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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 18th 2015
   • (edited Jan 18th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe note [Bryant 05](http://ncatlab.org/nlab/show/G2+manifold#Bryant05) uses the convention to call a differential 3-form $\sigma$ on a 7-manifold for which at each point there is a $GL(7)$-transformation that identifies it with the fixed associative 3-form on $\mathbb{R}^7$ 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 $\mathbf{\Omega}^3(\mathbb{D}^7)$-bundle associated to the frame bundle it gives a homotopy of the form $$ \array{ && \ast \\ & \nearrow & \downarrow^{\mathrlap{\phi}} \\ U &\swArrow & \mathbf{\Omega}^3(\mathbb{D}^7) \\ \downarrow && \downarrow \\ X &\stackrel{\sigma}{\longrightarrow} & \mathbf{\Omega}^3(\mathbb{D}^7)//GL(7) } $$ for a given atlas $U \to X$. This homotopy is in componentes just the $GL(7)$-valued function which transforms $\sigma|_U$ pointwise into $\phi$. This is a condition that makes sense much more generally, and I am looking for a terminology for functions $\sigma : X \to A$ on $V$-manifolds $X$ with values in some $\infty$-stack $A$ such that there is a homotopy of the form $$ \array{ && \ast \\ & \nearrow & \downarrow^{\mathrlap{\phi}} \\ U &\swArrow & A(\mathbb{D}^7) \\ \downarrow && \downarrow \\ X &\stackrel{\sigma}{\longrightarrow} & A(X)//GL(V) } $$ What would be a nice term for these? Would _definite_ work in this generality? I am inclined to say "$\sigma$ is definite on $\phi$", but I am not decided yet.

   The note Bryant 05 uses the convention to call a differential 3-form $\sigma$ on a 7-manifold for which at each point there is a $GL(7)$-transformation that identifies it with the fixed associative 3-form on $\mathbb{R}^7$ 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 $\mathbf{\Omega}^3(\mathbb{D}^7)$-bundle associated to the frame bundle it gives a homotopy of the form

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

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

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

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

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