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\section*{Poisson sigma-model}

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

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

[[!include infinity-Chern-Simons theory - contents]]

\hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory}

[[!include functorial quantum field theory - contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToQuantizationOfPoissonManifolds}{Relation to deformation quantization of Poisson manifolds}\dotfill \pageref*{RelationToQuantizationOfPoissonManifolds} \linebreak
\noindent\hyperlink{Branes}{Branes}\dotfill \pageref*{Branes} \linebreak
\noindent\hyperlink{HolographicDual}{Holographic dual}\dotfill \pageref*{HolographicDual} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{LitWithBranes}{With branes}\dotfill \pageref*{LitWithBranes} \linebreak
\noindent\hyperlink{recent_developments}{Recent developments}\dotfill \pageref*{recent_developments} \linebreak


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

The \emph{Poisson $\sigma$-model} is a 2-dimensional [[sigma-model]] [[quantum field theory]] whose target space is a [[Poisson Lie algebroid]]. It is a [[2-dimensional Chern-Simons theory]]. This may be thought of as encoding the [[quantum mechanics]] of a [[string theory|string]] propagating on the [[phase space]] of a system in [[classical mechanics]].

In his solution of the problem of [[deformation quantization]] [[Maxim Kontsevich]] showed that correlators for the 2-string interaction (the correlator on the worldsheet that is a disk with three marked points on its boundary) describe a product operation which is a deformation of the [[Poisson bracket]] on the target space. This solves the formal [[deformation quantization]] problem of the phase space in [[quantum mechanics]] by identifying the quantum algebra with the \emph{open string algebra} of a [[string theory]] on that target.

The principal variant of the nonlinear Poisson sigma model is sometimes called Cattaneo-Felder model who have shown the graphical expansion used in Kontsevich's approach to the deformation quantization is explained via a Feynman diagram expansion in this model.

If one considers [[brane]]s in the target space of the Poisson sigma-model, then then algebra of open strings that used to be just the deformation of the Poisson algebra becomes an [[A-infinity algebra|A-infinity algebroid]]. (See the references below).

\begin{quote}%
Probably something close to a [[Calabi-Yau category]], hence identifying the Poisson sigma-model as a [[TCFT]]. Does anyone know more in this direction?


\end{quote}
\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

The [[target space]] of a Poisson $\sigma$-model is any [[Poisson manifold]] $(X, \{\})$, or rather the [[Poisson Lie algebroid]] $\mathfrak{P}$ corresponding to that.

A field configuration on a 2-dimensional $\Sigma$ is a [[connection on an infinity-bundle|connection]]

\begin{displaymath}
(\phi,\eta) : \mathfrak{T}\Sigma \to \mathfrak{P}  
 \,.
\end{displaymath}
In components this is

\begin{enumerate}%
\item a [[smooth function]] $\phi : \Sigma \to X$;


\item a 1-form $\eta \in \Omega^1(\Sigma, \phi^* T X)$ with values in the pullback of the [[tangent bundle]] of $X$ along $\phi$.



\end{enumerate}
The [[action functional]] on the [[configuration space]] of all such connections for [[compact space|compact]] $\Sigma$ is defined to be

\begin{displaymath}
S : (\phi,\eta) \mapsto 
  \int_\Sigma
  \left(
     \langle \eta \wedge d_{dR}\phi\rangle
     + 
     \frac{1}{2}
     \phi^*\pi(\eta)
  \right)
  \,,
\end{displaymath}
where $\pi \in \wedge^2_{C^\infty(C)}\Gamma(T X)$ is the Poisson tensor of $(X, \{-,-\})$ and where $\langle -,-\rangle$ is the canonical [[invariant polynomial]] on the [[Poisson Lie algebroid]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{RelationToQuantizationOfPoissonManifolds}{}\subsubsection*{{Relation to deformation quantization of Poisson manifolds}}\label{RelationToQuantizationOfPoissonManifolds}

In (\hyperlink{CattaneoFelder}{Cattaneo-Felder}) it was shown that the [[n-point function|3-point function]] in the [[path integral quantization]] of the Poisson $\sigma$-model of a [[Poisson Lie algebroid]] associated with a [[Poisson manifold]] computes the [[star product]] in the [[deformation quantization]] of the Poisson manifold as given by (\hyperlink{Kontsevich}{Kontsevich}).

