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\begin{document}

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\section*{holographic principle of higher category theory}

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

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher category theory - contents]]

\hypertarget{functorial_quantum_field_theory}{}\paragraph*{{Functorial quantum field theory}}\label{functorial_quantum_field_theory}

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

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

\noindent\hyperlink{the_general_abstract_principle}{The general abstract principle}\dotfill \pageref*{the_general_abstract_principle} \linebreak
\noindent\hyperlink{examples_in_low_dimension}{Examples in low dimension}\dotfill \pageref*{examples_in_low_dimension} \linebreak
\noindent\hyperlink{Formalizations}{Formalizations}\dotfill \pageref*{Formalizations} \linebreak
\noindent\hyperlink{ApplicationsInQFT}{Application in functorial QFT}\dotfill \pageref*{ApplicationsInQFT} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


\hypertarget{the_general_abstract_principle}{}\subsection*{{The general abstract principle}}\label{the_general_abstract_principle}

In [[higher category theory]] it is easy to verify that a (strict) $1$-[[transfor]]mation $\lambda$ between [[n-functor]]s $F_1,F_2 : C \to D$ between [[strict n-categories]] $C$ and $D$

\begin{displaymath}
\lambda : F_1 \Rightarrow F_2
\end{displaymath}
is determined uniquely by an $(n-1)$-functor

\begin{displaymath}
\eta : C_{(n-1)} \to D^{\Delta^1}
\end{displaymath}
on the strict $(n-1)$-category obtained from $C$ by discarding the [[n-morphism]]s. (Of course, not every such $(n-1)$-functor determines such a transformation; the missing condition is ``naturality'' at the top level.)

Analogous statements hold for general (weak) [[n-categories]], although they are more complicated to formulate; see below.

As with various other easy facts about [[category theory]], these become interesting statements when realized in a concrete context where certain structures are modeled by $n$-[[functor categories]] for all $n$.

\hypertarget{examples_in_low_dimension}{}\subsubsection*{{Examples in low dimension}}\label{examples_in_low_dimension}

We spell out explicitly the $(n-1)$-functorial nature of transformation for low values of $n$.

\begin{itemize}%
\item \textbf{$(n=1)$} -- A [[natural transformation]] $\eta$ between [[functor]]s $F_1,F_2 : C \to D$ between ordinary [[categories]] consists of components which are given by a [[function]]

\begin{displaymath}
\eta : Obj(C) \to Mor(D)
\end{displaymath}
that sends [[object]]s of $C$ to [[morphism]]s in $D$

\begin{displaymath}
\eta : x \mapsto ( F_1(x) \stackrel{\eta(x)}{\to}  F_2(x))
  \,.
\end{displaymath}
Saying that such a function extends to a functor $C \to Arr(D)$:

\begin{displaymath}
\eta : 
  \left(
    \itexarray{
      x 
      \\
      \downarrow^{\mathrlap{\gamma_1}}
      \\
      y
      \\
      \downarrow^{\mathrlap{\gamma_2}}
      \\
      z
    }
  \right)
  \;\;\;
  \mapsto
  \;\;\;
  \left(
    \itexarray{
      F_1(x) &\stackrel{\eta(x)}{\to}& F_2(x)
      \\
      {}^{\mathllap{F_1(f)}}\downarrow
    &=& 
      \downarrow^{\mathrlap{F_2(f)}}
      \\
      F_1(y) &\stackrel{\eta(y)}{\to}& F_2(y)
      \\
      {}^{\mathllap{F_1(g)}}\downarrow
    &=& 
      \downarrow^{\mathrlap{F_2(g)}}
      \\
      F_1(z) &\underset{\eta(z)}{\to}& F_2(z)
    }
  \right)
  \,.
\end{displaymath}
is equivalent to saying that these components form a natural transformation. Since there are no nontrivial [[2-morphism]]s in $D$---in other words, the forgetful functor $Arr(D) \to D\times D$ is faithful---such an extension to a functor is necessarily unique.

So we may regard the component function of $\eta$ as a [[0-functor]]

\begin{displaymath}
\eta : \mathbf{sk}_0 C = Obj(C) \to D^{\Delta[1]} = Arr(D)
\end{displaymath}
from the [[discrete category]] on the set of objects of $C$ to the [[arrow category]] of $D$.


