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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 25th 2020
    • (edited Feb 25th 2020)

    Further towards homotopy-theoretic foundations for the M5-brane model in M-theory, we have a note observing emergence of twisted string structure (“non-abelian gerbe field”) on the M5-worldvolume, derived from twisted cohomotopy-theory, assuming Hypothesis H:

    This amounts to observing one big homotopy-commutative diagram, the outer part of which we had derived earlier. The observation that the inner part thus factors through BString c 2B String^{c_2} follows immediately. The article walks a spotlight through this big diagram, explaining all its constitutents.

    Comments are welcome. Please grab the latest version of the file from behind the above link.

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 25th 2020
    • (edited Feb 25th 2020)

    p. 3 ’Here we restrict attention to the subgroup … only for sake of exposition’ Is this because otherwise you’d have loads of Sp(1)Sp(1) factors all over, and these are making little difference to the mathematics?

    p. 6 The analogy between rational superspaces and topological spaces. What happens with the topology analogue of the 517-torus fibration? Do you need some form of supergeometry to generate it?

    Typos

    conisdering; Stringvstructure; quatization

    and

    In direct analogy with Spin cSpin^c-structure, is called String c 2String^{c_2}-structure…

    missing ’it’.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 25th 2020
    • (edited Feb 25th 2020)

    Is this because otherwise you’d have loads of Sp(1)Sp(1) factors all over, and these are making little difference to the mathematics?

    Yes, exactly

    p. 6 The analogy between rational superspaces and topological spaces. What happens with the topology analogue of the 517-torus fibration? Do you need some form of supergeometry to generate it?

    I think now the combination of the two will be

    exc,s 10,1|32String χ 8 c 2 \mathbb{R}^{10,1\vert \mathbf{32}}_{exc,s} \sslash String^{c_2}_{\chi_8}

    hence the “super-exceptional Poincaré String c 2String^{c_2} group”, if you wish.

    On this, the Hopf WZ term is going to be a universal 5-gerbe connection

    exc,s 10,1|32String χ 8 c 2Γ˜ 7B 6U(1) conn \mathbb{R}^{10,1\vert \mathbf{32}}_{exc,s} \sslash String^{c_2}_{\chi_8} \overset{\;\;\;\; \widetilde{\mathbf{\Gamma}}_7\;\;\;\;}{\longrightarrow} B^6 U(1)_{conn}

    whose curvature 7-form is the M5 super-cocycle

    exc,s 10,1|32String χ 8 c 2h 3 excμ M2+μ M6Ω cl 7 \mathbb{R}^{10,1\vert \mathbf{32}}_{exc,s} \sslash String^{c_2}_{\chi_8} \overset{h_3^{exc}\wedge \mu_{M2} + \mu_{M6}}{\longrightarrow} \mathbf{\Omega}^7_{cl}

    from FSS19c

    and whose integral class is the

    exc,s 10,1|32String χ 8 c 2BString χ 8 c 2Γ˜ 7H 3 univ(Γ˜ 4+12p 1)+2Γ 7K(,7) \mathbb{R}^{10,1\vert \mathbf{32}}_{exc,s} \sslash String^{c_2}_{\chi_8} \to B String^{c_2}_{\chi_8} \overset{ \widetilde \Gamma_7 \;\coloneqq\; H_3^{univ} \wedge (\widetilde \Gamma_4 + \tfrac{1}{2}p_1) + 2\Gamma_7 }{\longrightarrow} K(\mathbb{Z},7)

    from FSS19b.

    Something like this…

    Typos

    Thanks! Fixed now.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 25th 2020
    • (edited Feb 25th 2020)

    For what it’s worth, I think I’ve calculated that

    [S 5,BString c 2(4)](/2) 3 [S^5,B String^{c_2}(4)] \simeq (\mathbb{Z}/2)^3

    so that if your M5-brane is topologically a sphere, there are eight possible “sectors” for BString c 2(4)B String^{c_2}(4)-bundles on it. Does this show up in the physics?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeFeb 26th 2020
    • (edited Feb 26th 2020)

    The relation to “the physics” is this:

    1) in the bottom part of the big diagram the slicing over BSpin(8)B Spin(8) is what enforces the relation between the C-field and the field of gravity, which is the anomaly cancellation conditions, such as that [G 4+14p 1][G_4 +\tfrac{1}{4}p_1] is integral.

    2) the precomposition of that with the Borel-equivariant Hopf fibration encodes the relative shifted trivialization of the C-field on the M5-worldvolume

    The commutativity of the outer diagram follows, and hence the map to BString c 2(4)B String^{c_2}(4) (for instance as an element in [S 5,BString c 2(4)][S^5, B String^{c_2}(4)]) is determined by these two physics conditions: anomaly cancellation and relative trivialization of the C-field. These together serve to single out that twisted string class.