A [[higher geometric quantization]] that also yields the [[strict deformation quantization]] is discussed at \emph{[[extended geometric quantization of 2d Chern-Simons theory]]}.

One may think of this relation between the 2d Poisson sigma-model and [[quantum mechanics]] = 1d [[quantum field theory]] as an example of the Chern-Simons type [[holographic principle]]. For more along these lines see below at \emph{\hyperlink{HolographicDual}{holographic dual}}.

\hypertarget{Branes}{}\subsubsection*{{Branes}}\label{Branes}

The [[branes]] of the Poisson sigma model are related to [[coisotropic submanifolds]] of the underlying [[Poisson manifold]]. Notice that these are the [[Lagrangian dg-submanifolds]] of the [[Poisson Lie algebroid]]. (\hyperlink{CattaneoFelder03}{Cattaneo-Felder 03}).

\hypertarget{HolographicDual}{}\subsubsection*{{Holographic dual}}\label{HolographicDual}

By the Chern-Simons form of the [[holographic principle]] one expects the Poisson sigma-model to be related to a 1-dimensional [[quantum field theory]]. This is [[quantum mechanics]]. The \hyperlink{RelationToQuantizationOfPoissonManifolds}{above} relation to the deformation quantization of Poisson manifolds goes in this direction. More explicit realizations have been attempted, for instance (\hyperlink{Vassilevich}{Vassilevich}).

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

\begin{itemize}%
\item [[schreiber:∞-Chern-Simons theory]]


\item [[sigma-model]]

\begin{itemize}%
\item [[AKSZ sigma-model]]

\begin{itemize}%
\item \textbf{Poisson sigma-model}

\begin{itemize}%
\item [[A-model]], [[B-model]]

\end{itemize}

\item [[Courant sigma-model]]

\begin{itemize}%
\item [[Chern-Simons theory]]

\end{itemize}


\end{itemize}


\end{itemize}

\item [[schreiber:∞-Chern-Simons theory]]


\item [[higher dimensional Chern-Simons theory]]

\begin{itemize}%
\item [[1d Chern-Simons theory]]


\item [[2d Chern-Simons theory]]


\item [[3d Chern-Simons theory]]


\item [[4d Chern-Simons theory]]


\item [[5d Chern-Simons theory]]


\item [[6d Chern-Simons theory]]


\item [[7d Chern-Simons theory]]


\item [[infinite-dimensional Chern-Simons theory]]



\end{itemize}


\end{itemize}
[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

The Poisson sigma model was first considered in

\begin{itemize}%
\item [[Noriaki Ikeda]], \emph{Two-dimensional gravity and nonlinear gauge theory} , Ann.Phys.235(1994) 435- 464, hep-th/9312059.

\end{itemize}
and later independently by P. Schaller, T. Strobl, motivated from an attempt to unify several two-dimensional models of [[gravity]] and to cast them into a common form with [[Yang-Mills theory|Yang-Mills theories]].

\begin{itemize}%
\item P. Schaller, T. Strobl, \emph{Poisson structure induced (topological) field theories}, Modern Phys. Lett. A 9 (1994), no. 33, 3129--3136, \href{http://dx.doi.org/10.1142/S0217732394002951}{doi}; \emph{Introduction to Poisson $\sigma$-models}, Low-dimensional models in statistical physics and quantum field theory (Schladming, 1995), 321--333, Lecture Notes in Phys. \textbf{469}, Springer 1996.


\item Thomas Strobl, \emph{Gravity from Lie algebroid morphisms}, Comm. Math. Phys. \textbf{246} (2004), no. 3, 475--502, \emph{Algebroid Yang-Mills theories}, Phys. Rev. Lett. \textbf{93} (2004), no. 21, 211601, 4 pp. \href{http://dx.doi.org/10.1103/PhysRevLett.93.211601}{doi}


\item M. Bojowald, A. Kotov, T. Strobl, \emph{Lie algebroid morphisms, Poisson sigma models, and off-shell closed gauge symmetries}, J. Geom. Phys. 54 (2005), no. 4, 400--426, \href{http://dx.doi.org/10.1016/j.geomphys.2004.11.002}{doi}


