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\begin{document}

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\section*{QFT with defects}

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

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

[[!include physicscontents]]

\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{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{DefinitionGeneral}{General}\dotfill \pageref*{DefinitionGeneral} \linebreak
\noindent\hyperlink{DefectsFromBrokenSymmetry}{Topological defects from spontaneously broken symmetry}\dotfill \pageref*{DefectsFromBrokenSymmetry} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general_2}{General}\dotfill \pageref*{general_2} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{general_3}{General}\dotfill \pageref*{general_3} \linebreak
\noindent\hyperlink{in_2d_field_theory}{In 2d field theory}\dotfill \pageref*{in_2d_field_theory} \linebreak
\noindent\hyperlink{in_chernsimons_theory}{In Chern-Simons theory}\dotfill \pageref*{in_chernsimons_theory} \linebreak
\noindent\hyperlink{TopologicalDefectsInGaugeTheories}{Topological defects in gauge theories with broken symmetry}\dotfill \pageref*{TopologicalDefectsInGaugeTheories} \linebreak
\noindent\hyperlink{in_solid_state_physics}{In solid state physics}\dotfill \pageref*{in_solid_state_physics} \linebreak


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

A ([[prequantum field theory|pre]]-)[[quantum field theory|quantum]] [[field theory]] \emph{with defects} is, roughly a [[field theory]] that assigns data not just to plain [[manifolds]]/[[cobordisms]], but to spaces that may carry certain singularities and/or colorings. At the locus of such a singularity the [[bulk]] field theory may then undergo transitions.

Such defects are known by many names. In [[codimension]] 1 they are often called [[domain walls]]. If they are [[boundaries]] they are often called \emph{[[branes]]}, the corresponding domain walls are then sometimes called [[bi-branes]]. Examples of Dimension-1 defects are [[Wilson lines]] and [[cosmic strings]] (at least in [[gauge theory]]) and dimension-0 defects are often called [[monopoles]].

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

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

A plain $n$-dimensional local [[FQFT]] (a [[bulk]] field theory) is a [[monoidal (∞,n)-functor|symmetric monoidal]] [[(∞,n)-functor]] from the [[(∞,n)-category of cobordisms]].

\begin{displaymath}
Z : Bord_n \to \mathcal{C}^\otimes
  \,.
\end{displaymath}
If one replaces plain [[cobordisms]] here with cobordisms $Bord_n^{Def}$ ``with singularities'' including [[boundaries]] and [[corners]] but also partitions labeled in a certain index set $Def$, one calls a morphism

\begin{displaymath}
Z : Bord_n^{Def} \to \mathcal{C}
\end{displaymath}
a TQFT \emph{with defects}. A general formalization is in (\hyperlink{Lurie}{Lurie, section 4.3}), see at \emph{\href{cobordism+hypothesis#ForCobordismsWithSingularities}{Cobordism theorem -- For cobordisms with singuarities (boundaries/branes and defects/domain walls)}}. See also (\hyperlink{DavydovRunkelKong}{Davydov-Runkel-Kong} and \hyperlink{CarquevilleRunkelSchaumann}{Carqueville-Runkel-Schaumann}).

Such a morphism carries data as follows:

\begin{itemize}%
\item for each label in $Def$ of codimension 0 there is an ordinary [[bulk]] field theory;


\item for each label in $Def$ of codimension 1 data on how to ``connect'' the two TQFTs on both sides


\item etc.



\end{itemize}
So one may think of the codimension $k$ colors as \emph{defects} where the TQFT that one is looking at changes its nature.

In particular, when the QFT on one side of the defect is trivial, then the defect behaves like a \emph{boundary condition} for the remaining QFT. Since at least for $n=2$ QFT such boundary conditions are also called \emph{[[branes]]}, defects are also called \emph{[[bi-branes]]}.

The statement of the \textbf{cobordism theorem with singularities} (\hyperlink{Lurie}{Lurie, theorem 4.3.11}) is essentially the following:

Given a [[symmetric monoidal (∞,n)-category]] $\mathcal{C}^\otimes$, then for every choice of [[pasting diagram]] of [[k-morphisms]] for all $k$, there is a type of manifolds with singularity $Def$, such that $Bord_n^Def$ is the free symmetric monoidal $(\infty,n)$-category on this data, hence such that TFTs with defects $(Bord_n^Def)^\otimes \to \mathcal{C}^\otimes$ are equivalently given by realizing such a pasting diagram in $\mathcal{C}$, where each of the given [[k-morphism]] appears as the value of a codimension $(n-k)$-defect. (See also \hyperlink{Lurie}{Lurie, remark 4.3.14}).