    So what you might want to check is this:

    If we fix the worldvolume topology Σ \Sigma but allow the target 8-manifold X 8X^8 to vary, to which extent does the twisted String class on Σ \Sigma vary? That would be interesting to know!

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 26th 2020

    anomaly cancellation and relative trivialization of the C-field. These together serve to single out that twisted string class.

    hmm, that’s not as deep as I was thinking. :-)

    If we fix the worldvolume topology Σ \Sigma but allow the target 8-manifold X 8X^8 to vary, to which extent does the twisted String class on Σ \Sigma vary? That would be interesting to know!

    Hmm, yes, that would be interesting. I guess you mean not just X 8X^8 but the embedding ϕ:Σ 2,1×X 8\phi\colon \Sigma \to \mathbb{R}^{2,1} \times X^8? And maybe also bb?

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 26th 2020

    hmm, that’s not as deep as I was thinking. :-)

    let me rephrase that. What I was thinking was that is in principle possible for someone to have done some kind of toy-model calculation for a spherical M5-brane on a specific target space, and applying the anomaly cancellation and shifted trivialisation conditions, found three independent configurations (corresponding to the three factors of (/2) 3)(\mathbb{Z}/2)^3). One might expect such configurations to have more concrete interpretations from a physics point of view. Or maybe I’m being too naive, and the sort of “realistic” target 8-manifolds don’t particularly support M5-branes of that shape.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeFeb 26th 2020

    it is in principle possible for someone to have done some kind of toy-model calculation

    Yes, since it hit the arXiv 11 hours ago.

    Be the first.

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 26th 2020

    Isn’t the worldvolume of the M5-brane of dimension 6? Why the 5-sphere?

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeFeb 26th 2020

    David R. is thinking of Σ= 0,1×S 5\Sigma = \mathbb{R}^{0,1} \times S^5.

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 26th 2020

    I would be happy to make some small contribution, but I fear that I would be missing any crucial physical that may rule out my simplistic approach. Also, I wouldn’t quite know how to put such an idea is the language that M-theorists would be used to. Also, having a sensible and motivated target space in which to place the spherical M5-brane, along the lines of something already in the literature. I guess at the least one would want X 8X^8 to have nontrivial π 5\pi_5, so as to wrap an M5-brane of this form around it, and better, have a factor of something like 6{0}\mathbb{R}^6 \setminus \{0\}. But I know X 8X^8 is usually something rather rich

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeFeb 26th 2020

    On the contrary. The mathematical formulation allows to run without bothering about the folklore.

    • CommentRowNumber13.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 26th 2020
    • (edited Feb 26th 2020)

    Just idling about, I came across Hitchin’s SL(2) over the octonions where he mentions (p. 13) a couple of fibrations arising for quaternionic reasons that I hadn’t seen yet

    S 3×S 3Sp(2)S4S^3 \times S^3 \to Sp(2) \to S4, S 3Sp(2)S 7S^3 \to Sp(2) \to S^7.

    Feels like the second at least is in the vicinity.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeFeb 26th 2020
    • (edited Feb 26th 2020)

    Oh, but the second of these is the fiber of the projection map of the coset space realization Sp(2)/Sp(1)S 7Sp(2)/Sp(1) \simeq S^7; and the first composes that with the quaternionic Hopf fibration.

    • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 26th 2020

    Well, I can mock something up and run it by you, to see if you think it would pass muster in arXiv:hep-th

    • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 27th 2020

    Re #14, oh yes, of course.

    • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 28th 2020

    Does that idea of Atiyah and Witten to look at M-theory on G 2G_2-manifolds in relation to M-theory on the 8-manifold HP 2\mathbf{H P}^2, in the section on confinement, point perhaps to a broader relationship between M-theory on G2-manifolds and M-theory on 8-manifolds, something that could plug into the work you’ve done here?

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeMar 2nd 2020

    An insightful discussion of the relation between 7d G 2G_2-structure and 8d Spin(7)Spin(7)-structure is in arXiv:1405.3698

    [ I am on vacation with family for a few days, back next week ]

    • CommentRowNumber19.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 2nd 2020

    Thanks. Enjoy your break.

    • CommentRowNumber20.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 21st 2020
    • (edited Mar 22nd 2020)

    For what it’s worth, what I mentioned in #15 is now available here: Topological sectors for heterotic M5-brane charges under Hypothesis H (unstable link, ok for a few months). Thanks to Urs for lots of discussion and useful references. [edit: fixed link, it broke when updating to an edited version]