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\begin{document}

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\section*{higher geometric quantization}

\begin{quote}%
under construction, for a more coherent account see (\hyperlink{FiorenzaSchreiber}{hpqg}).


\end{quote}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{geometric_quantization}{}\paragraph*{{Geometric quantization}}\label{geometric_quantization}

[[!include geometric quantization - contents]]

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{ordinary_symplectic_manifolds}{Ordinary symplectic manifolds}\dotfill \pageref*{ordinary_symplectic_manifolds} \linebreak
\noindent\hyperlink{of_nonintegral_2forms}{Of non-integral 2-forms}\dotfill \pageref*{of_nonintegral_2forms} \linebreak
\noindent\hyperlink{of_2plectic_groupoids}{Of 2-plectic $\infty$-groupoids}\dotfill \pageref*{of_2plectic_groupoids} \linebreak
\noindent\hyperlink{in_codimension_2}{In codimension 2}\dotfill \pageref*{in_codimension_2} \linebreak
\noindent\hyperlink{in_codimension_1}{In codimension 1}\dotfill \pageref*{in_codimension_1} \linebreak
\noindent\hyperlink{chernsimons_theory}{$\infty$-Chern-Simons theory}\dotfill \pageref*{chernsimons_theory} \linebreak
\noindent\hyperlink{extended__abelian_chernsimons_theory}{Extended $(4k+3)d$ abelian Chern-Simons theory}\dotfill \pageref*{extended__abelian_chernsimons_theory} \linebreak
\noindent\hyperlink{extended_3d_chernsimons_theory}{Extended 3d $\mathrm{Spin}$-Chern-Simons theory}\dotfill \pageref*{extended_3d_chernsimons_theory} \linebreak
\noindent\hyperlink{extended_3d_chernsimons_theory_2}{Extended 3d $G \times G$-Chern-Simons theory}\dotfill \pageref*{extended_3d_chernsimons_theory_2} \linebreak
\noindent\hyperlink{extended_7d_chernsimons_theory}{Extended 7d $\mathrm{String}$-Chern-Simons theory}\dotfill \pageref*{extended_7d_chernsimons_theory} \linebreak
\noindent\hyperlink{_wesszuminowitten_theory}{$\infty$ Wess-Zumino-Witten theory}\dotfill \pageref*{_wesszuminowitten_theory} \linebreak
\noindent\hyperlink{ordinary_wzw_model}{Ordinary $G$-WZW model}\dotfill \pageref*{ordinary_wzw_model} \linebreak
\noindent\hyperlink{wzw_model}{$String$-WZW model}\dotfill \pageref*{wzw_model} \linebreak
\noindent\hyperlink{wzw_model_2}{$Fivebrane$-WZW model}\dotfill \pageref*{wzw_model_2} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea}

Higher geometric quantization is meant to complete this table:

\begin{tabular}{l|l|l}
[[classical mechanics]]&--[[quantization]]$\to$&[[quantum mechanics]]\\
\hline 
[[symplectic geometry]]&--[[geometric quantization]]$\to$&[[quantum field theory]]\\
[[higher symplectic geometry]]&--higher geometric quantization$\to$&[[extended quantum field theory]]\\
\end{tabular}

Being a concept in [[higher geometry]], higher geometric quantization is formulated naturally in [[(∞,1)-topos theory]]. More precisely, since it involves not just \emph{[[cohomology]]} but \emph{[[differential cohomology]]}, it is formulated in \emph{[[cohesive (∞,1)-topos|cohesive (∞,1)-topos theory]]} ([[cohesive homotopy type theory]]).

In this context, write $\mathbf{B}^n \mathbb{G}_{conn} \in \mathbf{H}$ for the cohesive [[moduli ∞-stack]] of [[circle n-bundles with connection]], in the ambient [[cohesive (∞,1)-topos]] $\mathbf{H}$. Then for $X \in \mathbf{H}$ any object to be thought of as the [[moduli ∞-stack]] of fields or as the [[target space]] for a [[sigma-model]], a morphism

\begin{displaymath}
\mathbf{c}_{conn} : X \to \mathbf{B}^n \mathbb{G}_{conn}
\end{displaymath}
modulates a [[circle n-bundle with connection]] on $X$. We regard this as a \emph{extended [[action functional]]} in that for $\Sigma_{k} \in \mathbf{H}$ of [[cohomological dimension]] $k \leq n$ and sufficiently compact so that [[fiber integration in ordinary differential cohomology]] $\exp(2 \pi i \int_{\Sigma}_k(-))$ applies, the [[transgression]] of $\mathbf{c}_{conn}$ to low [[codimension]] reproduces the traditional ingredients