\hypertarget{DefectsFromBrokenSymmetry}{}\subsubsection*{{Topological defects from spontaneously broken symmetry}}\label{DefectsFromBrokenSymmetry}

\begin{quote}%
under construction


\end{quote}
An old notion of \emph{defects} in field theory -- well preceeding the \hyperlink{DefinitionGeneral}{above} general notion in the context of [[FQFT]] -- is that of \textbf{[[topological defects]] in the [[vacuum]] structure of [[gauge theories]] that exhibit [[spontaneous symmetry breaking]] (such as a [[Higgs mechanism]]).}

A comprehensive review in in (\hyperlink{VilenkinShellard94}{Vilenkin-Shellard 94}. Steps towards conceptually systematizing these broken-symmetry defects and their interaction are made in \hyperlink{PreskillVilenkin92}{Preskill-Vilenkin 92}. We now discuss this may be translated to and formalized in the general [[FQFT]] definition \hyperlink{DefinitionGeneral}{above} along the lines of (\hyperlink{FiorenzaValentino}{Fiorenza-Valentino}, \hyperlink{FSS}{FSS}).

Let $G$ be a [[Lie group]], to be thought of as the (local or global) [[gauge group]] of some [[gauge theory]]. Let $H \hookrightarrow G$ a [[subgroup]], to be thought of as the subgroup of global symmetries preserved by some [[vacuum]] configuration (which ``[[spontaneous symmetry breaking|spontaneously breaks]]'' the symmetry from $G$ to $H$, the archetypical example is the [[Higgs mechanism]]).

Then the space of such vacuum configurations is the [[coset]] space $G/H$. So given a [[manifold]] $X$ ([[spacetime]]), [[vacuum]] configuratons are given by [[functions]] $X \to G/H$. Hence $G/H_1$ is the ``[[moduli space]] of vacua'' or ``vacuum space'' in this context.

The functions $X \to G/H$ are to be [[smooth functions]] in the [[bulk]] of spacetime. If they are allowed to be non-smooth or even non-[[continuous function|continuous]] along given strata of $X$, then these are called \textbf{defects} in the sense of broken gauge symmetry.

In particular (with counting adapted to $dim X = 4$)

\begin{itemize}%
\item if $X \to G/H$ is not smooth but is smooth on the pre-image of each element of $\pi_0(G/H)$ and becomes a smooth function on $X-S_1$ where $S_1 \hookrightarrow X$ is a [[codimension]]-1 [[submanifold]], then $S_1$ is said to be a [[domain wall]] for vacuum configurations.


\item if $X \to G/H$ is not smooth but becomes smooth on $X-S_2$, where $S_2$ is a codimension-2 submanifold, then $S_2$ is calld a [[cosmic string]]-defect of the vacuum configurations;


\item if $X \to G/H$ is not smooth but becomes smooth on $X-S_3$, where $S_3$ is a codimension-3 submanifold, then $S_3$ is calld a [[monopole]]-defect of the vacuum configurations.



\end{itemize}
\begin{quote}%
hm, need to fine-tune the technical conditions here, to make the following statement come out right\ldots{}


\end{quote}
So

\begin{itemize}%
\item [[domain walls]] can appear when $\pi_0(G/H)$ is non-trivial;


\item [[cosmic strings]] can appear when $\pi_1(G/H)$ is non-trivial;


\item [[monopoles]] can appear when $\pi_2(G/H)$ is non-trivial.



\end{itemize}
Next consider a sequence of [[subgroups]]

\begin{displaymath}
H_2 \hookrightarrow H_1 \hookrightarrow H_0 \coloneqq G
\end{displaymath}
to be thought of as coming from two consecutive steps of [[spontaneous symmetry breaking]], the first one down to $H_1$ at some [[energy]]-scale $E_1$, and the second at some lower energy scale $E_2 \lt E_1$.

Then we say that vacuum defects at energy $E_2$ of codimension-$k$ which wind around an element $\pi_k(H_1/H_2)$ are \textbf{metastable} if they become unstabe at energy $E_1$, hence if their image in $\pi_k(H_0/H_2)$ is trivial.

