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  1. Given a Lie group GG with Lie algebra 𝔤\mathfrak{g}, one can think of a 𝔤\mathfrak{g}-valued 1-form ω\omega on a differential manifold MM as a connection form on the trivial principal GG-bundle G×MMG\times M\to M. The curvature of ω\omega is a 𝔤\mathfrak{g}-valued 2-form on MM, and ω\omega is flat precisely when its curvature vanishes. These are in a sense the extreme cases: on the one side the curvature 2-form takes values in the whole of 𝔤\mathfrak{g}, on the other side the curvature takes values in the 00 Lie subalgebra of 𝔤\mathfrak{g}. I was therefore wondering whether given a Lie subalgebra 𝔥\mathfrak{h} of 𝔤\mathfrak{g} there is a meaningful notion of connection form ω\omega with 𝔥\mathfrak{h}-constrained curvature, i.e. whose curvature 2-form is 𝔥\mathfrak{h}-valued. This should be (hopefully) related to the homogeneous space G/HG/H, where HH is the Lie subgroup of GG corresponding to the Lie algebra 𝔥\mathfrak{h}.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 23rd 2011

    What you say reminds me a bit of Cartan connections.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 23rd 2011

    Have you read John Baez on torsoroids?

    • CommentRowNumber4.
    • CommentAuthordomenico_fiorenza
    • CommentTimeDec 23rd 2011
    • (edited Dec 23rd 2011)

    @David: I wasn’t aware of that. Thanks, that really seems something I have to investigate better.

    @Urs: I’ll have to look into that, too, thanks. What I’m after, as you can have imagined, is a Lie algebra valued forms local description of G/HG/H-bundles induced by a G-princiapal bundle which parallels our local description of principal GG-bundles in terms of 𝔤\mathfrak{g}-valued differential forms on Δ n×U\Delta^n\times U which are flat in the simplicial direction. I that can be done, then it would be another step in reproducing Brylinski-McLaughlin construction of differential characteristic classes via maps from Δ n×U\Delta^n\times U into homogeneous spaces in our Lie algebra integration language.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 23rd 2011
    • (edited Dec 23rd 2011)

    It is of course possible to consider the sub-simplicial presheaf of exp(𝔤) conn\exp(\mathfrak{g})_{conn} for which the curvature forms satisfy an extra constraint. But now I am not sure what you have in mind with this. To me the problem we’d have to solve is how to get a resolution of BG\mathbf{B}G by a simplicial object of homology-chains in G/HG/H.

    • CommentRowNumber6.
    • CommentAuthorjim_stasheff
    • CommentTimeDec 23rd 2011
    @Urs: how to get a resolution of BG by a simplicial object of homology-chains in G/H.

    What kind of resolution?
    Do you have in mind H --> G --> G/H --> BH --> BG ??
  2. @Urs: what I have in mind is not a subsheaf of exp(𝔤) connexp(\mathfrak{g})_{conn} but a supersheaf: in exp(𝔤) connexp(\mathfrak{g})_{conn} we impose curvature to vanish in the vertical direction, and this is the origin of the topological constrain. Namely, to give a flat 𝔤\mathfrak{g}-connection on Δ n\Delta^n is the same thing as giving a based map from Δ n\Delta^n to GG. This means that if we are given a flat 𝔤\mathfrak{g}-connection on Δ n\partial \Delta^n, the obstruction to extending it to a flat 𝔤\mathfrak{g}-connection on Δ n\Delta^n is a topological one: it is an element in π n(G,e)\pi_n(G,e). Yet, assume we weaken our request and ask to extend the flat 𝔤\mathfrak{g}-connection on Δ n\partial \Delta^n to an 𝔥\mathfrak{h}-curvature constrained connection on Δ n\Delta^n. Maybe this can be done and if we are lucky the obstruction to this is an element in π n(G/H,[e])\pi_n(G/H,[e]). Surely thsi would not solve the issue with Pontryagin classes (since there is still a finite group there whicih is not seen by these Lie theoretic construtions), but this could hopefully handle Chern classes for princiapal U(n)U(n)-bundles (the Stiefel manifolds U(n)/U(k)U(n)/U(k) are connected-to-the-point-one-is-interested-in).

    @jim: welcome in this discussion: I really hoped you would have joined here

    • CommentRowNumber8.
    • CommentAuthorjim_stasheff
    • CommentTimeDec 24th 2011
    @Domenico:don't know if it's relevant, but just before Serre's thesis diverted attention from de Rham cohomology,
    Cartan et Cie were working on bundles with G/H fibers
  3. I got an excellent answer from Robert Bryant here

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeDec 28th 2011
    • (edited Dec 28th 2011)

    Hi Domenico,

    sorry for being silent for so long. I was utterly exhausted after the last two months and needed a good break in the last week, just family, and offline :-) Hope you all had good Christmas holidays.

    But now some thoughts.

    First: it seems to me that for the construction that we need, the question of how to most suitably parameterize maps into a coset is secondary. The primary problem is to exhibit a useful resolution of BG\mathbf{B}G in terms of homology cycles in cosets, whichever way these are described.

    Since we are discussing this out in the open anyway, let me briefly recall for other readers what this is all about, since otherwise this discussion must seem mysterious to everybody else:

    Context.

