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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMar 28th 2014

started something at splitting principle

(wanted to do more, but need to interrupt now)

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMar 29th 2014

Added some lines on Examples but am running out of steam now.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeFeb 16th 2018

Is there any general discussion of splitting principles for twisted cohomology?

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 16th 2018

Something here perhaps.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeFeb 16th 2018
• (edited Feb 16th 2018)

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.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeFeb 20th 2018

I have touched the Idea-section here, trying to better bring out the main point.

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 20th 2018

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeFeb 20th 2018
• (edited Feb 20th 2018)

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! :-)

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeFeb 20th 2018
• (edited Feb 20th 2018)

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)

• CommentRowNumber10.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 20th 2018

You could try a bounty on MO. That seems to motivate some people.

• CommentRowNumber11.
• CommentAuthorDavidRoberts
• CommentTimeFeb 20th 2018

I do wonder whether Ganter’s categorical tori, which sit inside eg the String 2-groups, exhibit a form of the splitting principle

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeFeb 21st 2018
• (edited Feb 21st 2018)

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

1. $\mathfrak{m}2\mathfrak{brane}$ 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 $\mu_{M2} = \tfrac{i}{2} \overline{\psi}\Gamma_{a_1 a_2} \psi \wedge e^{a_1} \wedge e^{a_2}$

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.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeJan 3rd 2021