\begin{tabular}{l|l|l}
$k =$&[[transgression]] of $\mathbf{c}_{conn}$ to $[\Sigma_{n-1},X]$&meaning in [[geometric quantization]]\\
\hline 
$n$&$\exp(2 \pi i S(-)) : [\Sigma_n, X] \stackrel{[\Sigma_n, \mathbf{c}_{conn}]}{\to} [\Sigma_n, \mathbf{B}^n \mathbb{G}_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_n}(-))}{\to}  \mathbb{G}$&[[action functional]]\\
$n-1$&$\exp(2 \pi i S(-)) : [\Sigma_{n-1}, X] \stackrel{[\Sigma_{n-1}, \mathbf{c}_{conn}]}{\to} [\Sigma_{n-1}, \mathbf{B}^n \mathbb{G}_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_{n-1}}(-))}{\to}  \mathbf{B}\mathbb{G}_{conn}  \,$&ordinary (off-shell) [[prequantum circle bundle]]\\
\end{tabular}

The idea is to consider the higher geometric quantization not just of the low codimension transgressions, but of all transgressions of $\mathbf{c}_{conn}$. The basic constructions that higher geometric quantization is concerned with are indicated in the following table. All of them have also a fundamental interpretation in [[twisted cohomology|twisted]] [[cohomology]] (independent of any interpretation in the context of [[quantization]]) this is indicated in the right column of the table:

\newline | [[Planck's constant]] $\hbar$ | $\tfrac{1}{\hbar}\mathbf{c}_{conn} : X \to \mathbf{B}^n \mathbb{G}_{conn}$ | divisibility of twisting class | | [[quantomorphism group]] $\superset$ [[Heisenberg group]] | $\mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn}) = \left\{ \itexarray{ X &&\stackrel{\simeq}{\to}&& X \\ & {}_{\mathllap{\mathbf{c}_{conn}}}\searrow &\swArrow_\simeq& \swarrow_{\mathrlap{\mathbf{c}_{conn}}} \\ && \mathbf{B}^n \mathbb{G}_{conn} }  \right\}$ | twist [[automorphism ∞-group]] | | [[Hamiltonian]] [[quantum operator (in geometric quantization)|quantum observables]] with [[Poisson bracket]] | $Lie(\mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn}))$ | [[infinitesimal cohesion|infinitesimal]] twist automorphisms | | [[Hamiltonian actions]] of a [[smooth ∞-group]] $G$ / dual [[moment maps]]| $\mu : \mathbf{B}G \to \mathbf{B}\mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn})$ | $G$-[[∞-action]] on the twisting | | [[gauge reduction]] | $\mathbf{c}_{conn}//G \,:\, X//G \to \mathbf{B}^n \mathbb{G}_{conn}$ | $G$-[[∞-quotient]] of the twisting | | [[Hamiltonian symplectomorphisms]] | [[∞-image]] of $\mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn}) \to \mathbf{Aut}_{/\Omega^{n+1}_{cl}(-,\mathbb{G})}(\omega)$ | twists in de Rham cohomology that lift to differential cohomology | | [[∞-representation]] of [[n-group]] $\mathbf{B}^{n-1}\mathbb{G}$ on $V_n$| $\itexarray{ V_n &\to& V_n//\mathbf{B}^{n-1}\mathbb{G} \\ && \downarrow^{\mathbf{p}} \\ && \mathbf{B}^n \mathbb{G} }$ | [[local coefficient bundle]] | |[[prequantum space of states]] | $\mathbf{\Gamma}_X(E) := [\mathbf{c},\mathbf{p}]_{/\mathbf{B}^n \mathbb{G}} = \left\{ \itexarray{ X &&\stackrel{\sigma}{\to}&& V//\mathbf{B}^{n-1}\mathbb{G} \\ & {}_{\mathllap{\mathbf{c}}}\searrow &\swArrow_{\simeq}& \swarrow_{\mathrlap{\mathbf{p}}} \\ && \mathbf{B}^n \mathbb{G} } \right\}$ | [[cocycles]] in $[\mathbf{c}]$-[[twisted cohomology|twisted V-cohomology]] | | [[prequantum operator]] | $\widehat{(-)} : \mathbf{\Gamma}_X(E) \times \mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn}) \to \mathbf{\Gamma}_X(E)$ | [[∞-action]] of twist automorphisms on twisted cocycles | | [[trace]] to higher [[dimension]] | $\itexarray{ [S^1, V_n//\mathbf{B}^{n-1}\mathbb{G}_{conn}] &\stackrel{tr\,hol_{S^1}}{\to}& V_{n-1}//\mathbf{B}^{n-2}\mathbb{G}_{conn}  \\  \downarrow^{\mathrlap{\mathbf{p}^{V_n}_{conn}}} && \downarrow^{\mathrlap{\mathbf{p}^{V_{n-1}}_{conn}}} \\ \mathbf{B}^n \mathbb{G}_{conn} &\stackrel{\exp(2 \pi i \int_{S^1}(-))}{\to}& \mathbf{B}^{n-1} \mathbb{G}_{conn} }$ | [[fiber integration in ordinary differential cohomology]] adjoined with one in nonabelian differential cohomology |

