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 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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