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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 14th 2011
   • (edited Sep 14th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexsomething 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 [[schreiber:Higher Chern-Weil Derivation of AKSZ Sigma-Models]] 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 structure]]s 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.

   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.