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{ordinary_symplectic_manifolds}{}\subsubsection*{{Ordinary symplectic manifolds}}\label{ordinary_symplectic_manifolds}

\begin{itemize}%
\item [[prequantum circle bundle]]

$X \to \mathbf{B} U(1)_{conn}$


\item [[local coefficient bundle]]

\begin{displaymath}
\itexarray{
    \mathbb{C} &\to& \mathbb{C}//U(1)
    \\
    && \downarrow
    \\
    && \mathbf{B} U(1)
  }
\end{displaymath}


\end{itemize}
\hypertarget{of_nonintegral_2forms}{}\subsubsection*{{Of non-integral 2-forms}}\label{of_nonintegral_2forms}

While only integral [[presymplectic forms]] have a [[prequantization]] to a prequantum [[circle bundle with connection]], hence to a $(\mathbb{Z} \to \mathbb{R})$-[[principal 2-bundle]], a general 2-form has a higer prequantization given by a [[connection on a 2-bundle]] on a [[principal 2-bundle]] with structure-[[2-group]] that coming from the [[crossed module]] $(\Gamma \hookrightarrow \mathbb{R})$, where $\Gamma$ is the [[discrete group]] of [[periods]] of the 2-form.

This is discussed further at \emph{[[prequantization of non-integral 2-forms]]}.

\hypertarget{of_2plectic_groupoids}{}\subsubsection*{{Of 2-plectic $\infty$-groupoids}}\label{of_2plectic_groupoids}

\hypertarget{in_codimension_2}{}\paragraph*{{In codimension 2}}\label{in_codimension_2}

\begin{itemize}%
\item [[prequantum circle n-bundle|prequantum circle 2-bundle]]


\item [[local coefficient bundle]]

\begin{displaymath}
\itexarray{
    \mathbf{B}U(n) &\to& \mathbf{B}PU(n)
    \\
    && \downarrow^{\mathrlap{\mathbf{dd}_n}}
    \\
    && \mathbf{B}^2 U(1)
  }
\end{displaymath}

\item [[space of states]]: [[twisted bundles]]



\end{itemize}
\hypertarget{in_codimension_1}{}\paragraph*{{In codimension 1}}\label{in_codimension_1}

\textbf{Proposition} There is a lift of coefficient bundles to [[loop space]]

\begin{displaymath}
\itexarray{
    [S^1,(\mathbf{B}U(n))//(\mathbf{B}U(1))_{conn}] &\stackrel{tr hol_{S^1}}{\to}& \mathbb{C}//U(1)_{conn}
    \\
    \downarrow^{\mathrlap{\mathbf{p}^{\mathbf{B}U}}}
    &&
    \downarrow^{\mathrlap{\mathbf{p}^{\mathbb{C}}}}
    \\
    [S^1,\mathbf{B}^2 U(1)_{conn}]
    &\stackrel{\exp(2 \pi i \int_{S^1}(-))}{\to}& \mathbf{B}U(1)_{conn}
  }
\end{displaymath}
where on the left we have [[loop space objects]] formed in $\mathbf{H}$ and on the bottom we have [[fiber integration in ordinary differential cohomology]].

Forming the pasting composite with this sends 2-states and 2-operators in codimension 2 to ordinary states and operators in codimension 1.

In particular it sends [[twisted bundles]] to sections of a line bundle.

For $X$ a [[D-brane]] and $\mathbf{c}_{conn}$ the [[B-field]], this reproduces [[Freed-Witten anomaly cancellation]] mechanism.

