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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 $B String^{c_2}$ 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.
p. 3 ’Here we restrict attention to the subgroup … only for sake of exposition’ Is this because otherwise you’d have loads of $Sp(1)$ 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 $Spin^c$-structure, is called $String^{c_2}$-structure…
missing ’it’.
Is this because otherwise you’d have loads of $Sp(1)$ 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
$\mathbb{R}^{10,1\vert \mathbf{32}}_{exc,s} \sslash String^{c_2}_{\chi_8}$hence the “super-exceptional Poincaré $String^{c_2}$ group”, if you wish.
On this, the Hopf WZ term is going to be a universal 5-gerbe connection
$\mathbb{R}^{10,1\vert \mathbf{32}}_{exc,s} \sslash String^{c_2}_{\chi_8} \overset{\;\;\;\; \widetilde{\mathbf{\Gamma}}_7\;\;\;\;}{\longrightarrow} B^6 U(1)_{conn}$whose curvature 7-form is the M5 super-cocycle
$\mathbb{R}^{10,1\vert \mathbf{32}}_{exc,s} \sslash String^{c_2}_{\chi_8} \overset{h_3^{exc}\wedge \mu_{M2} + \mu_{M6}}{\longrightarrow} \mathbf{\Omega}^7_{cl}$from FSS19c
and whose integral class is the
$\mathbb{R}^{10,1\vert \mathbf{32}}_{exc,s} \sslash String^{c_2}_{\chi_8} \to B String^{c_2}_{\chi_8} \overset{ \widetilde \Gamma_7 \;\coloneqq\; H_3^{univ} \wedge (\widetilde \Gamma_4 + \tfrac{1}{2}p_1) + 2\Gamma_7 }{\longrightarrow} K(\mathbb{Z},7)$from FSS19b.
Something like this…
Typos
Thanks! Fixed now.
For what it’s worth, I think I’ve calculated that
$[S^5,B String^{c_2}(4)] \simeq (\mathbb{Z}/2)^3$so that if your M5-brane is topologically a sphere, there are eight possible “sectors” for $B String^{c_2}(4)$-bundles on it. Does this show up in the physics?
The relation to “the physics” is this:
1) in the bottom part of the big diagram the slicing over $B Spin(8)$ is what enforces the relation between the C-field and the field of gravity, which is the anomaly cancellation conditions, such as that $[G_4 +\tfrac{1}{4}p_1]$ 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 $B String^{c_2}(4)$ (for instance as an element in $[S^5, B String^{c_2}(4)]$) 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 $\Sigma$ but allow the target 8-manifold $X^8$ to vary, to which extent does the twisted String class on $\Sigma$ vary? That would be interesting to know!
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 $\Sigma$ but allow the target 8-manifold $X^8$ to vary, to which extent does the twisted String class on $\Sigma$ vary? That would be interesting to know!
Hmm, yes, that would be interesting. I guess you mean not just $X^8$ but the embedding $\phi\colon \Sigma \to \mathbb{R}^{2,1} \times X^8$? And maybe also $b$?
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 $(\mathbb{Z}/2)^3)$. 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.
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.
Isn’t the worldvolume of the M5-brane of dimension 6? Why the 5-sphere?
David R. is thinking of $\Sigma = \mathbb{R}^{0,1} \times S^5$.
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 $X^8$ to have nontrivial $\pi_5$, so as to wrap an M5-brane of this form around it, and better, have a factor of something like $\mathbb{R}^6 \setminus \{0\}$. But I know $X^8$ is usually something rather rich
On the contrary. The mathematical formulation allows to run without bothering about the folklore.
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
$S^3 \times S^3 \to Sp(2) \to S4$, $S^3 \to Sp(2) \to S^7$.
Feels like the second at least is in the vicinity.
Oh, but the second of these is the fiber of the projection map of the coset space realization $Sp(2)/Sp(1) \simeq S^7$; and the first composes that with the quaternionic Hopf fibration.
Well, I can mock something up and run it by you, to see if you think it would pass muster in arXiv:hep-th
Re #14, oh yes, of course.
Does that idea of Atiyah and Witten to look at M-theory on $G_2$-manifolds in relation to M-theory on the 8-manifold $\mathbf{H P}^2$, 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?
An insightful discussion of the relation between 7d $G_2$-structure and 8d $Spin(7)$-structure is in arXiv:1405.3698
[ I am on vacation with family for a few days, back next week ]
Thanks. Enjoy your break.
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]
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