Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 27th 2014

    I’ve made comments at Fivebrane, fivebrane 6-group, Fivebrane structure and differential fivebrane structure regarding the fact String(n)String(n) is not 6-connected for n6n \leq 6 (though trivially so for n=2n=2).

    There will be a bunch of interesting invariants for manifolds of dimensions 3-6. This includes, for instance, on 6-dimensional spin manifolds a non-torsion class corresponding to the obstruction to lifting the tangent bundle to the 6-connected group covering String(6)String(6) (here π 5(String(6))=\pi_5(String(6)) = \mathbb{Z}). This is the only non-torsion example, but should be given by a U(1)U(1)-4-gerbe, I think, which will have a 6-form curvature. Since H 6H^6 won’t be vanishing for oriented manifolds, one has some checking to do. For instance, the frame bundle of S 6S^6 lifted to a String(6)String(6)-bundle (I plan to write a paper on this 2-bundle) will not lift to a Fivebrane(6)Fivebrane(6)-bundle, because it won’t even lift to a String(6)˜\widetilde{String(6)}-bundle (i.e. the 6-connected cover of String(6)String(6)), since the transition function S 5BString(6)S^5 \to BString(6) is the generator. Thus the 6-form curvature of this 4-gerbe should be the volume form on the 6-sphere.

    Another point that occurs to me is that there are two copies of \mathbb{Z} to kill off in String(8)String(8) to get Fivebrane(8)Fivebrane(8), so one gets a U(1)×U(1)U(1)\times U(1) higher gerbe. I suspect this larger π 7\pi_7 is why there are so many more exotic spheres in dimension 15 than in neighbouring dimensions (16256 vs 2 in d=14 and 16); it’s certainly the case for exotic spheres in dimension 7 that π 3(SO(4))=×\pi_3(SO(4)) = \mathbb{Z}\times \mathbb{Z} helps.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2014

    Thanks for adding that! Long overdue.

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 27th 2014
    • (edited Nov 27th 2014)

    Regarding my last comment regarding exotic 15-spheres, lemma 2 in http://www.maths.ed.ac.uk/~aar/papers/shimada.pdf is relevant. And I now believe the 992 exotic spheres in dimension 11 are due, at least in part, to π 5(String(6))=\pi_5(String(6)) = \mathbb{Z}. This is pretty cool, if this is really higher bundles ’detecting’ exotic spheres.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2014
    • (edited Nov 27th 2014)

    This sounds interesting. Could you elaborate just a tad more on this comment? There is that article by Witten arguing about gravitational instantons via exotic spheres and it was somehow related to the ninebrane story. But I need to remind myself. Not now, though, now I need to rush off.

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 28th 2014

    Hmm, there’s a recent question on physicsoverflow that asks whether these ideas are still up for grabs.

    If we consider S 3S^3-fibre bundles over S 4S^4 with structure group SO(4)SO(4), they are classified by [S 3,SO(4)]= 2[S^3,SO(4)]=\mathbb{Z}^2. Given (n,m)(n,m) describing such a map, if n+m=1n+m=1 then the total space of the bundle is homeomorphic to a 7-sphere, and p 12=nm\frac{p_1}{2} = n-m. The differentiable structure is only standard for (nm) 21=0(mod7)(n-m)^2 - 1 = 0 (mod 7). I wish I knew what that mod 7 was meant to be. It comes in Milnor’s proof because of the factor of 7 in Hirzebruch’s signature theorem for 8-manifolds, but that’s not immensely satisfying.

    I don’t know how to describe the exotic spheres in dimension 11 yet.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeNov 28th 2014

    Ah, thanks! If you now add a pointer to the literature to this paragraph, then that’s something to be copied into some nnLab entry.