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\begin{document}

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\section*{sigma-model -- exposition of a general abstract formulation}

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

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

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

\hypertarget{phyiscs}{}\paragraph*{{Phyiscs}}\label{phyiscs}

[[!include physicscontents]]

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

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

\begin{quote}%
This is a sub-entry of [[sigma-model]]. See there for further background and context.


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{exposition_of_a_general_abstract_formulation}{Exposition of a general abstract formulation}\dotfill \pageref*{exposition_of_a_general_abstract_formulation} \linebreak
\noindent\hyperlink{ExpositionQuantumFieldTheory}{Quantum field theory}\dotfill \pageref*{ExpositionQuantumFieldTheory} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{ExpositionQFTExamples}{Examples}\dotfill \pageref*{ExpositionQFTExamples} \linebreak
\noindent\hyperlink{ExpositionClassicalFieldTheory}{Classical field theory}\dotfill \pageref*{ExpositionClassicalFieldTheory} \linebreak
\noindent\hyperlink{ExpositionQuantization}{Quantization}\dotfill \pageref*{ExpositionQuantization} \linebreak
\noindent\hyperlink{ExpositionClassicalSigmaModels}{Classical $\sigma$-models}\dotfill \pageref*{ExpositionClassicalSigmaModels} \linebreak
\noindent\hyperlink{ExpositionQuantumSigmaModels}{Quantum $\sigma$-models}\dotfill \pageref*{ExpositionQuantumSigmaModels} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{exposition_of_a_general_abstract_formulation}{}\subsection*{{Exposition of a general abstract formulation}}\label{exposition_of_a_general_abstract_formulation}

We give a leisurely exposition of a general abstract formulation $\sigma$-models, aimed at readers with a background in [[category theory]] but trying to assume no other prerequisites.

What is called an \emph{$n$-dimensional $\sigma$-model} is first of all an instance of an $n$-dimensional [[quantum field theory]] (to be explained). The distinctive feature of those quantum field theories that are $\sigma$-models is that

\begin{enumerate}%
\item these arise from a simpler kind of field theory -- called a [[classical field theory]] -- by a process called [[quantization]]


\item moreover, this simpler kind of field theory encoded by[[geometry|geometric data]] in a nice way: it describes physical [[configuration space]]s that are [[mapping space]]s into a [[geometry|geometric]] [[space]] equipped with some [[synthetic differential geometry|differential geometric]] [[structure]].



\end{enumerate}
We give expositions of these items step-by-step:

\begin{enumerate}%
\item \hyperlink{ExpositionQuantumFieldTheory}{Quantum field theory}


\item \hyperlink{ExpositionClassicalFieldTheory}{Classical field theory}


\item \hyperlink{ExpositionQuantization}{Quantization}


\item \hyperlink{ExpositionClassicalSigmaModels}{Classical sigma-models}


\item \hyperlink{ExpositionQuantumSigmaModels}{Quantum sigma-models}



\end{enumerate}
We draw from (\hyperlink{FHLT}{FHLT, section 3}).

\hypertarget{ExpositionQuantumFieldTheory}{}\subsubsection*{{Quantum field theory}}\label{ExpositionQuantumFieldTheory}

\hypertarget{definition}{}\paragraph*{{Definition}}\label{definition}

For our purposes here, a [[quantum field theory]] of [[dimension]] $n$ is a [[symmetric monoidal functor]]

\begin{displaymath}
Z : Bord_n^S \to \mathcal{C}
  \,,
\end{displaymath}
where

\begin{itemize}%
\item $Bord_n^S$ is a [[cobordism category|category of n-dimensional cobordisms]] that are equipped with some [[structure]] $S$;

\begin{itemize}%
\item its [[object]]s are $(n-1)$-[[dimension]]al [[topological manifold]]s;


\item its [[morphism]]s are [[isomorphism class]]es of $n$-[[dimension]]al [[cobordism]]s between such manifolds


\item and all manifolds are equipped with \emph{$S$-[[structure]]} . For instance $S$ could be \emph{[[Riemannian structure]]} . Then we would call $Z$ a [[Euclidean quantum field theory]] (confusingly). If $S$ is ``no structure'' then we call $Z$ a [[topological quantum field theory]].


\item the [[monoidal category]]-structure is given by [[disjoint union]] of manifolds

\begin{displaymath}
\Sigma_1 \otimes \Sigma_2 := \Sigma_1 \coprod \Sigma_2    
  \,;
\end{displaymath}


\end{itemize}

\item $\mathcal{C}$ is some [[symmetric monoidal category]].