    In

    • Brylinski, McLaughlin, Čech cocycles for characteristic classes (EUCLID)

    the authors constructed what the title announces: Cech cohomology representatives of all classical characteristic classes.

    In

    • Fiorenza, Schreiber, Stasheff, Čech cocycles for differential characteristic classes (nLab)

    we observed that for degree-4 classes this construction naturally refines to a morphism of smooth higher moduli stacks BG connB 3U(1) conn\mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn} from the moduli stack of GG-principal connections to that of circle 3-connections.

    More generally, for any degree (n+1)(n+1)-class, this refinement generalizes to a “higher Chern-Weil homomorphism” of this form where, however, GG is now required to be an n-group that extends the original Lie group.

    For instance this way the second Pontryagin class is obtained not as a morphism on the moduli stack of Spin\mathrm{Spin}-connections, but on that of String-connections 16p^ 2:BString connB 7U(1) conn\frac{1}{6} \hat \mathbf{p}_2 : \mathbf{B} String_{conn} \to \mathbf{B}^7 U(1)_{conn}.

    Therefore there is the evident

    Open question

    Can the construction of Brylisnki-MacLaughlin be refined to a morphism of smooth higher moduli stacks BG connB nU(1) conn\mathbf{B}G_{conn} \to \mathbf{B}^{n} U(1)_{conn} with GG a Lie group also for the case of higher characteristic classes?

    Or in other words, can we “push down” the universal construction of Fiorenza-Schreiber-Stasheff to lower connected covers? For instance, can we find the extension p^ 2\hat \mathbf{p}_2 in

    BString conn 16p^ 2 B 7U(1) conn 6 BSpin conn p^ 2 B 7U(1) conn. \array{ \mathbf{B} String_{conn} &\stackrel{\frac{1}{6}\hat \mathbf{p}_2}{\to}& \mathbf{B}^7 U(1)_{conn} \\ \downarrow && \downarrow_{\cdot 6} \\ \mathbf{B} Spin_{conn} &\stackrel{\hat \mathbf{p}_2}{\to}& \mathbf{B}^7 U(1)_{conn} } \,.

    ??

    (We have the corresponding extension on the level of topological universal characteristic maps. The task is here to first refine to smooth higher stacks, and then further to their differential refinement.)

    I want to state an attempt to answer this question. But I’ll do so in the next comment.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeDec 28th 2011
    • (edited Dec 28th 2011)

    Hi Domenico,

    let me try to play around with a suggestion for how to proceed.

    Let G=Spin(n)G = Spin(n). We need a simplicial presheaf resolution QG ×=:BG chQ \stackrel{\simeq}{\to} G^{\times \bullet} =: \mathbf{B}G_{ch} that supports that BMcL-construction not just for k=1k = 1 but also for higher kk.

    For k=1k = 1 we used

    Q:=cosk 3exp(𝔰𝔬(n)) Q := \mathbf{cosk}_3 \exp(\mathfrak{so}(n))

    which we may think of as “cosk 3Sing based,smooth(G)\mathbf{cosk}_3 Sing_{based, smooth}(G)”, the simplicial complex of smooth simplices in GG that are based at the neutral element. The coskeleton construction at some point does away with the singular simplices and fills up everything with unique abstract simplices.

    Our task is: replace the blunt coskeleton construction by a construction that instead fills in all higher simplicial spheres by homology chains in the coset Spin(n)/Spin(q)Spin(n)/Spin(q).

    There is an obvious guess for how to do this:

    Let QQ be given as follows

    • a single object;

    • 1-morphisms are smooth based paths in Spin(n)Spin(n);

    • 2-morphisms are smooth based triangles in Spin(n)Spin(n);

    now:

    • 3-morphisms are smooth based 3-chains over \mathbb{R} in Spin(n)/Spin(q)Spin(n)/Spin(q), whose faces have representatives by maps Δ 2Spin(n)\Delta^2 \to Spin(n);

    • 4-morphisms are smooth based 4-chains over \mathbb{R} whose faces are of the above form

    and so on until

    • (4k2)(4k-2)-morphisms are smooth based (4k2)(4k-2)-chains etc.

    and then complete by the coskeleton

    • there are unique higher cells filling any bounding ball.

    The evident projection QBG chQ \to \mathbf{B}G_{ch} should be a weak equivalence, since, by that table on p. 3 of BMcL, the integral homology of the coset space is all torsion in the relevant range, and hence vanishes over the reals.

    Finally, QQ should have a differential refinement Q connQ_{conn} and the BMcL-construction should extend to a morphism of simplicial presheaves p^ k:Q connB 4k1U(1) conn\hat \mathbf{p}_k : Q_{conn} \to \mathbf{B}^{4k-1} U(1)_{conn}. I’d hope.

    That would be my suggestion. I haven’t fully thought this through. I am saying this in order to indicate in which direction I think we should be pushing.

    [edit: after a bit of fiddling, I am not sure if I can make this work…]

  4. Hi Urs,

    thanks for this additions and suggestions. I’m now away for winter holidays and so I will be for other 4-5 days. After that I’ll come back to this. But as you see I’m giving a glance to what’s happening on nforum, so if this discussions evolves while I’m out I’ll nevertheless be able to follow it at least a bit