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have added a minimum on the level decompositon of the first fundamental rep of $E_{11}$ here.
Could that Mysterious Duality found by Vafa et al. feature in your account?
A beautiful duality was discovered by Iqbal, Neitzke and Vafa between compactifications of M-theory on tori and the second cohomology of some associated del Pezzo surfaces. Now the full cohomology of theses surfaces spans the root lattice of a Borcherds superalgebra. Henry-Labordere Julia and Paulot have shown that some truncations of these Borchers algebras provide a classification of p-forms coming from tori reduction of (massive) maximal supergravity. This classification matches the one of the E11 conjecture of Peter West. The Borcherds description was recently proven to be systematically derived from the split real form of E11 by Henneaux, Julia and Levie. (MathOverflow comment).
Presumably that last result is in E_11, Borcherds algebras and maximal supergravity.
Thanks for the pointer! HJL 10 is most useful. The discussion in 1.3 on p. 6 makes me hopeful that it’s their Borcherds’ algebra which is the right thing to act on the exceptional tangent bundle in 11d. For the fact that there is the immense richness of higher levels in $E_{11}$ is good for campfire speculation as to M-theory-the-grandiose, but that Borcherds algebra seems to contain exactly what we need for M-theory-the-concrete, which is really good to know.
I am trying to understand if there is any algebraic structure, Borcherds or otherwise, that is represented on a low level truncation of the fundamental of $E_{11}$. Some text seemed to suggest this, but maybe not.
Shouldn’t it say ’0d supergravity’ (whatever that is) rather than
As U-duality group of 1d supergravity
Woops. Yes! Thanks for catching this silly typo. Fixed now.
added this pointer, which had been missing:
Used unicode subscripts for indices of exceptional Lie groups including title and links. When not linked, usual formulas are used. See discussion here. Links will be re-checked after all titles have been changed. (Removed two redirects for “E11” from the top and added one for “E11” at the bottom of the page.)
I saw your post on X. Do you have a preliminary write-up I could read that touches on the significance of this fact?
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).
Is this the same 528 as in ’528-toroidal T-duality’, here?
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:
$\mathrm{dim}\Big( \mathbb{R}^{1,10} \oplus \wedge^2 (\mathbb{R}^{1,10})^\ast \oplus \wedge^5 (\mathbb{R}^{1,10})^\ast \Big) \;\;\simeq\;\; \left(11 \atop 1\right) + \left(11 \atop 2\right) + \left(11 \atop 5\right) \;\; = \;\; 528$For this is it important to have a Lorentzian signature or does it work for more general signatures?
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 :-)
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