Calculation and philosophy

In a comment at the sub-reddit /r/ThePortal, a forum originally devoted to Eric Weinstein's podcast of that name, I mentioned how uplifting well-known physical solutions like electromagnetic plane waves and black hole metrics, to their 14-D counterparts in GU, would be a way to advance the (classical bosonic)  theory. 

Meanwhile, a Youtube discussion led to the idea that quantum GU would be advanced by trying to embed processes like QCD confinement and the Higgs mechanism into the GU framework. 

I'll also note that in a slide on his talk in April, "From Dark to Geometric Energy", Eric provides the most concrete explanation we have yet seen, as to how the Seiberg-Witten equations relate to GU. See the bottom of the screen at 2:30 - a Rosetta stone relating general relativity, Geometric Unity, and the Seiberg-Witten equations. 

Another important part of the lecture, is where he motivates the use of the Hodge star operator and the shiab operator, to construct a higher-dimensional gauge-invariant analogue of the tensor contraction used to define the Einstein tensor in general relativity. 

Crossover with Sezgin 1997

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. 

The plan II

From a Youtube comment I posted today. "Here" refers to Curt Jaimungal's 3-hour "iceberg" video about GU. 

If you want to study it, I would suggest the following sequence of topics: the 14-dimensional field content and equations of motion (layer 2 here), viewed classically; the derivation of standard 4-dimensional field equations from the 14-dimensional equations (layer 3 here); and finally the quantum theory. This "classical first" approach means focusing on the bosonic field content. It would be nice if we knew something, analytical or numerical, about solutions to classical GU - they would be relevant for any quantum version too, e.g. by way of their role in a path integral. 

Another angle you could take, is to use aspects of Seiberg-Witten theory as a guide. Eric says he came up with his version of the Seiberg-Witten equations while contemplating moduli spaces of solutions to a toy model of GU. You could also look at his dissertation, and at the paper by Baulieu et al ("Special Quantum Field Theories in Eight and Other Dimensions"), that cites his dissertation for more inspiration of this kind. Baulieu et al is also of interest because it describes field theories with a boson-fermion relationship that is not a standard supersymmetry, but rather a form of BRST symmetry, and Eric may have this in mind when he talks about a kind of SUSY for his 14-dimensional field theory. 

What I'm saying here, is that studying moduli spaces of solutions to all these simpler theories, may set you up to tackle solutions of full GU. At this point I'm describing a research program rather than just a study plan, but since GU has at this point a mostly formal existence, this is what's required to advance it to the level of calculation.

The plan

I've been studying Eric Weinstein's Geometric Unity on and off for a few years. The latest blitz of videos, especially from Curt Jaimungal, has revealed enough information that it may finally be possible for me to make sense of it as a whole. 

I anticipate that the three main topics here will be GU as a classical theory, GU as a quantum theory, and how 4-dimensional physics is obtained from 14-dimensional physics.