I am not sure, actually. One would probably need a theory where gauge transformations with values in $G$ somehow get identified modulo a subgroup $H$. Maybe an $H$-equivariant version of $G$-CS theory.

]]>The claims in the section on “Ologs” seem surprising. If you could add a reference substantiating this?

]]>Ologs: Rewrite the example olog to go in roughly the correct direction, to use standard biology jargon, and to follow the “good practices” from Spivak & Kent 2012; merge explanatory paragraph and Spivak & Kent 2012 ref with ontology log.

]]>Document the rules for English and flesh out the idea a bit more. I want to avoid parroting the hype around this stuff.

There are rules for other languages too, but English is easy to add since it’s already in Spivak & Kent.

]]>Here what plays the analogous role of Chern-Simons in the 3d-2d CS/WZW correspondence?

]]>Clarify which weak countable choice.

]]>Typo fix in leading diagram.

]]>Yes, Lorentzian signature is crucial for this story.

On the U-duality side this is baked into the structure of the exceptional Lie algebra series: Together with the information about all of 11d Sugra, the over-extended $\mathfrak{e}_n$-algebras know its Lorentzian spacetime signature (see e.g. p. 11 in Nicolai’s 2009 lecture notes here). For this remarkable reason the Kac-Moody algbebra $\mathfrak{e}_{10}$ is also called a “hyperbolic Kac-Moody algebra” and $\mathfrak{e}_{11}$ a “Lorentzian Kac-Moody algebra”.

On the supersymmetry side this is baked into the Clifford representation theory: It is specifically the $\mathbf{32}$ of $Spin(1,10)$ whose symmetric square gives the exceptional 528-dimensional tangent space: $\mathbf{32} \otimes_{sym} \mathbf{32} \simeq \mathbf{11} \oplus \mathbf{55} \oplus \mathbf{462}$.

$\,$

Of course you may speculate that an entirely different series of Lie algebras will analogously serve as the U-duality for supergravity in other signature and/or dimension.

I don’t know, but intuitively I’d doubt it (based on a sentiment of exceptional naturalism :-)

]]>For this is it important to have a Lorentzian signature or does it work for more general signatures?

]]>Yes, section 4.6 there (pp 40) is where, I guess, we started suggesting the hidden M-algebra as the super-exceptional tangent space.

This 528 is the dimension of the 11d tangent space plus the central M-brane charges of the extended susy algebra:

$\mathrm{dim}\Big( \mathbb{R}^{1,10} \oplus \wedge^2 (\mathbb{R}^{1,10})^\ast \oplus \wedge^5 (\mathbb{R}^{1,10})^\ast \Big) \;\;\simeq\;\; \left(11 \atop 1\right) + \left(11 \atop 2\right) + \left(11 \atop 5\right) \;\; = \;\; 528$ ]]>Is this the same 528 as in ’528-toroidal T-duality’, here?

]]>Am working on it as we speak – hoping to have a readable version ready in a few days.

But the point is that the previous pattern of exceptional tangent bundles (as in Section 4 of Hull 2007) seemed to break beyond $n = 7$, because the global U-dualities $\mathfrak{e}_{n(n)}$ are no longer represented.

But consider that in general we should ask for the *local* U-duality symmetry to be represented, which is the “maximal compact” sub-symmetry $\mathfrak{k}_{n(n)}$.

Amazingly, that fixes the pattern completely: There is a $\mathbf{528}$ of the maximal compact $\mathfrak{k}_{11(11)}$ which *is* the bosonic part $\mathbb{R}^{528} \simeq \mathbf{32} \otimes_{sym} \mathbf{32}$ of the M-algebra and hence *is* the full exceptional tangent space (previously seemingly way too small, compared to the hugely infinite-dimensional basic rep of the Kac-Moody parent $\mathfrak{e}_{11(11)}$) — and under the chain of inclusions $\mathfrak{k}_{8(8)} \hookrightarrow \mathfrak{k}_{9(9)} \hookrightarrow \mathfrak{k}_{10(10)} \hookrightarrow \mathfrak{k}_{11(11)}$ this branches exactly through the previously seemingly broken sequence of exceptional tangent spaces for lower $n$.

The upshot is that the “hidden M-algebra” is thus indeed the “correct” model space which unifies the exceptional-geometric and the super-geometric formulation of 11d SuGra to “super-exceptional geometry”.

Assuming this was our starting point for the “super-exceptional M5-brane model” here and here, but back then we didn’t justify this assumption more deeply.

But now I finally had the simple idea that from our hypothesis I can just *predict* which irreps of $\mathfrak{k}_{n(n)}$ ought to exist. By just googling for my predicted three numbers I found exactly three existing articles (all from the exceptional Potsdam group) where exactly these irreps are noted (all in passing side remarks, the relevance was apparently not realized before).

added pointer to

- Axel Kleinschmidt, Ralf Köhl, Robin Lautenbacher, Hermann Nicolai:
*Representations of involutory subalgebras of affine Kac-Moody algebras*, Commun. Math. Phys.**392**(2022) 89–123 [arXiv:2102.00870, doi:10.1007/s00220-022-04342-9]

I saw your post on X. Do you have a preliminary write-up I could read that touches on the significance of this fact?

]]>added pointer to:

- Christopher L. Rogers, Jesse Wolfson:
*Lie’s Third Theorem for Lie $\infty$-Algebras*[arXiv:2409.08957]

(here and at *Lie integration*)

Added a reference by Cristina Pedicchio.

]]>as expected (from the pattern here) there seems to be a 528-dimensional irrep of the maximal compact subalgebra of $\mathfrak{e}_{11(11)}$ (according to p. 29 of arXiv:1809.09171)

have made a brief note here, hoping to later expand further

]]>have added some minimum of references (there were none before)

but I hope to find the time to put some actual content into the entry:

the sequence of exceptional tangent bundles used to be truncated, and the other day I saw (cf. nForum discussion here and here) how to complete it, using recent results.

a pdf note is now here (just 1 page)

]]>pointer

- John Baez.
*What is Entropy?*(2024). (arXiv:2409.09232).

added pointer to today’s

- Fernando Quevedo, Andreas Schachner:
*Cambridge Lectures on The Standard Model*[arXiv:2409.09211]

This is for olog-specific stuff which wouldn’t be appropriate for biology.

]]>Fixing a broken link

Natalie Stewart

]]>Fixing a broken link

Natalie Stewart

]]>I have checked with the authors, and it’s indeed true. This is remarkable.

Have made a brief note of the matter here, will expand tomorrow.

]]>