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something like this:
let $(\mathfrak{P}, \omega)$ be a symplectic Lie n-algebroid. Then by the discussion at symplectic infinity-groupoid the Lie integration
$\exp(\mathfrak{P})$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
$\exp(\mathfrak{P})_{conn} \stackrel{\exp(cs,\omega)}{\to} \exp(b^{n+1} \mathbb{R})_{conn}$whose holonomy is the AKSZ-action functional and whose underlying cocycle
$\exp(\mathfrak{P}) \stackrel{\exp(\pi)}{\to} \exp(b^{n+1}\mathbb{R})$classifies a higher central extension $\widehat{\exp(\mathfrak{P})}$ given by the homotopy pullback
$\array{ \widehat {\exp(\mathfrak{P})} &\to& * \\ \downarrow && \downarrow \\ \exp(\mathfrak{P}) & \stackrel{\exp(\pi)}{\to}& \exp(b^{n+1}\mathbb{R}) } \,.$The underlying $L_\infty$-algebroid $\hat \mathfrak{P}$ of this is the string-like higher extension
$b^{n}\mathbb{R} \to \hat \mathfrak{P} \to \mathfrak{P}$classified by the cocycle $\pi$ that $\omega$ transgresses to.
The corresponding quantum algebra is the “irreducible and polarized” $\infty$-representation of this $\widehat {\exp(\mathfrak{P})}$.
Accordingly, the corresponding AKSZ sigma-model with action functional being the image under $[\Sigma,-]$ of $\exp(cs,\omega)$
$S_{AKSZ} : \tau_0 [\Sigma.\exp(\mathfrak{P})_{conn}] \stackrel{}{\to} conc \tau_0 [\Sigma,\exp(b^{n+1} \mathbb{R})_{conn}]$computes aspects of this quantization by the universal property of the $\infty$-limit, which gives the homotopy pullback
$\array{ [\Sigma,\widehat {\exp(\mathfrak{P})}] &\to& * \\ \downarrow && \downarrow \\ [\Sigma,\exp(\mathfrak{P})] & \stackrel{[\Sigma,\exp(\pi)]}{\to}& [\Sigma,\exp(b^{n+1}\mathbb{R})] }$sitting inside
$\array{ [\Sigma,\widehat {\exp(\mathfrak{P})}_{conn}] &\to& * \\ \downarrow && \downarrow \\ [\Sigma,\exp(\mathfrak{P})_{conn}] & \stackrel{[\Sigma,\exp(cs,\omega)]}{\to}& [\Sigma,\exp(b^{n+1}\mathbb{R})_{conn}] }$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 $\omega$.
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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