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something like this:
let (π,Ο) be a symplectic Lie n-algebroid. Then by the discussion at symplectic infinity-groupoid the Lie integration
exp(π)is the corresponding higher symplectic Lie groupoid . Following Higher Chern-Weil Derivation of AKSZ Sigma-Models (schreiber) we obtain a canonically induced infinity-Chern-Weil homomorphism
exp(π)connexp(cs,Ο)βexp(bn+1β)connwhose holonomy is the AKSZ-action functional and whose underlying cocycle
exp(π)exp(Ο)βexp(bn+1β)classifies a higher central extension ^exp(π) given by the homotopy pullback
^exp(π)β*ββexp(π)exp(Ο)βexp(bn+1β).The underlying Lβ-algebroid ^π of this is the string-like higher extension
bnββ^πβπclassified by the cocycle Ο that Ο transgresses to.
The corresponding quantum algebra is the βirreducible and polarizedβ β-representation of this ^exp(π).
Accordingly, the corresponding AKSZ sigma-model with action functional being the image under [Ξ£,β] of exp(cs,Ο)
SAKSZ:Ο0[Ξ£.exp(π)conn]βconcΟ0[Ξ£,exp(bn+1β)conn]computes aspects of this quantization by the universal property of the β-limit, which gives the homotopy pullback
[Ξ£,^exp(π)]β*ββ[Ξ£,exp(π)][Ξ£,exp(Ο)]β[Ξ£,exp(bn+1β)]sitting inside
[Ξ£,^exp(π)conn]β*ββ[Ξ£,exp(π)conn][Ξ£,exp(cs,Ο)]β[Ξ£,exp(bn+1β)conn]which, on 1-cells, picks critical points of the action, hence the covariant phase space of the system. Equivalently this are the corresponding analogs of the differential string structures for the invariant polynomial Ο.
So therefore now the quantization of the AKSZ model in one dim higher knows about the quantization of the original symplectic Lie n-algebroid.
something like this.
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