\item \textbf{$(n=2)$} A [[pseudonatural transformation]] $\eta$ between (strict, say, for ease of of notation) [[2-functor]]s $F_1,F_2 : C \to D$ between ([[strict 2-category|strict]], for simplicity) [[2-categories]] is in components a 1-[[functor]] that functorially assigns pseudonaturality squares:

\begin{displaymath}
\eta : 
  \left(
    \itexarray{
      x 
      \\
      \downarrow^{\mathrlap{\gamma_1}}
      \\
      y
      \\
      \downarrow^{\mathrlap{\gamma_2}}
      \\
      z
    }
  \right)
  \;\;\;
  \mapsto
  \;\;\;
  \left(
    \itexarray{
      F_1(x) &\stackrel{\eta(x)}{\to}& F_2(x)
      \\
      {}^{\mathllap{F_1(f)}}\downarrow
    &\swArrow_{\eta(f)}& 
      \downarrow^{\mathrlap{F_2(f)}}
      \\
      F_1(y) &\stackrel{\eta(y)}{\to}& F_2(y)
      \\
      {}^{\mathllap{F_1(g)}}\downarrow
    &\swArrow_{\eta(g)}&
      \downarrow^{\mathrlap{F_2(g)}}
      \\
      F_1(z) &\underset{\eta(z)}{\to}& F_2(z)
     }
  \right)
\end{displaymath}
We may regard this as a 1-functor

\begin{displaymath}
\eta : \mathbf{sk}_1 C \to Arr(D)
\end{displaymath}
from the underlying 1-category of $C$ to the arrow category of $D$, whose objects are morphisms in $D$, whose morphisms are squares in $D$, and whose composition is [[pasting]] of such squares (see [[double category]] for details).

Again, saying that this 1-functor extends to a 2-functor from $C$ to the arrow 2-category of $D$ says precisely that these components form a pseudonatural transformation, and any such extension is unique when it exists since the forgetful 2-functor $Arr(D)\to D\times D$ is locally faithful.


\item \textbf{$(n=3)$} -- A transformation between 3-functors is in components a [[2-functor]] that sends [[2-morphism]]s in $C$ to cyclinders in $D$. This is shown in the $(n=3)$-row of the following diagram



The pseudonaturality condition on $\eta$, which is componentwise the equation



and the fact that there are only identity 3-morphisms in $D$ implies that this already uniquely extends to a 2-functor

\begin{displaymath}
\eta : C \to Arr(D)
  \,,
\end{displaymath}
where on the right we have the 2-category whose objects are morphisms in $D$, whose morphisms are squares in $D$ and whose 2-morphisms are cylinders bounded by these squares.



\end{itemize}
\hypertarget{Formalizations}{}\subsubsection*{{Formalizations}}\label{Formalizations}

For [[strict ∞-categories]] modeled as globular [[strict ∞-categories]] we have the following simple statement of the general principle.

\begin{uLemma}
For $C,D \in Str n Cat$ and $F_1, F_2 : C \to D$ two strict $n$-functors, [[transfor]]mations $\eta : F_1 \Rightarrow F_2$ which are in components given by $n$-functors

\begin{displaymath}
\eta : C \to D^{G_1}
\end{displaymath}
are entirely specified by their underlying $(n-1)$-functors

\begin{displaymath}
\eta : C_{n-1} \to D^{G_1}
  \,.
\end{displaymath}
\end{uLemma}
For weak $n$-categories analogous statements hold, but may have a less straightforward formulation. What is always true is that the transformation $\eta$ is specified by its values on $(n-1)$-morphisms (and below) and will be functorial in a weak sense on these, but these $(n-1)$-morphisms and below will usually not form an $(n-1)$-category themselves, since they will compose coherently only up to $n$-morphisms.

One way to bring the general case into the above simple form is to invoke models by [[semi-strict ∞-categories]]. By [[Simpson's conjecture]], every [[∞-category]] has a model in which everything is strict except possibly the [[identities]] and their unitalness [[coherence law]]s. This means that if $C$ is such a semistrict model of an $n$-category, then $C_{n-1}$ is an $(n-1)$-[[semicategory]] and the transformation

\begin{displaymath}
\eta : C_{n-1} \to D^{\Delta[1]}
\end{displaymath}
is an $n$-functor on that. (By naturalness we have that $\eta$ is guaranteed also to respect the weak identities in $C$ in some way, but that way is not so easy to formalize.)

More generally, for any algebraic notion of weak $n$-category, there is a corresponding algebraic ``$(n-1)$-dimensional'' structure containing only the operations on $(n-1)$-dimensional cells and below in the given notion of weak $n$-category. This is not in general a notion of weak $(n-1)$-category, but it may suffice to formulate the above principle precisely. If the starting notion of $n$-category had strict associativity and interchange, then the resulting $(n-1)$-dimensional structure will be a notion of $(n-1)$-semicategory.

\hypertarget{ApplicationsInQFT}{}\subsection*{{Application in functorial QFT}}\label{ApplicationsInQFT}

For instance in [[FQFT]] one models $n$-dimensional [[topological quantum field theories]] as [[(∞,n)-functor]]s on a flavor of an [[(∞,n)-category of cobordisms]]

\begin{displaymath}
Z : Bord_{n}^S \to \mathcal{C}
\end{displaymath}
(where the superscript $S$ is to remind us that this may be [[cobordism]]s equipped with some extra [[stuff, structure, property|structure]]).

