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started something at splitting principle
(wanted to do more, but need to interrupt now)
Added some lines on Examples but am running out of steam now.
Is there any general discussion of splitting principles for twisted cohomology?
Something here perhaps.
Thanks. I am asking because, following a suggestion of Hisham’s, it occurs to me that the general idea of the splitting principle is the right perspective to understand how the M-theory super Lie algebra arises from the supergravity Lie 3-algebra:
The CE-algebra of the latter is that of the $D = 10+1$, $N = \mathbf{32}$ super-translation Lie algebra equipped with one more generator $c_3$ of degree 3, which trivializes the M2-brane cocycle . We may read this as saying that $c_3$ is a 3-cocyle in $\mu_{M2}$-twisted cohomology.
Now the “M-theory super Lie algebra” is the answer to the question: Is there, rationally, a twisted toroidal geometry such that the twisted higher-degree cohomology of the supergravity Lie 3-algebra injects into it? And there is.
Except for all the twists flying around, this is in the spirit of the splitting principle in its broad form: Given a classifying space with some higher degree cohomology classes, find a torus classifying space such that these higher classes inject into its cohomology.
I am still not exactly sure where to take this splitting principle-perspective on the M-theory super Lie algebra, but I have a strong feeling that this is finally the right abstract perspective to understand what it is.
I have touched the Idea-section here, trying to better bring out the main point.
Did you advance with #5?
No, I am stuck on this, yet I keep feeling that it’s the right idea.
Here I was about to chat with somebody else about the idea, and in pointing to the entry, I realized that the idea-section could be improved.
If I had the answer to that H-cohomology issue (here), I would get a statement at least close to what I need. But I am stuck on that, too! :-)
It’s getting all the more interesting, in that just three weeks back a new, alternative “splitting principle” of the M5-7-cocyle-twisted cohomology on the M2-brane extension of $\mathbb{R}^{10,1\vert\mathbf{32}}$ was found (not presented in this perspective, of course) in Ravera 18a, with the most curious property that now the super Lie 1-algebra is non-abelian, in fact a super-extension of $Lie(Spin(10,1))$.
This is exactly what I want to see appear in section 2.4 of From higher to exceptional geometry (schreiber)
You could try a bounty on MO. That seems to motivate some people.
I do wonder whether Ganter’s categorical tori, which sit inside eg the String 2-groups, exhibit a form of the splitting principle
That might be interesting.
But, just to highlight, what I am after in #5 above here is crucially not a variant of the splitting principle where we ask whether it generalizes tori to higher tori.
Instead, I am trying to see if the role of approximation of ordinary tori, hence approximation by homotopy 1-types, is a way to understand conceptually what the DF-algebra is doing.
The logic in the supergravity literature going back to D’Auria-Fré 82, section 6 is as follows:
First they show that 11d SuGra is governed by the supergravity Lie 3-algebra, only that that’s not what they really say, since they have no concept of higher (super-)Lie algebra. Accordingly, next they insist that they must force it to become an ordinary super Lie 1-algebra.
If we write
$\mathfrak{m}2\mathfrak{brane}$ for the super Lie 3-algebra, which is the higher central extension of $\mathbb{R}^{10,1\vert\mathbf{32}}$ by the M2-brane 4-cocyle $\mu_{M2} = \tfrac{i}{2} \overline{\psi}\Gamma_{a_1 a_2} \psi \wedge e^{a_1} \wedge e^{a_2}$
$T_{exc,s}\mathbb{R}^{10,1\vert \mathbf{32}}$ for the super Lie 1-algebra whose CE-algebra is the DF-algebra at parameter $s$ (Bandos-Azcarraga-Izquierdo-Picon-Varela 04) (a fermionic extension of the “M-theory super Lie algebra”, but introduced long before the latter got a name)
then what they show is that there is a homomorphism
$\array{ T_{exc,s}\mathbb{R}^{10,1\vert\mathbf{32}} && \overset{comp}{\longrightarrow} && \mathfrak{m}2\mathfrak{brane} \\ & \searrow && \swarrow \\ && \mathbb{R}^{10,1\vert\mathbf{32}} }$such that pullback $comp^\ast$ along it injects the degree-3 generator $c \in CE(\mathfrak{m}2\mathfrak{brane})$ which witnesses the higher central extension, in that $d c = \mu_{M2}$.
Moreover, from the details of the construction it seems clear that at $s = -6$ the left hand $T_{exc,s}\mathbb{R}^{10,1\vert\mathbf{32}}$ is the smallest super Lie 1-algebra that has this property, though I don’t have a rigorous proof for this.
So, you see, the key point here is that a super Lie 1-algebra, hence from the point of view of rational super homotopy theory a super torus, “approximates” a higher super homotopy type, where the nature of “approximation” might remind one of the splitting principle.
Concretely, there is a 7-cocycle $\tilde \mu_{M5}$ on $\mathfrak{m}2\mathfrak{brane}$, which is also injected into the cohomology of $T_{exc,s}\mathbb{R}^{10,1\vert\mathbf{32}}$ now, under $comp^\ast$, and in the given applications one would really like to have that under $comp^\ast$ the $\tilde \mu_{M5}$-twisted rational cohomology of $\mathfrak{m}2\mathfrak{brane}$ injects into the $comp^\ast(\tilde \mu_{M5})$-twisted rational cohomology of $T_{exc,s}\mathbb{R}^{10,1\vert \mathbf{32}}$.
This really makes the analogy to the standard splitting principle clear, I think. Still, it’s all a bit different, due to the twists, but mostly due to an overall shift of degree as compared to the standard story.
added pointer to
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