\hypertarget{chernsimons_theory}{}\subsubsection*{{$\infty$-Chern-Simons theory}}\label{chernsimons_theory}

[[schreiber:∞-Chern-Simons theory]]

$G$ a [[smooth ∞-group]],

\begin{displaymath}
\mathbf{c}_{conn}
  : 
  \mathbf{B}G_{conn}
  \to 
  \mathbf{B}^n U(1)_{conn}
\end{displaymath}
a universal differential characteristic map.

The following examples are of this form.

\hypertarget{extended__abelian_chernsimons_theory}{}\paragraph*{{Extended $(4k+3)d$ abelian Chern-Simons theory}}\label{extended__abelian_chernsimons_theory}

[[higher dimensional Chern-Simons theory]]

[[prequantum circle n-bundle|prequantum circle (4k+3)-bundle]] from [[Beilinson-Deligne cup product]]

\begin{displaymath}
\mathbf{B}^{2k+1}U(1)_{conn}
  \stackrel{(-)\cup (-)}{\to}
  \mathbf{B}^{4k+3}U(1)_{conn}
\end{displaymath}
The quantomorphism $\infty$-group of this should be

\begin{displaymath}
\mathbb{Z}_2 \simeq Aut(U(1))
  \,.
\end{displaymath}
For there is, up to equivalence, a unique autoequivalence

\begin{displaymath}
\mathbf{B}^{2k+1}U(1)_{conn} \stackrel{\simeq}{\to}
  \mathbf{B}^{2k+1}U(1)_{conn}
  \,,
\end{displaymath}
the one induced by the nontrivial automorphism of $U(1)$. Since the cup-product is strictly invariant under this, this extends to

\begin{displaymath}
\itexarray{
     \mathbf{B}^{2k+1}U(1)_{conn} &&\stackrel{\simeq}{\to}&&
     \mathbf{B}^{2k+1}U(1)_{conn}
     \\
     & {}_{\mathllap{(-)\cup(-)}}\searrow &\swArrow_\simeq& \swarrow_{\mathrlap{(-)\cup(-)}}
     \\
     && \mathbf{B}^{4k+3}U(1)_\conn
  }
  \,.
\end{displaymath}
But for any further nontrivial such autoequivalence in the slice we would need in particular a gauge transformation parameterized by $(2k+1)$-forms over test manifolds from $C \wedge d C$ to itself. But the only closed $2k$-forms that we can produce naturally from $C$ are multiples of $C \wedge C$. But these all vanish since $C$ is of odd degree $2k+1$.

For $k = 1$ the total space of the [[prequantum circle n-bundle|prequantum circle 3-bundle]] of $U(1)$-Chern-Simons theory over the point is the [[smooth infinity-groupoid|smooth]] [[moduli infinity-stack|moduli 2-stack]] of [[T-Duality and Differential K-Theory|differential T-duality structures]].

\hypertarget{extended_3d_chernsimons_theory}{}\paragraph*{{Extended 3d $\mathrm{Spin}$-Chern-Simons theory}}\label{extended_3d_chernsimons_theory}

\begin{itemize}%
\item [[prequantum circle n-bundle|prequantum circle 3-bundle]]:

[[differential string structure|differential first fractional Pontryagin class]]

$\tfrac{1}{2}\hat \mathbf{p}_1 : \mathbf{B}Spin_{conn} \to \mathbf{B}^3 U(1)_{conn}$



\end{itemize}
So [[Planck's constant]] here is $\hbar = 2$ (relative to the naive multiple).

The total space of the prequantum 3-bundle is

\begin{displaymath}
\itexarray{
    \mathbf{B}String_{conn'} &\to& \Omega^{1 \leq \bullet \leq 3} &\to& *
    \\
    \downarrow && \downarrow && \downarrow 
    \\
    \mathbf{B}Spin_{conn} &\stackrel{\tfrac{1}{2}\hat \mathbf{p}_1}{\to}& \mathbf{B}^3 U(1)_{conn} &\to& \mathbf{B}^3 U(1)
  }
\end{displaymath}
as it appears in [[schreiber:The moduli 3-stack of the C-field]].

But the quantomorphism group of this will be small, as the [[Chern-Simons form]] is far from being gauge invariant.

See the discussion at \emph{\href{Chern-Simons%20theory#GeometricQuantHigher}{Chern-Simons theory -- Geometric quantization -- In higher codimension}}.

