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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2014
    • (edited Dec 21st 2014)

    I have been added a first approximation to an Idea-section to torsion of a G-structure -

    Have also added a pointer to Lott 90 and started a stub torsion constraints in supergravity, for the moment only to record some references.

    Have also further touched related entries such as torsion of a Cartan connection.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2014
    • (edited Dec 21st 2014)

    Let me try to restate synthetically the characterization of integrable/torsion-free GG-structure the way Lott says it on p. 4 of arXiv:0108125:

    Fix a model space 𝔸 n\mathbb{A}^n with first order infinitesimal disk around the origin denoted 𝔻 n𝔸 n\mathbb{D}^n \hookrightarrow \mathbb{A}^n. Assume that the canonical 𝔻 n\mathbb{D}^n-bundle of 𝔸 n\mathbb{A}^n is trivial.

    Write GL(n):=Aut(𝔻 n)GL(n) := \mathbf{Aut}(\mathbb{D}^n).

    Then (as we discussed in another thread recently) if any XX has a formally étale cover by 𝔸 n\mathbb{A}^n-s then it carries a canonical frame bundle, modulated by some map τ X:XBGL(n)\tau_X : X \longrightarrow \mathbf{B} GL(n).

    Now fix any group GG and a map GStruc:BGBGL(n)G\mathbf{Struc} : \mathbf{B}G \longrightarrow \mathbf{B}GL(n).

    Then given an 𝔸 n\mathbb{A}^n-manifold XX as above, a GG-structure on XX is equivalently a morphism

    c:τ XGStruc(X)\mathbf{c} : \tau_X \longrightarrow G\mathbf{Struc}(X)

    in the slice over BGL(n)\mathbf{B}GL(n).

    So far this is clear. Now regarding how to say synthetically that this GG-structure is integrable/torsion free.

    To that end, fix a GG-structure on the model space

    c 0:τ 𝔸 nGStruc\mathbf{c}_0 : \tau_{\mathbb{A}^n} \longrightarrow G\mathbf{Struc}.

    Now I suppose we should say: the GG-structure c\mathbf{c} on XX is integrable/torsion-free if there exists a formally étale cover i𝔸 nX\coprod_i \mathbb{A}^n \longrightarrow X such that this extends to a morphism of GG-structures, i.e. a morphism

    ic 0c \coprod_i \mathbf{c}_0 \longrightarrow \mathbf{c}

    in the slice over GStrucG\mathbf{Struc} (which itself is in the slice over BGL(n)\mathbf{B}GL(n)).

    So this just expresses that along each patch inclusion 𝔸 i nX\mathbb{A}^n_i \longrightarrow X the GG-structure on XX restricts to the fixed one on the model space, up to equivalence

    τ 𝔸 i n τ X c 0 c GStruc \array{ \tau_{\mathbb{A}^n_i} && \longrightarrow && \tau_X \\ & {}_{\mathllap{\mathbf{c}_0}}\searrow &\swArrow_\simeq& \swarrow_{\mathrlap{\mathbf{c}}} \\ && G \mathbf{Struc} }
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