So if we add to the singular cobordism category the $k$-morphism which is the $k$-dimenional unit cube with an open $k$-[[ball]] removed, then the boundary field data for metastable codimension $n-k$-defects is

\begin{displaymath}
\itexarray{
     [0,1]^k - D^k &\to & \Pi(H_1/H_2)
     \\
     \downarrow &\swArrow& \downarrow
     \\
     [0,1]^k &\to& \Pi(H_0/H_2) &\to& \Pi(H_0/H_1)
  }
\end{displaymath}
we have a [[homotopy fiber sequence]]

\begin{displaymath}
\Pi(H_1/H_2) \to \Pi(H_0/H_2) \to \Pi(H_0/H_1)
  \,.
\end{displaymath}
This induces a [[long exact sequence of homotopy groups]]

\begin{displaymath}
\cdots \to \pi_{k+1}(H_0/H_1) \to \pi_k(H_1/H_2) \to \pi_k(H_0/H_2) \to \pi_k(H_0/H_1)
  \to 
  \pi_{k-1}(H_1/H_2) \to \cdots
  \,.
\end{displaymath}
So for every metastable defect of codimension $n-k$ given by $c \in ker(\pi_k(H_1/H_2) \to \pi_k(H_0/H_2))$ there is an element in $\pi_{k+1}(H_0/H_1)$ of one codimension higher. One says (\hyperlink{PreskillVilenkin92}{Preskill-Vilenkin 92}) that the codimenion $(n-k)$-defect may end on that codimension $(n-k-1)$-defect.

(\ldots{})

In order to formalize this we introduce, following the [[cobordism theorem]] with singularities, cells in $Span_n(\mathbf{H})$ which label spontaneous-symmetriy-breaking defects as well as their defects-of-defects which exhibit their decay by higher codimension defects.

Consider a span of the form

\begin{displaymath}
\itexarray{
    [\Pi(S^{k-1}), \Pi(H_1/H_2)]
    &\leftarrow&
    [\Pi(S^{k-1}), \Pi(H_1/H_2)]
    &\rightarrow&  
    \ast
  }
  \,.
\end{displaymath}
Comparing this to the span that comes from the ``cap'' $S^{k-1} \to D^{k} \leftarrow \emptyset$ in the theory at energy level $E_2$, which is just

\begin{displaymath}
\itexarray{
    [\Pi(S^{k-1}), \Pi(H_1/H_2)]
    &\leftarrow&
    [\Pi(D^{k}), \Pi(H_1/H_2)]
    &\rightarrow&  
    \ast
  }
\end{displaymath}
shows that the former models a $k$-disk which is not filled with spacetime, but nevertheless closes the disk. This is the defect given as a removal of a piece of spacetime.

In order to formalize how these defects may decay at higher energy, consider next a [[span]] of [[field (physics)|field]] configurations of the form

\begin{displaymath}
\itexarray{
    [\Pi(S^{k-1}), \Pi(H_1/H_2)]
    &\leftarrow&
    [\Pi(S^{k}), \Pi(H_0/H_1)]
    &\rightarrow&  
    [\ast, \Pi(H_0/H_2)]
  }
  \,.
\end{displaymath}
Comparing again to the span that comes from the ``cap'' $S^{k-1} \to D^{k} \leftarrow \emptyset$ in the theory at energy level $E_2$, which is just

\begin{displaymath}
\itexarray{
    [\Pi(S^{k-1}), \Pi(H_1/H_2)]
    &\leftarrow&
    [\Pi(D^{k}), \Pi(H_1/H_2)]
    &\rightarrow&  
    \ast
  }
\end{displaymath}
shows that the former models a $k$-disk whose center point carries a singularity: the fields at the bounding $S^{k-1}$ take values in the moduli space of the ambient theory $\Pi(H_1/H_2)$, but then at the tip of the ``cap'' there is a ``field insertion'' of a field with values in $\Pi(H_0/H_2)$. Hence this labels a defect of codimension $k$.