\item Ctirad Klim\'i{}k, T. Strobl, \emph{WZW-Poisson manifolds}, J. Geom. Phys. \textbf{43} (2002), no. 4, 341--344, 



\end{itemize}
The detailed argument by Cattaneo and Felder on how [[Maxim Kontsevich]]`s formula for the [[deformation quantization]] star product is the 3-point function of the Poisson sigma-model is in

\begin{itemize}%
\item [[Alberto Cattaneo]], [[Giovanni Felder]], \emph{A path integral approach to the Kontsevich quantization formula}, Commun. Math. Phys. 212, 591--611 (2000) \href{http://dx.doi.org/10.1007/s002200000229}{doi}, \href{http://arxiv.org/abs/math/9902090}{math.QA/9902090}.


\item [[Alberto Cattaneo]], [[Giovanni Felder]], \emph{Poisson sigma models and deformation quantization}, Mod. Phys. Lett. A 16, 179--190 (2001) \href{http://arxiv.org/abs/hep-th/0102208}{hep-th/0102208}.



\end{itemize}
See also

\begin{itemize}%
\item [[Alberto Cattaneo]], [[Giovanni Felder]], \emph{Poisson sigma models and symplectic groupoids} , (ed. [[Klaas Landsman]], M. Pflaum, M. Schlichenmeier), Progress in Mathematics 198, 61--93 (Birkh\"a{}user, 2001) \href{http://arxiv.org/abs/math/0003023}{math.SG/0003023}.


\item [[Alberto Cattaneo]], [[Giovanni Felder]], \emph{On the AKSZ formulation of the Poisson sigma model}, Lett. Math. Phys. 56, 163--179 (2001) \href{http://arxiv.org/abs/math/0102108}{math.QA/0102108}.



\end{itemize}
The interpretation in terms of [[schreiber:infinity-Chern-Simons theory]] is discussed in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:AKSZ Sigma-Models in Higher Chern-Weil Theory]]} (2011)

\end{itemize}
Discussion in terms of [[holography]] is in

\begin{itemize}%
\item D. V. Vassilevich, \emph{Holographic duals to Poisson sigma models} (\href{http://arxiv.org/abs/1301.7029}{arXiv:1301.7029})

\end{itemize}
\hypertarget{LitWithBranes}{}\subsubsection*{{With branes}}\label{LitWithBranes}

The study of [[branes]] in the Poisson sigma-model has been started in

\begin{itemize}%
\item Damien Calaque, [[Giovanni Felder]], Andrea Ferrario, Carlo A. Rossi, \emph{Bimodules and branes in deformation quantization} (\href{http://arxiv.org/abs/0908.2299}{arXiv:0908.2299})


\item Damien Calaque, [[Giovanni Felder]], Carlo A. Rossi, \emph{Deformation quantization with generators and relations} (\href{http://arxiv.org/abs/0911.4377}{arXiv:0911.4377})


\item [[Alberto Cattaneo]], [[Giovanni Felder]], \emph{Coisotropic submanifolds in Poisson geometry and branes in the Poisson $\sigma$-model}, Lett.Math.Phys. 69 (2004) 157-175 (\href{http://arxiv.org/abs/math/0309180}{arXiv:0309180})



\end{itemize}
\begin{itemize}%
\item Andrea Ferrario, \emph{Poisson Sigma Model with branes and hyperelliptic Riemann surfaces} (\href{http://arxiv.org/abs/0709.0635}{arXiv:0709.0635})

\end{itemize}
A review is in

\begin{itemize}%
\item F. Falceto, \emph{Branes in Poisson sigma models} (2009) (\href{http://benasque.org/2009gph/talks_contr/094Falceto.pdf}{pdf})

\end{itemize}
\hypertarget{recent_developments}{}\subsubsection*{{Recent developments}}\label{recent_developments}

\begin{itemize}%
\item Francesco Bonechi, [[Alberto Cattaneo]], [[Pavel Mnev]], \emph{The Poisson sigma model on closed surfaces} (\href{http://arxiv.org/abs/1110.4850}{arXiv:1110.4850})

\end{itemize}
[[!redirects Poisson sigma model]]



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