I have a question about the use of the term “connection” on this page, but first I think that here $\eta$ should be valued in $\phi^*T^*X,$ right?

About the use of the term “connection” (I found this paper https://arxiv.org/pdf/1011.4735.pdf). I’ve been thinking about what it should mean, and my feeling was that a map $(\phi,\eta)$ should define a connection in the case that $\phi_*(V)=\pi(\eta(V),\cdot),$ ie. it satisfies the natural compatibility condition with respect to the anchor map of the Lie algebroid. I think in such a case you get an Ehresmann connection on $\phi^*G,$ where $G\rightrightarrows X$ is a groupoid integrating the Poisson structure, which we are thinking of as a principal groupoid bundle for $G$ itself.

This is a special case of the definition given in the paper, but with the more general definition, do you get an Ehresmann connection on some principal $G$-bundle over $\Sigma?$

]]>In a logical way it is more consistent the way you done it – because the version with Greek letter had a hyphen, but somehow this easier no-hyphen way got statistically prevalent in present day references with word-spelled form of sigma. Don’t worry.

]]>Hm, okay, thanks. I am always unsure about how to typeset such compunds. Maybe I should change that throughout….

]]>Sigma model is much more common spelling then sigma-model, as far as the main contemporary references and google search show, as well as scholarpedia and wikipedia entries. Of course, if spelled out the Greek letter than it has a dash.

]]>I have spelled out the definition of the action functional of the Poisson sigma-model

]]>I think this is true, MK was imagining diagram expansion for a particular field theory, though his purpose was deformation quantization, hence maybe did not develop full theory of the model, which had for him then just a heuristic purpose.

]]>Thanks!

By the way, what's the full story: Kontsevich didn't mention the Poisson sigma-model in his work, first. Then Cattaneo-Felder guessed that this was really where he got his diagrammatics from, secretly. I have been told that after they figured this out they asked him if he obtained his idea this way, without saying so, and allegedly he just said something like "Sure."

Do you know more?

]]>In related entry AKSZ model I have changed the link to action functional which actually was directing to path integral to the true link action functional, which itself has a paragraph on path integral containing a link path integral. I added some references to Poisson sigma model.

]]>stub for Poisson sigma-model. Needs references.

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