To construct such a decay-process span that captures the above story from \hyperlink{PreskillVilenkin92}{Preskill-Vilenkin 92}, consider the following diagram:

\begin{displaymath}
\itexarray{
     && [\Pi(S^{k}), \Pi(H_0/H_1)]
     \\
     && \downarrow
     \\
     && [\Pi(S^{k-1}), \Omega\Pi(H_0/H_1)]
     \\
     & \swarrow && \searrow
     \\
     [\Pi(S^{k-1}), \Pi(H_1/H_2)] && (pb) && [\Pi(D^k), \Pi(H_0/H_2)] & \simeq & [\ast, \Pi(H_0/H_2)]
     \\
     & \searrow & & \swarrow
     \\
     && [\Pi(S^{k-1}), \Pi(H_0/H_2)]
  }
\end{displaymath}
This may be read as follows:

\begin{enumerate}%
\item on the far left $[\Pi(S^k), \Pi(H_1/H_2)]$ is the space of fields of the ambient theory at energy scale $E_2$ around the defect;


\item the bottom left map is the ``fluctuation'' map that sends these fields to fields at the higher energy scale $E_1$;


\item the bottom right map exhibt the possible ``decays'': a lift through this map takes a field configuration that winds around a $k$-ball and contracts it through that $k$-ball, hence going forth and back through the bottom two maps corresponds to carrying a defect over the energy barrier from $E_2$ to $E_1$ and there having it decay away.


\item these decaying configurations are therefore given by the [[homotopy fiber product]] of the bottom two functions, which is $[\Pi(S^{k-1}), \Omega\Pi(H_0/H_1)]$, as indicated. But this space is really given by field configurations at energy scale $E_1$ that wind around a $(k+1)$-ball, as shown at the very top.



\end{enumerate}
Hence the top part of this diagram is a span that exhibits a defect-of-defects which tells just the story that (\hyperlink{PreskillVilenkin92}{Preskill-Vilenkin 92}) is telling: a codimension-$k$ defect of the low energy theory decays at higher energy, and the decay is witnessed by the appearance of a codimension $(k+1)$-defect of the high energy theory.

\begin{displaymath}
\itexarray{
     && {high\;energy \atop codim-(k+1)\;defects}
     \\
     && \downarrow
     \\
     && {low\;energy\;codim-k\;defects \atop with\;their\;decay\;processes}
     \\
     & \swarrow && \searrow
     \\
     {low\;energy \atop codim-k\;defects} && (pb) && {high\;energy \atop decay\;processes}
     \\
     & {}_{\mathllap{tunnel}}\searrow & & \swarrow_{\mathrlap{apply}}
     \\
     && {codim-k\;defects \atop raised\;to\;higher\;energy}
  }
\end{displaymath}
(\ldots{})

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

[[!include field theory with boundaries and defects - table]]

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

\hypertarget{general_2}{}\subsubsection*{{General}}\label{general_2}

A general formulation via an [[(∞,n)-category of cobordisms]] with defects is in section 4.3 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[On the Classification of Topological Field Theories]]}

\end{itemize}
Defect TQFTs as 1-functors on stratified decorated bordisms are discussed in

\begin{itemize}%
\item [[Alexei Davydov]], [[Ingo Runkel]], [[Liang Kong]], \emph{Field theories with defects and the centre functor} in [[Hisham Sati]], [[Urs Schreiber]] (eds.) \emph{[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]} AMS, 2011


\item [[Nils Carqueville]], [[Ingo Runkel]], [[Gregor Schaumann]], \emph{Orbifolds of n-dimensional defect TQFTs}, (\href{http://arxiv.org/abs/1705.06085}{arXiv:1705.06085})



\end{itemize}
Details in dimension 2 and 3 are discussed in

\begin{itemize}%
\item [[Chris Schommer-Pries]], \emph{Topological Defects and Classification of Local TQFTs in Low Dimension}, [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]] (\href{https://ncatlab.org/nlab/files/SchommerPriesDefects.pdf}{pdf})

\end{itemize}
Discussion of defects in [[prequantum field theory]], hence for [[coefficients]] in an [[(∞,n)-category of spans]] is in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Alessandro Valentino]], \emph{Boundary conditions in local TFTs} (in preparation)

\end{itemize}
\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]] et. al, \emph{[[schreiber:Higher Chern-Simons local prequantum field theory]]}

\end{itemize}
\hypertarget{examples}{}\subsubsection*{{Examples}}\label{examples}

\hypertarget{general_3}{}\paragraph*{{General}}\label{general_3}

Examples in physics of interaction of defects of various dimension is discussed in

\begin{itemize}%
\item Muneto Nitta, \emph{Defect formation from defect--anti-defect annihilations}, Phys. Rev. D85:101702,2012 (\href{http://arxiv.org/abs/1205.2442}{arXiv:1205.2442})

