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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 3rd 2011
   • (edited Aug 3rd 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexWith Domenico Fiorenza and Chris Rogers we are beginning to bring parts of our notes on [[schreiber:infinity-Chern-Simons theory]] into shape in order to eventually publish them. We have dediced to split off the discussion of [[AKSZ theory]] as a separate writeup that just explains and then proves the following statement: **Proposition.** The action functional of the AKSZ [[sigma-model]] with [[target space]] a [[symplectic Lie n-algebroid]] $\mathfrak{P}$ is the $\infty$-Chern-Simons functional assigned by $\infty$-Chern-Weil theory to the canonical [[invariant polynomial]] $\omega$ on $\mathfrak{P}$. A version of the notes so far is here: [oo-Chern-Simons theory -- Examples -- AKSZ theory](http://ncatlab.org/schreiber/show/infinity-Chern-Simons+theory+--+examples#ASKZTheory).

   With Domenico Fiorenza and Chris Rogers we are beginning to bring parts of our notes on infinity-Chern-Simons theory (schreiber) into shape in order to eventually publish them.

   We have dediced to split off the discussion of AKSZ theory as a separate writeup that just explains and then proves the following statement:

   Proposition. The action functional of the AKSZ sigma-model with target space a symplectic Lie n-algebroid $\mathfrak{P}$ is the $\infty$-Chern-Simons functional assigned by $\infty$-Chern-Weil theory to the canonical invariant polynomial $\omega$ on $\mathfrak{P}$.

   A version of the notes so far is here:

   oo-Chern-Simons theory – Examples – AKSZ theory.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeAug 3rd 2011
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   Author: zskoda
   Format: MarkdownItextypo -- link on symplectic

   typo – link on symplectic

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeAug 3rd 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks! I have fixed it.

   Thanks! I have fixed it.

4. • CommentRowNumber4.
   • CommentAuthorjim_stasheff
   • CommentTimeAug 4th 2011
   • (edited Aug 4th 2011)
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   Author: jim_stasheff
   Format: Textwhich is THE canonical invariant polynomial ω on P? and AKSZ means the super version?

   which is THE canonical invariant polynomial ω on P?
   
   and AKSZ means the super version?
5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeAug 4th 2011
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   Author: Urs
   Format: MarkdownItex> which is THE canonical invariant polynomial ω on P? That $\mathfrak{P}$ is a [[symplectic Lie n-algebroid]] means that on its [[Chevalley-Eilenberg algebra]] there is a graded Poisson structure which is symplectic and comes from a symplectic 2-form on the corresponding "symplectic dg-manifold". Simply re-interpreting this in $L_\infty$-algebraic language one sees that this symplectic form $\omega$ is in $L_\infty$-language an invariant polynomial on an $L_\infty$-algebroid.

   which is THE canonical invariant polynomial ω on P?

   That $\mathfrak{P}$ is a symplectic Lie n-algebroid means that on its Chevalley-Eilenberg algebra there is a graded Poisson structure which is symplectic and comes from a symplectic 2-form on the corresponding “symplectic dg-manifold”. Simply re-interpreting this in $L_\infty$-algebraic language one sees that this symplectic form $\omega$ is in $L_\infty$-language an invariant polynomial on an $L_\infty$-algebroid.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeAug 5th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have now put an early version of our pdf writeup online. It can be found at [[schreiber:AKSZ Sigma-Models in Higher Chern-Weil Theory]]

   I have now put an early version of our pdf writeup online. It can be found at

   AKSZ Sigma-Models in Higher Chern-Weil Theory (schreiber)

7. • CommentRowNumber7.
   • CommentAuthorjim_stasheff
   • CommentTimeAug 5th 2011
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   Author: jim_stasheff
   Format: TextTHE canonical invariant polynomial ω on P so the symplectic form! why not say THE canonical invariant polynomial which is the symplectic formω on P

   THE canonical invariant polynomial ω on P
   
   so the symplectic form! why not say
   THE canonical invariant polynomial which is the symplectic formω on P
8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeAug 5th 2011
   • (edited Aug 5th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexJim, I think we do that in the entry and in the file. Here I was just dropping a brief sentence indicating what I have done elsewhere!

   Jim, I think we do that in the entry and in the file. Here I was just dropping a brief sentence indicating what I have done elsewhere!

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeAug 5th 2011
   • (edited Aug 5th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexJim, I have just uploaded a new version of our file, which is now fairly complete (albeit not fully polished) see the pdf-link [here](http://ncatlab.org/schreiber/show/AKSZ+Sigma-Models+in+Higher+Chern-Weil+Theory)

   Jim, I have just uploaded a new version of our file, which is now fairly complete (albeit not fully polished)

   see the pdf-link here

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeAug 6th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexi have made this a blog post on the $n$Café [here](http://golem.ph.utexas.edu/category/2011/08/aksz_model_in_higher_chernweil.html)

    i have made this a blog post on the $n$Café here