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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2015
    • (edited Jan 18th 2015)

    The note Bryant 05 uses the convention to call a differential 3-form σ\sigma on a 7-manifold for which at each point there is a GL(7)GL(7)-transformation that identifies it with the fixed associative 3-form on 7\mathbb{R}^7 a “definite 3-form”.

    One may observe that in differential cohesion being a definite 3-form means that pulled back to the infinitesimal disk bundle and then regarded as a section of the Ω 3(𝔻 7)\mathbf{\Omega}^3(\mathbb{D}^7)-bundle associated to the frame bundle it gives a homotopy of the form

    * ϕ U Ω 3(𝔻 7) X σ Ω 3(𝔻 7)//GL(7) \array{ && \ast \\ & \nearrow & \downarrow^{\mathrlap{\phi}} \\ U &\swArrow & \mathbf{\Omega}^3(\mathbb{D}^7) \\ \downarrow && \downarrow \\ X &\stackrel{\sigma}{\longrightarrow} & \mathbf{\Omega}^3(\mathbb{D}^7)//GL(7) }

    for a given atlas UXU \to X. This homotopy is in componentes just the GL(7)GL(7)-valued function which transforms σ| U\sigma|_U pointwise into ϕ\phi.

    This is a condition that makes sense much more generally, and I am looking for a terminology for functions σ:XA\sigma : X \to A on VV-manifolds XX with values in some \infty-stack AA such that there is a homotopy of the form

    * ϕ U A(𝔻 7) X σ A(X)//GL(V) \array{ && \ast \\ & \nearrow & \downarrow^{\mathrlap{\phi}} \\ U &\swArrow & A(\mathbb{D}^7) \\ \downarrow && \downarrow \\ X &\stackrel{\sigma}{\longrightarrow} & A(X)//GL(V) }

    What would be a nice term for these? Would definite work in this generality? I am inclined to say “σ\sigma is definite on ϕ\phi”, but I am not decided yet.