nForum - Discussion Feed (symplectic infinity-groupoid) 2023-06-04T18:17:08+00:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher Urs comments on "symplectic infinity-groupoid" (25666) https://nforum.ncatlab.org/discussion/3088/?Focus=25666#Comment_25666 2011-09-02T21:42:23+00:00 2023-06-04T18:17:08+00:00 Urs https://nforum.ncatlab.org/account/4/ I am starting an entry symplectic infinity-groupoid. This is still in the making. Currently there are two things: A little general indication of what this is supposed to be about; A proof of an ...

I am starting an entry symplectic infinity-groupoid.

This is still in the making. Currently there are two things:

1. A little general indication of what this is supposed to be about;

2. A proof of an assertion that serves to justify the whole concept.

Namely, the literature already knows the concept of a symplectic groupoid. This plays a big role in Weistein’s program and in particular in geometric quantization of symplectic groupoids, which induces, among other things, a notion of geometric quantization of Poisson manifolds.

As far as I am aware (though I might not have been following the latest developments here, would be grateful for comments) it is generally expected that symplectic groupoids are formally the Lie integration of Poisson Lie algebroids, but there is no proof or even formalization of this in the literature.

In the entry I indicate such a formalization and give the respective proof.

The idea is that this is a special case of the general machine of infinity-Chern-Weil theory:

namely: the symplectic form on a symplectic Lie $n$-algebroid such as the Poisson Lie algebroid is Lie theoretically an invariant polynomial. So the $\infty$-Chern-Weil homomorphism produces a corresponding morphism from the integrating smooth $\infty$-groupoid to de Rham coefficients. This is a differential form in the world of smooth $\infty$-groupoids.

The assertion is: this comes out right. Feed a Poisson Lie algebroid with its canonical invariant polynomial into $\infty$-Chern-Weil theory, out comes the “classical” symplectic Lie groupoid.

(I do this for the case that the Poisson manifold is in fact itself symplectic, which is the only case I remember having seen discussed in earlier literature. But I think I can generalize this easily.)

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