\end{itemize}
We think of data as follows:

\begin{itemize}%
\item $Bord_n^S$ is a model for \emph{being} and \emph{becoming} in [[physics]] (following [[Bill Lawvere]]`s terminology): the objects of $Bord_n^S$ are archetypes of \emph{physical spaces that are} and the morphisms are \emph{physical spaces that evolve} ;


\item the [[object]] $Z(\Sigma)$ that $Z$ assigns to any $(n-1)$-manifold $\Sigma$ is to be thought of as the \emph{[[space]] of all possible [[state]]s} over the space $\Sigma$ of a the physical system to be modeled;


\item so $\mathcal{C}$ is the category of [[n-vector space]]s among which the \emph{spaces of [[state]]s} of the quantum theory can be picked;


\item the [[morphism]] $Z(\hat \Sigma) : Z(\Sigma_{in}) \to Z(\Sigma_{out})$ that $\Sigma$ assigns to any [[cobordism]] $\hat \Sigma$ with incoming [[boundary]] $\Sigma_{in}$ and outgoing boundary $\Sigma_{out}$ is the [[propagator]] along $\hat \Sigma$: it maps every state $\psi \in Z(\Sigma_{in})$ of the system over $\Sigma_{in}$ to the state $Z(\Psi) \in \Sigma_{out}$ that is the result of the evolution of $\psi$ along $\hat \Sigma$ by the \emph{dynamics} of the system. Or conversely: the action of $Z$ encodes what this dynamics is supposed to be.



\end{itemize}
Notice that since $Z$ is required to be a symmetric monoidal functor it sends disjoint unions of manifolds to [[tensor product]]s

\begin{displaymath}
F(\Sigma_1 \coprod \Sigma_2) \simeq Z(\Sigma_1) \otimes Z(\Sigma_2)
  \,.
\end{displaymath}
Moreover, for $\hat \Sigma$ a closed [[cobordism]], hence a morphism $\emptyset \stackrel{\hat \Sigma}{\to} \emptyset$ from the [[empty set|empty]] manifold to itself, we have that

\begin{itemize}%
\item $Z(\emptyset) = \mathbf{1}$ is the tensor unit of $\mathcal{C}$;


\item $Z(\hat \Sigma)  \in End(\mathbb{1})$ is an [[endomorphism]] of this tensor unit, a \emph{number} as seen internal to $\mathcal{C}$ -- this is the \emph{invariant} associated to $\hat \Sigma$ by $Z$, called the [[partition function]] of $Z$ over $\hat \Sigma$. We can think of $Z$ as being a rule for computing such invariants by building them up from smaller pieces. This is the \emph{locaity} of quantum field theory.



\end{itemize}
\hypertarget{ExpositionQFTExamples}{}\paragraph*{{Examples}}\label{ExpositionQFTExamples}

\begin{itemize}%
\item A simple but archetypical example is this: let $S := Riem$ be [[Riemannian manifold|Riemannian structure]]. Then the category $Bord_1^{Riem}$ of 1-dimensional cobordisms equipped with Riemannian structure is generated (as a [[symmetric monoidal category]]) from [[interval]]s

\begin{displaymath}
\bullet \stackrel{t}{\to} \bullet
\end{displaymath}
equipped with a [[length]] $t \in \mathbb{R}_+$. Composition is given by addition of lengths

\begin{displaymath}
(\bullet \stackrel{t_1}{\to} \bullet \stackrel{t_2}{\to})
  = 
  (\bullet \stackrel{t_1 + t_2}{\to} \bullet)
  \,.
\end{displaymath}
Therefore a 1-dimensional [[Euclidean quantum field theory]]

\begin{displaymath}
Z : Bord_1^{Riem} \to Vect
\end{displaymath}
is specified by

\begin{itemize}%
\item a [[vector space]] $\mathcal{H}$ (``of [[state]]s'') assigned to the point;


\item for each $t \in \mathbb{R}_+$ a linear endomorphism

\begin{displaymath}
U(t) : \mathcal{H} \to \mathcal{H}
\end{displaymath}
such that

\begin{displaymath}
U(t_1 + t_2) = U(t_2) \circ U(t_1)
  \,.
\end{displaymath}


\end{itemize}
This is just a system of [[quantum mechanics]]. If we demand that $Z$ respects the [[smooth structure]] on the space of morphisms in $Bord_1^{Riem}$ then there will be a linear map $i H : \mathcal{H} \to \mathcal{H}$ such that

\begin{displaymath}
U(t) = \exp(i H t)
   \,.
\end{displaymath}
This $H$ is called the [[Hamilton operator]] of the system.

