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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 14th 2011
    • (edited Sep 14th 2011)

    something like this:

    let (𝔓,ω)(\mathfrak{P}, \omega) be a symplectic Lie n-algebroid. Then by the discussion at symplectic infinity-groupoid the Lie integration

    exp(𝔓) \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(𝔓) connexp(cs,ω)exp(b n+1) conn \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(𝔓)exp(π)exp(b n+1) \exp(\mathfrak{P}) \stackrel{\exp(\pi)}{\to} \exp(b^{n+1}\mathbb{R})

    classifies a higher central extension exp(𝔓)^\widehat{\exp(\mathfrak{P})} given by the homotopy pullback

    exp(𝔓)^ * exp(𝔓) exp(π) exp(b n+1). \array{ \widehat {\exp(\mathfrak{P})} &\to& * \\ \downarrow && \downarrow \\ \exp(\mathfrak{P}) & \stackrel{\exp(\pi)}{\to}& \exp(b^{n+1}\mathbb{R}) } \,.

    The underlying L L_\infty-algebroid 𝔓^\hat \mathfrak{P} of this is the string-like higher extension

    b n𝔓^𝔓 b^{n}\mathbb{R} \to \hat \mathfrak{P} \to \mathfrak{P}

    classified by the cocycle π\pi that ω\omega transgresses to.

    The corresponding quantum algebra is the “irreducible and polarized” \infty-representation of this exp(𝔓)^\widehat {\exp(\mathfrak{P})}.

    Accordingly, the corresponding AKSZ sigma-model with action functional being the image under [Σ,][\Sigma,-] of exp(cs,ω)\exp(cs,\omega)

    S AKSZ:τ 0[Σ.exp(𝔓) conn]concτ 0[Σ,exp(b n+1) conn] 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

    [Σ,exp(𝔓)^] * [Σ,exp(𝔓)] [Σ,exp(π)] [Σ,exp(b n+1)] \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

    [Σ,exp(𝔓)^ conn] * [Σ,exp(𝔓) conn] [Σ,exp(cs,ω)] [Σ,exp(b n+1) conn] \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 ω\omega.

    So therefore now the quantization of the AKSZ model in one dim higher knows about the quantization of the original symplectic Lie nn-algebroid.

    something like this.