincare group).
D) Bonus: triality speculations
There are a number of phenomenological "E8 models" in the literature (similar to Garrett Lisi's theory), in which the decomposition of the e8 Lie algebra includes an so(3,3) or sl(4,R) subalgebra. These models never seem to explain how the elements labelled as "fermions", actually acquire fermionic statistics. However, there is a 14-dimensional grading of E8(C), so I may need to consider whether one of these can also dovetail with a Sezgin-GU crossover.
Meanwhile, in his paper Sezgin speculates that triality, another common theme of these E8 models, may also be lurking somewhere in his own construct.
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