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Given a spectral triple (A,H,D) in the sense of Connes, what would be the right notion of a fiber bundle or a principal fiber bundle on it? An example of this type is the Connes' cosphere algebra S*A, which is the noncommutative analogue of the cosphere bundle of a Riemannian manifold. It is defined as the image in the Calkin algebra Q(H) of the C*-algebra generated by the compact opertators in H and all dialations of A by the continuous 1-parameter family of unitaries generated by the the derivation [|D|,.] on B(H). Connes showed that when the spectral triple is the canonical Dirac triple of a compact spin^c Riemannian manifold, the cosphere algebra is the C*-algebra of continuous functions on the cosphere bundle of the manifold.

The question is: can we put this into a more general framework by finding the right notion of a fiber bundle on a spectral triple?

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See the paper arxiv.org/abs/math/0701033 by Paul Baum et al. I haven't looked into it but I'm sure there is something relevant there for you. –  MTS Apr 2 '10 at 22:32
    
I have seen the paper. It is very interesting, but it studies the commutative case. At least it is not obvious to me how one could use their idea to find something for noncommutative cases, especially when a spectral triple is around. –  Kamran Reihani Apr 12 '10 at 19:10
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4 Answers

One can consider torsors (principal bundles) within any sufficiently nice category and with respect to descent with respect to given Grothendieck topology and given fibered category over the ground site. See for example

  • Tomasz Brzeziński, On synthetic interpretation of quantum principal bundles, AJSE D - Mathematics 35(1D): 13-27, 2010 arxiv:0912.0213.

where in the main motivational part the codomain fibration and regular epimorphism topology are implicitly used. For the fibered category of modules (viewed as quasicoherent sheaves) over noncommutative rings, the faithfully flat Hopf-Galois extensions are the answer provided we accept Hopf coactions as dual representations of group actions. The tensor product is not a monoidal product in the category of associative algebras so this is a bit of a problem. Next thing is that one needs to consider nonaffine objects, if one is in algebraic framework, what can allow for more general concept of noncommutative principal bundles over the covers by noncommutative localizations which are analogues of covers in a Grothendieck topology. This kind of noncommutative principal bundles were skecthed in my articles

  • Z. Škoda, Localizations for construction of quantum coset spaces, math.QA/0301090, Banach Center Publ. 61, pp. 265--298, Warszawa 2003;

  • Z. Škoda, Coherent states for Hopf algebras, Letters in Mathematical Physics 81, N.1, pp. 1-17, July 2007. (earlier arXiv version: math.QA/0303357),

based on general picture of actions in noncommutative algebraic geometry as explained in the newer survey

  • Z. Škoda, Some equivariant constructions in noncommutative algebraic geometry, Georgian Mathematical Journal 16 (2009), No. 1, 183--202, arXiv:0811.4770.

See also the nlab:noncommutative principal bundle. I will hopefully release 2 more articles in this direction within next month or two.

Yes, for the spaces of sections of associated bundles with structure Hopf algebra one uses the cotensor product construction in the affine case; this is well known in the literature; the recipe can also be globalized by gluing along localizations. However this does not give the total spaces of associated bundles (in the category of noncommutative spaces) in satisfactory way in general, but only the spaces of sections.

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Non conmutative geometry makes heavy use of vector bundles formulated as finite projective modules over the corresponding non commutative algebra. Thus one should expect that any extension of NCG to principal fiber bundles needs to be consistent with this identification.

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The question is about principal bundles (and associated fibre bundles, not only vector bundles). They should be in the same category, plus having sort of G-action. Thus this is not an answer in the wanted generality. –  Zoran Skoda Jun 29 '10 at 11:25
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It might not be the context you want (although if you think about this, it might be very close to what you want) but for some people a non-commutative bundle is a Banach bundle with non-commutative fibres. unlike a fibre bundle, the fibres are not necessarily isomorphic. Also bundles related to categories of Morita equivalence bimodules might be related at to the context I'm referring to because Connes' differential calculus is built into Morita equiv bimodules. The latter is a way to generalise the "tangent bundle" over the space described by A. I'm not an expert. Anyway, this approach seems to be working.

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It's a matter of opinion whether this is really the "right" notion of fibre bundle on a spectral triple but it is "a" notion. One has to relate the geometrical data in the Dirac operator to a generalised connection on the bundle. Is this close to your thoughts? –  Rachel Dec 19 '12 at 11:37
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I guess that a noncommutative principal fiber bundle is a Hopf-Galois extension. See e.g. this entry of the nLab. There are many references on this subject now... I particularly like the work of Thomas Aubriot (see here for a nice presentation), who obtained some results related to (but weaker than) the ones of Victor Ostrik in a more "down-to-earth" approach.

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