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nLab > Latest Changes: splitting principle

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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+1D = 10+1, N=32N = \mathbf{32} super-translation Lie algebra equipped with one more generator c 3c_3 of degree 3, which trivializes the M2-brane cocycle . We may read this as saying that c 3c_3 is a 3-cocyle in μ M2\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

    Did you advance with #5?

    • 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 ℝ 10,1|32\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))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 21st 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. 𝔪2𝔟𝔯𝔞𝔫𝔢\mathfrak{m}2\mathfrak{brane} for the super Lie 3-algebra, which is the higher central extension of ℝ 10,1|32\mathbb{R}^{10,1\vert\mathbf{32}} by the M2-brane 4-cocyle μ M2=i2ψ¯Γ a 1a 2ψ∧e a 1∧e a 2\mu_{M2} = \tfrac{i}{2} \overline{\psi}\Gamma_{a_1 a_2} \psi \wedge e^{a_1} \wedge e^{a_2}

    2. T exc,sℝ 10,1|32T_{exc,s}\mathbb{R}^{10,1\vert \mathbf{32}} for the super Lie 1-algebra whose CE-algebra is the DF-algebra at parameter ss (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

    T exc,sℝ 10,1|32 ⟶comp 𝔪2𝔟𝔯𝔞𝔫𝔢 ↘ ↙ ℝ 10,1|32 \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 *comp^\ast along it injects the degree-3 generator c∈CE(𝔪2𝔟𝔯𝔞𝔫𝔢)c \in CE(\mathfrak{m}2\mathfrak{brane}) which witnesses the higher central extension, in that dc=μ M2d c = \mu_{M2}.

    Moreover, from the details of the construction it seems clear that at s=−6s = -6 the left hand T exc,sℝ 10,1|32T_{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 μ˜ M5\tilde \mu_{M5} on 𝔪2𝔟𝔯𝔞𝔫𝔢\mathfrak{m}2\mathfrak{brane}, which is also injected into the cohomology of T exc,sℝ 10,1|32T_{exc,s}\mathbb{R}^{10,1\vert\mathbf{32}} now, under comp *comp^\ast, and in the given applications one would really like to have that under comp *comp^\ast the μ˜ M5\tilde \mu_{M5}-twisted rational cohomology of 𝔪2𝔟𝔯𝔞𝔫𝔢\mathfrak{m}2\mathfrak{brane} injects into the comp *(μ˜ M5)comp^\ast(\tilde \mu_{M5})-twisted rational cohomology of T exc,sℝ 10,1|32T_{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

    added pointer to

    diff, v22, current

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