@MPorter2025
In GU, the physical metric on Y is not the canonical metric
@MPorter2025
More precisely: you have a canonical metric on the fibers, but you also need to choose a specific metric on the base space, to define the "horizontal" part of the metric on Y that connects the fibers.
@YinMills
Hey, which part of @TimHenke9 's criticisms is that addressed to? The initial reply where he said that it's cool that Met(M) seems to come free with a canonical metric? I'm just trying to figure out what's going on here.
@MPorter2025
Mostly I'm stating my understanding of g_Y in Geometric Unity. You need a choice of g_X to determine the horizontal parts of g_Y... Incidentally, just to bring in physics, the Pati-Salam group is supposed to emerge in GU as a compact part of the isometry group of the fiber metric
@YinMills
Right, so you have the space Y of all metrics on X, and you are saying you need to pick a metric on X to get a metric of metrics on Y? Do I have that right? As an aside, why did he define ι like that? Why couldn't he just have it be any embedding of X into Y?
@MPorter2025
Y is metric bundle over X, which is organized differently to space of all metrics on X. Space of all metrics is ∞-D, each X-metric is just a point in that space. Metric bundle is 14-D, consists of fibers above the points of X, each X-metric is a "section" through all fibers...
@YinMills
I keep being imprecise with words, I'm sorry. By "the standard embedding of X in Y", I didn't really mean anything specific, I just meant why does he define this weird construction that takes various open sets in X to Y, what does he gain as opposed to just taking a section?
@MPorter2025
Possibly a nod towards the most general version of his overall idea. See 2021 paper, pages 16-17. (And compare with 01:06:19 in Oxford lecture.) In principle GU via the metric bundle is just one form, in practice it's the only one he talks about. "Open sets U" never show up again
@YinMills
That is, why doesn't he just say that the observerse is a space X and a bundle Y on X, and a section on that bundle ι?
@MPorter2025
In practice, that's what it is. Please see the references I gave, if you want to see him talking about how bundle-theoretic GU is just one possible version. If (like me) your main interest is the actual proposed theory, stick with the idea that ι is a section of the metric bundle
@YinMills
Alright, that makes things simpler. So in practice we just have a space X and a metric on it, that can be many things. What would you say the juice is? It's very confusing to read his paper because it's full of very math-y terminology and weird definitions like this.
@MPorter2025
I found it hard to get into too. My primary focus has been on forming a picture of what is coupled to what. You have Y, you have X which is a brane or a defect in Y, you have fields on Y. g_Y is a kind of uplift of g_X, and...
@MPorter2025
the 4-D non-gravitational fields are the bulk fields from Y restricted to X. I also had to suppose an Einstein-like coupling between those fields and g_X in order to close the loop. That's basically how I think of it at present...
@MPorter2025
and then it's just a matter of trying to fill in the details. GU's quantum theory is up in the air, but I think the classical theory of a GU bosonic sector could be defined and solutions studied, right now. Part of the interest is simply that this is an unfamiliar form of theory,
@MPorter2025
so I'd like to see how it fits into the broader world of field theories. The bulk theory on Y is like an uplift of Poincare gauge theory onto the metric bundle, the shiab operator is meant to imitate the Einstein tensor in a gauge-invariant way inspired by Chern-Simons theory,
@MPorter2025
there are various ways you can try to solve the problems like anomalies and nonrenormalizability, there's just lots to think about.
@YinMills
Alright, but does that actually give the fields of the SM with the correct interactions? If it does then it's interesting, otherwise I'm not sure what utility it has. I mean, anyone can decide to use any old space as base space for a field theory.
@MPorter2025
I can tell you some of how it's supposed to work. There's a gauge field on Y (with a large complexified gauge group), coupled to scalar, spinor, and spinor-vector fields. Part of the gauge field corresponds to spin group of the fibers above X, which have metric signature (6,4)...
@MPorter2025
... thus Spin(6,4). The spinorial fields are supposed to give rise to the known fermion generations when restricted to X, and the hope is that some mechanism reduces the spin group to its compact part Spin(6) x Spin(4), which is the Pati-Salam group.
@YinMills
Is it supposed to have a graviton too?
@MPorter2025
Maybe only in 4D EFT. The 14D gauge field is presumably more fundamental, e.g. in GU, the 4D Einstein tensor descends from the "augmented torsion" of that 14D field (augmented torsion being a gauge-covariant form of torsion).
@YinMills
Does the 14D theory have some notion of Poincare invariance?
@MPorter2025
GU is a 14D gravi-GUT theory in which the 14D gauge group includes Y's local Spin(7,7) symmetry, which splits into a local Spin(1,3) symmetry on X, and Spin(6,4) on the fibers. g_Y has (7,7) signature and is a block-diagonal product of g_X and the fibers' canonical metric...
@MPorter2025
The 14D gauge group is a semidirect product of GL(128,C) with translations in field space. This is done so that a gauge-covariant torsion can be defined. Apparently something similar is done with torsion in Cartan gravity...
@MPorter2025
This semidirect product structure is why I said it's like an uplift of Poincare gauge theory to the metric bundle, though as far as I can tell there's no relationship between the "translations in field space" and spatial translations on X.
@YinMills
I'm asking because there might be problems with Coleman-Mandula and the like.
@MPorter2025
There's Weinberg-Witten to think about too! But I'm still digesting the big picture in GU.
No comments:
Post a Comment