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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 30th 2014
    • (edited Dec 30th 2014)

    I have fine-tuned the definition of manifolds in differential cohesion a bit more (here).

    I think now a good axiomatization is like this:

    Let VV be a differentially cohesive homotopy type equipped with a framing. Then a VV-manifold is an object XX such that there exists a VV-cover, namely a correspondence

    U V X \array{ && U \\ & \swarrow && \searrow \\ V && && X }

    such that both morphisms are formally étale morphisms and such that UXU \to X is in addition an effective epimorphism.

    This style of definition very naturally leads to a good concept of integrable G-structures (in differential cohesion).

    What I find particularly charming is that if we take such a correspondence and “prequantize” it in the sense of prequantized Lagrangian correspondences, i.e. if we pick a differential coefficient object B𝔾 conn\mathbf{B}\mathbb{G}_{conn} and complete to a correspondence in the slice

    U V X L WZW L WZW X B𝔾 conn \array{ && U \\ & \swarrow && \searrow \\ V && \swArrow_{\simeq} && X \\ & {}_{\mathllap{\mathbf{L}_{WZW}}}\searrow && \swarrow_{\mathrlap{\mathbf{L}^X_{WZW}}} \\ && \mathbf{B}\mathbb{G}_{conn} }

    then this captures precisely the globalization problem of WZW terms that we have been discussing elsewhere: on the left we pick a WZW term on the model space, and completing the diagram to the right means finding a globalization of this term to XX that locally restricts to the canonical term, up to equivalence.

    I think I have now full proof of one direction of the corresponding obstruction (details in this pdf):

    Theorem Given VV a differentially cohesive \infty-group, XX a VV-manifold, and L WZW\mathbf{L}_{WZW} an equivariant WZW-term on VV, then an obstruction to L WZW X\mathbf{L}_{WZW}^X to exist as above is the existence of an integrable QuantMorph(L WZW 𝔻 V)QuantMorph(\mathbf{L}_{WZW}^{\mathbb{D}^V})-structure on XX,

    (i.e. a lift of the structure group of the frame bundle to the quantomorphism n-group of the restriction L WZW 𝔻 e V\mathbf{L}_{WZW}^{\mathbb{D}^V_e} of the WZW term to the infinitesimal neighbourhood of the neutral element in VV, such that this lift restricts over a VV-cover UU to the canonical QuantMorph(L WZW 𝔻 e V)QuantMorph(\mathbf{L}_{WZW}^{\mathbb{D}^V_e})-structure on VV).

    I still need to prove that this is not just a necessary but also a sufficient condition. This is harder…

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeFeb 12th 2015
    • (edited Feb 12th 2015)

    With the definition of VV-manifolds and their frame bundles in differential cohesion here and here, it is immediate that forming frame bundles extends to an \infty-functor τ ()\tau_{(-)} from the sub-\infty-category of the \infty-topos H\mathbf{H} on the VV-manifold with local diffeos between them, to the slice \infty-topos H /BGL(V)\mathbf{H}_{/\mathbf{B}GL(V)}.

    But I need this functor internally: for XX a VV-manifold I need a morphism in H\mathbf{H} of the form

    Aut(X)BGL(V)Aut(τ X) \mathbf{Aut}(X) \longrightarrow \underset{\mathbf{B}GL(V)}{\prod} \mathbf{Aut}(\tau_X)

    where on the right Aut(τ X)H /BGL(V)\mathbf{Aut}(\tau_X) \in \mathbf{H}_{/\mathbf{B}GL(V)} denotes the internal automorphism \infty-group of τ XH /BGL(V)\tau_X \in \mathbf{H}_{/\mathbf{B}GL(V)}.

    First I thought it’s obvious, but now I seem to be stuck…

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