It follows that with $Z_1, Z_2$ two such $n$-dimensional QFTs, a transformation $B : Z_1 \Rightarrow Z_2$ does look in components itself like an QFT -- which is \emph{twisted} by $Z_1$ and $Z_2$ in some sense (see \href{}{below}) -- , but in dimension $(n-1)$.

More specifically, if $\mathcal{C}$ is a [[symmetric monoidal (∞,n)-category]] with tensor unit $1$ there is the trivial FQFT $\mathbf{1}$ given by the constant $(\infty,n)$-functor $\mathbf{1} : Bord_n \to \mathcal{C}$.

One can see in examples that the transformations

\begin{displaymath}
B : Z \Rightarrow \mathbf{1}
\end{displaymath}
encode \emph{boundary conditions} on cobordisms with boundary for the theory $Z$. Conversely, this means that one discovers on the boundary of the $n$-dimensional QFT $Z$ the $(n-1)$-dimensional QFT $B$. Or rather, this is the case if instead of [[natural transformation]]s $\eta$ one uses [[canonical transformation]]s: those component maps $\eta : C_{n-1} \to D^{I}$ that are required to be natural only with respect to the invertible $(n-1)$-morphisms in $C$.

For the case of $n=2$ and 2-dimensional cobordisms without any extra structure, a detailed version of these statements are given in (\hyperlink{SchommerPries}{Schommer-Pries}). For $n=3$ and the holographic relation between [[Reshetikhin?Turaev model]] and rational 2d [[CFT]] in [[FFRS-formalism]] some remarks are in (\hyperlink{Schreiber}{Schreiber}).

In the study of [[quantum field theory]] and [[string theory]] such kinds of relations between $n$-dimensional QFTs and $(n-1)$-dimensional QFTs on their boundary have been called the \emph{[[holographic principle]]} . The degree to which this principle has been formalized and the degree to which this formalization has been verified varies greatly. Examples include

\begin{itemize}%
\item The boundary theory of 3-dimensional [[Chern-Simons theory]] is the 2-dimensional [[WZW-model]].

This is probably the oldest known holographic relation between QFTs.


\item In the 2-d QFT called the [[Poisson sigma-model]] with target the [[Poisson Lie algebroid]] coming from a [[Poisson manifold]] the boundary 3-point function computes the [[deformation quantization]] of the classical system described by that Poisson manifold.

(This was in fact used implicitly by [[Maxim Kontsevich]] to solve deformation quantization. The relation to the Poisson sigma-model was made explicit by Cattaneo and Felder.)


\item The boundary theory of the 2-dimensional [[A-model]] QFT on a target space that is a complexification of a classical [[phase space]] is the 1-dimensional QFT (= [[quantum mechanics]]) that is the [[geometric quantization]] of this phase space.

This is described at [[quantization via the A-model]].


\item The [[AdS/CFT correspondence]] conjectures specifically a holographic relation between [[quantum gravity]] on a 10-dimensional asymptotically [[anti-de Sitter spacetime]] and a superconformal [[Yang-Mills theory]] on its asymptotic boundary. More generally, this specific construction is supposed to apply, with variations, in many different dimensions.



\end{itemize}
See also [[higher category theory and physics]].

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

The discussion of transformations between 2d FQFTs and how these encode boundary 1-[[brane]]s and defect 1-[[bi-brane]]s is in

\begin{itemize}%
\item [[Chris Schommer-Pries]], \emph{Topological defects and classifying local topological field theories in low dimension} ([[SchommerPriesDefects.pdf:file]])

\end{itemize}
from slide 65 on.

A formally comparatively well understood case of QFT holography is the relation between 3-dimensional [[Chern-Simons theory]] and the 2-dimensional [[WZW-model]]. This is formalized by the [[Reshetikhin?Turaev model]] on the 3-dimensional side and the [[Fuchs-Runkel-Schweigert construction]] on the 2-dimensional side.

Remarks on how the relation between Reshitikhin-Turaev and FSR seem to have an interpretation in terms transformations between 3-functors are at

\begin{itemize}%
\item [[Urs Schreiber]], \emph{Towards 2-functorial CFT} (\href{http://golem.ph.utexas.edu/category/2007/08/dbranes_from_tin_cans_part_x.html}{blog entry})

\end{itemize}
There is it discussed how the basic [[string diagram]] that in FSR formalism encodes a field insertion on, possibly, a defect line and encodes the disk amplittudes of the CFT is the string diagram Poincar\'e{}-dual to the cylinder in a 3-category of [[n-vector space|3-vector space]]s.

For references on the [[holographic principle]] in QFT, see there.



\end{document}