(We are glossing here over some technical fine print in the definition of $Bord_1^{Riem}$. Done right we have that $\mathcal{H}$ may indeed be an infinite-dimensional vector space. See [[(1,1)-dimensional Euclidean field theories and K-theory]])



\end{itemize}
\hypertarget{ExpositionClassicalFieldTheory}{}\subsubsection*{{Classical field theory}}\label{ExpositionClassicalFieldTheory}

A special class of examples of $n$-dimensional quantum field theories, as discussed \hyperlink{ExpositionQuantumFieldTheory}{above}, arise as [[deformation quantization|deformations]] or \emph{[[path integral|averages]]} of similar, but simpler structure: \emph{[[classical field theories]]} . The process that constructs a quantum field theory out of a classical field theory is called \emph{[[quantization]]} . This is discussed \hyperlink{ExpositionQuantization}{below}. Here we describe what a classical field theory is. We shall inevitably oversimplify the situation such as to still count as a leisurely exposition. The kind of examples that the following discussion applies to strictly are field theories . But despite its simplicity, this case accurately reflects most of the general abstract properties of the general theory.

For our purposes here, a [[classical field theory]] of dimension $n$ is

\begin{itemize}%
\item a [[symmetric monoidal functor]]

\begin{displaymath}
\exp(i S(-)) : Bord_n^S \to Span(Grpd, \mathcal{C})
  \,,
\end{displaymath}
where

\begin{itemize}%
\item $Bord_n^S$ is the same [[category of cobordisms]] as before;


\item $Span(Grpd, \mathcal{C})$ is the [[category of spans]] of [[groupoid]]s over $\mathcal{C}$:

\begin{itemize}%
\item [[object]]s are [[groupoid]]s $K$ equipped with [[functor]]s $\phi : K \to \mathcal{C}$;


\item [[morphism]]s $(K_1, \phi_1) \to (K_2, \phi_2)$ are [[diagram]]s

\begin{displaymath}
\itexarray{
&& \hat K
\\
& \swarrow && \searrow
\\
K_1 &&\swArrow&& K_2
\\
& \searrow && \swarrow
\\
&& \mathcal{C}
  }
  \,,
\end{displaymath}
where in the middle we have a [[natural transformation]];


\item [[composition]] of morphism is by forming [[2-pullback]]s:

\begin{displaymath}
(\hat K_2 \circ \hat K_1) = \hat K_1 \prod_{K_2} \hat K_2
  \,.
\end{displaymath}


\end{itemize}


\end{itemize}


\end{itemize}
Let $\hat \Sigma : \Sigma_1 \to \Sigma_2$ be a [[cobordism]] and

\begin{displaymath}
\exp(i S(-))_{\Sigma} =
  \left(
        \itexarray{
           && Conf_{\hat \Sigma}
           \\
           & {}^{\mathllap{(-)|_{in}}}\swarrow && \searrow^{\mathrlap{(-)|_{out}}}
           \\
           Conf_{\Sigma_1} &&\swArrow_{\exp(i S(-)_{\hat \Sigma})}&& Conf_{\Sigma_2}
           \\
           & {}_{V_{\Sigma_1}}\searrow && \swarrow_{\mathrlap{V_{\Sigma_2}}}
           \\
           && \mathcal{C}
        }
  \right)
\end{displaymath}
the value of a classical field theory on $\hat \Sigma$. We interpret this data as follows:

\begin{itemize}%
\item $Conf_{\Sigma_1}$ is the [[configuration space]] of a [[classical field theory]] over $\Sigma_1$: [[object]]s are ``field configurations'' on $\Sigma_1$ and [[morphism]]s are [[gauge transformation]]s between these. Similarly for $Conf_{\Sigma_2}$.

Here a ``physical field'' can be something like the [[electromagnetic field]]. But it can also be something very different. For the special case of $\sigma$-models that we are eventually getting at, a ``field configuration'' here will instead be a way of an particle of shape $\Sigma_1$ sitting in some [[target space]].