\hypertarget{extended_3d_chernsimons_theory_2}{}\paragraph*{{Extended 3d $G \times G$-Chern-Simons theory}}\label{extended_3d_chernsimons_theory_2}

However, when we consider $G \times G$ CS theory given by

\begin{displaymath}
\mathbf{B}(G \times G)_{conn}
  \stackrel{\mathbf{c}^1_{conn}- \mathbf{c}^2_{conn}}{\to}
  \mathbf{B}^3 U(1)_{conn}
\end{displaymath}
then diagonal gauge transformations $\mathbf{B}(G \times G)_{conn}
 \to \mathbf{B}(G \times G)_{conn}$ have interesting extensions to quantomorphisms, because for $g : U \to G$ the given gauge transformation at stage of definition $U$, the Chern-Simons form transforms by an exact term

\begin{displaymath}
CS(A_1^g,A_2^g)
  =
  CS(A_1,A_2)
  + 
  d \langle A_1 - A_2, g^* \theta\rangle
  \,.
\end{displaymath}
\hypertarget{extended_7d_chernsimons_theory}{}\paragraph*{{Extended 7d $\mathrm{String}$-Chern-Simons theory}}\label{extended_7d_chernsimons_theory}

\begin{itemize}%
\item [[prequantum circle n-bundle|prequantum circle 7-bundle]]:

[[differential fivebrane structure|differential second fractional Pontryagin class]]

$\tfrac{1}{6}\hat \mathbf{p}_2 : \mathbf{B}String_{conn} \to \mathbf{B}^z U(1)_{conn}$



\end{itemize}
So [[Planck's constant]] here is $\hbar = 6$ (relative to the naive multiple).

The total space of the prequantum 7-bundle is

\begin{displaymath}
\itexarray{
    \mathbf{B}Fivebrane_{conn'} &\to& \Omega^{1 \leq \bullet \leq 7} &\to& *
    \\
    \downarrow && \downarrow && \downarrow 
    \\
    \mathbf{B}String_{conn} &\stackrel{\tfrac{1}{6}\hat \mathbf{p}_2}{\to}& \mathbf{B}^7 U(1)_{conn} &\to& \mathbf{B}^7 U(1)
  }
\end{displaymath}
\hypertarget{_wesszuminowitten_theory}{}\subsubsection*{{$\infty$ Wess-Zumino-Witten theory}}\label{_wesszuminowitten_theory}

\begin{itemize}%
\item [[schreiber:∞-Wess-Zumino-Witten theory]]

\end{itemize}
Differentially twisted looping of $\infty$-Chern-Simons theory

\begin{displaymath}
\Omega \mathbf{c} : G \to \mathbf{B}^{n-1}\mathbb{G}
\end{displaymath}
\hypertarget{ordinary_wzw_model}{}\paragraph*{{Ordinary $G$-WZW model}}\label{ordinary_wzw_model}

\begin{itemize}%
\item [[WZW model]]

\end{itemize}
\begin{displaymath}
\tilde\Omega \tfrac{1}{2}\mathbf{p}_1 :  G \to \mathbf{B}^2 U(1)_{conn}
\end{displaymath}
studied in (\hyperlink{RogersPhD}{Rogers PhD, section 4.2}).

\hypertarget{wzw_model}{}\paragraph*{{$String$-WZW model}}\label{wzw_model}

(\ldots{})

\hypertarget{wzw_model_2}{}\paragraph*{{$Fivebrane$-WZW model}}\label{wzw_model_2}

(\ldots{})

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

[[!include Isbell duality - table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

2-geometric quantization over [[smooth manifolds]] is discussed in section 6 and section 7 of

\begin{itemize}%
\item [[Chris Rogers]], \emph{Higher symplectic geometry} PhD thesis (2011) (\href{http://arxiv.org/abs/1106.4068}{arXiv:1106.4068})

\end{itemize}
with further indications in

\begin{itemize}%
\item [[Chris Rogers]], \emph{Higher geometric quantization}, at \emph{Higher Structures 2011} ([[RogersGottingen11.pdf:file]])

\end{itemize}
The special case of geometric quantization over infinitesimal [[action groupoids]] can be described in terms of [[BRST complexes]]. For references on this see \emph{\href{http://ncatlab.org/nlab/show/geometric+quantization#ReferencesBRST}{Geometric quantization -- References -- Geometric BRST quantization}}.

Higher geometric quantization in a [[cohesive (∞,1)-topos]] over [[smooth ∞-groupoids]] is discussed in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Urs Schreiber]], [[Chris Rogers]], \emph{[[schreiber:infinity-geometric prequantization]]}

\end{itemize}
and the examples of higher Chern-Simons theories in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]] \emph{[[schreiber:Higher geometric prequantum theory]]}

\end{itemize}
[[!redirects higher geometric prequantization]]

[[!redirects extended geometric quantization]]



\end{document}