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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeAug 5th 2011

I added a Definition-section to AKSZ sigma-model with a bit of expanded discussion

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeNov 28th 2012

• Theodore Th. Voronov, Vector fields on mapping spaces and a converse to the AKSZ construction, arxiv/1211.6319
• CommentRowNumber3.
• CommentAuthorjim_stasheff
• CommentTimeNov 28th 2012
The action functional of AKSZ theory is that of ∞-Chern-Simons theory induced from the Chern-Simons element that correspondonds to the invariant polynomial ω. Details on this are at ∞-Chern-Simons theory – Examples – AKSZ theory.

That's a little misleading - AKSZ is much more general than that one example.
• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeNov 28th 2012

No, it’s not the one example of ordinary 3d Chern-Simons theory. The statement is that every AKSZ theory is an example of an $\infty$-Chern-Simons theory (as linked to in the entry). Indeed, the AKSZ theories are precisely those $\infty$-Chern-Simons theories that are induced form invariant polynomials on $L_\infty$-algebroids which are

1. binary

2. non-degenerate .

• CommentRowNumber5.
• CommentAuthorjim_stasheff
• CommentTimeNov 30th 2012
I'd not absorbed the magnificaent? generality of the name ∞-Chern-Simons theory. I still doubt that AKSZ requires invariant polynomials on L∞-algebroids. See Thedya Voronov's recent posting to the arXiv.
• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeNov 30th 2012

I still doubt that AKSZ requires invariant polynomials on L∞-algebroids.

That’s how its defined, by target spaces which are symplectic dg-manifolds. The symplectic form is equivalently an invariant polynomial on the dg-manifold regarded as an $L_\infty$-algebroid.

• CommentRowNumber7.
• CommentAuthorjim_stasheff
• CommentTimeDec 1st 2012
I see no mention of an invariant polynomial in e.g. Thedya Voronov's
Vector fields on mapping spaces...
Is it hiding there somewhere?
• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeDec 2nd 2012
• (edited Dec 2nd 2012)

Is it hiding there somewhere?

It’s the object denoted $\omega$, as usual.

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeSep 27th 2018

This recent paper

• Hyungrok Kim, AKSZ-type Topological Quantum Field Theories and Rational Homotopy Theory_, ( arXiv:1809.09583)

makes a claim as to the close relation between its two parts. Is this something to mention at AKSZ sigma-model?

Does it suggest something AKSZ-like going on in your recent papers on rational homotopy theory?

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeSep 27th 2018
• (edited Sep 27th 2018)

Thanks for the reminder. I had seen this when it came out and had wanted to email the author, but then forgot about it.

As we had shown in our article that he cites, AKSZ field theories are a special class of $L_\infty$ higher Chern-Simons theories (namely precisely those induced from binary invariant polynomials, in contrast to higher ary invariant polynomials, as e.g. for 7d Chern-Simons theory of String-connections).

However, the super-cocycles that control the fundamental super $p$-branes are not transgressive (do not correspond to invariant polynomials). The higher $L_\infty$-field theories which these induce are not of Chern-Simons type (hence not of AKSZ type) but of WZW type. That was the content of the last section of our Super Lie n-algebra extensions, higher WZW models, ….

Finally, that RHT is around once one is playing with dgc-algebras/$L_\infty$-algebras is a tautology ever since Quillen 69 introduced the subject. Just saying this in 2018 is not an achievement, the task is to put it some to some use in modern QFT.

• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeSep 27th 2018

I see, thanks.