\item $Conf_{\hat \Sigma}$ is similarly the groupoid of field configurations on the whole cobordism, $\hat \Sigma$. If we think of an object in $Conf_{\hat \Sigma}$ of a way of a [[brane]] of shape $\Sigma_1$ sitting in some [[target space]], then an object in $Conf_{\hat Sigma}$ is a \emph{trajectory} of that brane in that target space, along which it evolves from shape $\Sigma_1$ to shape $\Sigma_2$.


\item $V_{\Sigma_i} : Conf_{\Sigma_i} \to \mathcal{C}$ is the [[classifying space|classifying map]] of a kind of [[vector bundle]] over configuration space: a [[state]] $\psi \in Z(\Sigma_1)$ of the [[quantum field theory]] that will be associated to this [[classical field theory]] by [[quantization]] will be a [[section]] of this vector bundle. Such a section is to be thought of as a generalization of a [[probability distribution]] on the space of classical field configurations. The [[generalized element]]s of a [[fiber]] $V_{c}$ of $V_{\Sigma_1}$ over a configuration $c \in Conf_{\Sigma_1}$ may be thought of as an \emph{internal state} of the brane of shape $\Sigma_1$ sitting in target space.


\item $\exp(i S(-))_{\hat \Sigma}$ is the [[action functional]] that defines the classical field theory: the component

\begin{displaymath}
\exp(i S(\gamma))_{\hat \Sigma} 
  : 
  V_{\gamma|_{in}}
  \to
  V_{\gamma|_{out}}
\end{displaymath}


\end{itemize}
of this [[natural transformation]] on a trajectory $\gamma \in Conf_{\hat \Sigma}$ going from a configuration $\gamma|_{in}$ to a configuration $\gamma|_{out}$ is a morphism in $\mathcal{C}$ that maps the internal states of the ingoing configuration $\gamma|_{\Sigma_1}$ to the internal states of the outgoing configuration $\gamma|_{\Sigma_2}$. This evolution of internal states encodes the \emph{classical dynamics} of the system.

Notice that this way a classical field theory is taken to be a special case of a \hyperlink{ExpositionQuantumFieldTheory}{quantum field theory}, where the codomain of the symmetric monoidal functor is of the special form $Span(Grpd, \mathcal{C})$. For more on this see [[classical field theory as quantum field theory]].

\hypertarget{ExpositionQuantization}{}\subsubsection*{{Quantization}}\label{ExpositionQuantization}

We assume now that $\mathcal{C}$ has [[colimit]]s and in fact [[biproduct]]s.

Then for every [[functor]] $\phi : K \to \mathcal{C}$ the [[colimit]]

\begin{displaymath}
\int^{K} \phi \in \mathcal{C}
\end{displaymath}
exists, and (using the existence of [[biproduct]]s) this construction extends to a [[functor]]

\begin{displaymath}
\int : Span(Grpd, \mathcal{C}) \to \mathcal{C}
  \,.
\end{displaymath}
We call this the \emph{[[path integral]]} functor.

For

\begin{displaymath}
\exp(i S(-)) : Bord_n^S \to Span(Grpd, \mathcal{C})
\end{displaymath}
a \hyperlink{ExpositionClassicalFieldTheory}{classical field theory}, we get this way a \hyperlink{ExpositionQuantumFieldTheory}{quantum field theory} by forming the composite functor

\begin{displaymath}
Z 
    := 
  \int \circ \exp(i S(-)) 
   : 
  Bord_n^S 
    \stackrel{\exp(i S(-))}{\to}
  Span(Grpd, \mathcal{C})
    \stackrel{\int}{\to}
  \mathcal{C}
  \,.
\end{displaymath}
This $Z$ we call the \emph{[[quantization]]} of $\exp(i S(-))$.

It acts

\begin{itemize}%
\item on [[object]]s by sending

\begin{displaymath}
\begin{aligned}
    \Sigma_{in} & \mapsto (V_{\Sigma_{in}} : Conf_{\Sigma_{in}} \to \mathcal{C}) 
    \\
    & \mapsto \mathcal{H}_{\Sigma_{in}} := \int^K V_{\Sigma_{in}}
  \end{aligned}
\end{displaymath}
the vector bundle on the [[configuration space]] over some boundary $\Sigma_{in}$ of [[worldvolume]] to its space $\mathcal{H}_{\Sigma_{in}}$ of [[gauge transformation|gauge invariant]] [[section]]s. In typical situations this $\mathcal{H}_{\Sigma_{in}}$ is the famous [[Hilbert space]] of [[state]]s in [[quantum mechanics]], only that here it is allowed to be any object in $\mathcal{C}$;


\item on [[morphism]]s by sending a [[natural transformation]]

\begin{displaymath}
\begin{aligned}   
    \hat \Sigma & \mapsto 
     (\exp(i S(-))_{\hat \Sigma} : \gamma \mapsto 
     V_{\gamma|_{in}} \to V_{\gamma|_{out}})
     \\   
     & \mapsto
     (\int^K \exp(i S(-))_{\hat \Sigma} 
      : 
      \mathcal{H}_{\Sigma_1} \to \mathcal{H}_{\Sigma_2}
     )
  \end{aligned}
\end{displaymath}
to the [[integral transform]] that it defines, weighted by the [[groupoid cardinality]] of $Conf_{\hat \Sigma}$ : the \emph{[[path integral]]} .



\end{itemize}
\hypertarget{ExpositionClassicalSigmaModels}{}\subsubsection*{{Classical $\sigma$-models}}\label{ExpositionClassicalSigmaModels}

A \emph{classical $\sigma$-model} is a \hyperlink{ExpositionClassicalFieldTheory}{classical field theory} such that

\begin{itemize}%
\item the [[configuration space]]s $Conf_{\Sigma}$ are [[mapping spaces]] $\mathbf{H}(\Sigma,X)$ in some suitable category -- some [[cohesive (infinity,1)-topos|higher topos]] in fact -- , for $X$ some fixed [[object]] of that category called \emph{[[target space]]} ;


\item the bundles $V_{\Sigma} : Conf_{\Sigma} \to \mathcal{C}$ ``of internal states'' over these mapping spaces are

\begin{itemize}%
\item the [[transgression]] to these [[mapping space]]s\ldots{}


\item \ldots{}of an [[associated infinity-bundle|associated higher bundle]]\ldots{}


\item \ldots{}asscociated to a [[circle n-bundle with connection]] on [[target space]]\ldots{}


\item \ldots{}encoded by a classifying morphism

\begin{displaymath}
\alpha : X \to \mathbf{B}^{n} U(1)
\end{displaymath}
into the [[circle n-group]] in $\mathbf{H}$; equipped with an [[connection on an infinity-bundle|n-connection]] $\nabla$\ldots{}


\item \ldots{}where the [[associated bundle|association]] is via a [[representation]]

\begin{displaymath}
\rho : \mathbf{B}^{n+1} U(1) \to (n+1) Vect
\end{displaymath}
on [[n-vector spaces]], which is usually taken to be the canonical 1-dimensional one.



\end{itemize}
One calls $(\alpha,\nabla)$ the [[background gauge field]] of the $\sigma$-model.


\item The [[action functional]]s $\exp(i S(-))_{\hat \Sigma}$ are given by the [[higher parallel transport]] of $\nabla$ over $\hat \Sigma$.



\end{itemize}
So an $n$-dimensional $\sigma$-model is a classical field theory that is [[representable functor|represented]], in a sense, by a [[circle n-bundle with connection]] on some [[target space]].

More specifically and more simply, in cases where $X$ is just a [[discrete ∞-groupoid]] -- the case of , every [[principal ∞-bundle]] on $X$ is necessarily , hence the [[background gauge field]] is given just by the morphism

\begin{displaymath}
\alpha : X \to \mathbf{B}^{n} U(1)
  \,.
\end{displaymath}
Then for $\hat \Sigma$ a closed $n$-dimensional manifold, the [[action functional]] of the sigma-model on $\Sigma$ on a field configuration $\gamma : \hat \Sigma \to X$ has the value

\begin{displaymath}
\exp(i S(\gamma))_{\hat \Sigma}
  = 
  \int_{\hat \Sigma} [\alpha]
\end{displaymath}
being the evaluation of $[\alpha]$ regarded as a class in [[ordinary cohomology]] $H^n(\hat \Sigma, U(1))$ evaluated on the [[fundamental class]] of $X$.

One says that $[\alpha]$ is the [[Lagrangian]] of the theory.

\hypertarget{ExpositionQuantumSigmaModels}{}\subsubsection*{{Quantum $\sigma$-models}}\label{ExpositionQuantumSigmaModels}

(\ldots{})

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

\begin{itemize}%
\item [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], \emph{[[Topological Quantum Field Theories from Compact Lie Groups]]} (\href{http://arxiv.org/abs/0905.0731}{arXiv})

\end{itemize}


\end{document}