\end{itemize}
\hypertarget{in_2d_field_theory}{}\paragraph*{{In 2d field theory}}\label{in_2d_field_theory}

\begin{itemize}%
\item Defects in 2-dimensional [[conformal field theory]] have a long history in real-world application, for instance in [[Kramers-Wannier duality]]

\begin{itemize}%
\item [[Jürg Fröhlich]], [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], \emph{Kramers-Wannier duality from conformal defects} (\href{http://arxiv.org/abs/cond-mat/0404051}{arXiv:cond-mat/0404051})


\item Fr\"o{}hlich, Fuchs, Runkel, Schweigert, \emph{Duality and defects in rational conformal field theory} (\href{http://arxiv.org/abs/hep-th/0607247}{arXiv})



\end{itemize}

\item Defects in 2-dimension TFT have been studied a lot in the context of genus-0 TFT, where they are described using the language of [[planar algebra]]s.

\begin{itemize}%
\item See the discussion at \href{http://golem.ph.utexas.edu/category/2008/09/planar_algebras_tfts_with_defe.html}{Planar Algebras, TFTs with Defects} for a start.

\end{itemize}


\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Nils Carqueville]], \emph{Lecture notes on 2-dimensional defect TQFT} (\href{http://arxiv.org/abs/1607.05747}{arXiv:1607.05747})

\end{itemize}
\hypertarget{in_chernsimons_theory}{}\paragraph*{{In Chern-Simons theory}}\label{in_chernsimons_theory}

\begin{itemize}%
\item An old example is the class of [[Turaev-Reshetikhin TQFT]], which is a functor on 3-dimensional [[cobordisms]] with codimension 3 and 2 defects. This is supposed to be the would-be result of [[Chern-Simons theory]], where the defect lines are the original [[Wilson lines]] in this context.

\end{itemize}
Defects in [[Chern-Simons theory]] and related systems are discussed in

\begin{itemize}%
\item [[Davide Gaiotto]], [[Edward Witten]], \emph{S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory} (\href{http://arxiv.org/abs/0807.3720}{arXiv:0807.3720})


\item [[Anton Kapustin]], Mikhail Tikhonov, \emph{Abelian duality, walls and boundary conditions in diverse dimensions}, JHEP 0911:006,2009 (\href{http://arxiv.org/abs/0904.0840}{arXiv:0904.0840})


\item [[Anton Kapustin]], [[Natalia Saulina]], \emph{Surface operators in 3d TFT and 2d Rational CFT} in [[Hisham Sati]], [[Urs Schreiber]] (eds.) \emph{[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]} AMS, 2011



\end{itemize}
In the context of the [[3d-3d correspondence]]:

\begin{itemize}%
\item Dongmin Gang, Nakwoo Kim, Mauricio Romo, Masahito Yamazaki, \emph{Aspects of Defects in 3d-3d Correspondence}, J. High Energ. Phys. (2016) (\href{https://arxiv.org/abs/1510.05011}{arXiv:1510.05011})

\end{itemize}
Defects in [[higher dimensional Chern-Simons theory]] on [[manifolds with corners]] are discussed in

\begin{itemize}%
\item [[Hisham Sati]], \emph{Corners in M-theory}, J.Phys.A44:255402,2011 (\href{http://arxiv.org/abs/1101.2793}{arXiv:1101.2793})

\end{itemize}
\hypertarget{TopologicalDefectsInGaugeTheories}{}\paragraph*{{Topological defects in gauge theories with broken symmetry}}\label{TopologicalDefectsInGaugeTheories}

The following references discuss the traditional notion of \emph{[[topological defects]] in the [[vacuum]] structure of gauge theory with [[spontaneous symmetry breaking]]} such as [[domain walls]], [[cosmic strings]] and [[monopoles]].

Discussion of ``topological defects in [[gauge theory]]'' in higher codimension is in

\begin{itemize}%
\item [[John Preskill]], [[Alexander Vilenkin]], \emph{Decay of Metastable Topological Defects}, Phys. Rev. D47 : 2324-2342 (1993) (\href{http://arxiv.org/abs/hep-ph/9209210}{arXiv:hep-ph/9209210})

\end{itemize}
\begin{itemize}%
\item [[Alexander Vilenkin]], E.P.S. Shellard, \emph{Cosmic strings and other topological defects}, Cambridge University Press (1994)

\end{itemize}
\hypertarget{in_solid_state_physics}{}\paragraph*{{In solid state physics}}\label{in_solid_state_physics}

Defects field theory motivated from [[solid state physics]] is discussed in

\begin{itemize}%
\item [[Alexei Kitaev]], [[Liang Kong]], \emph{Models for gapped boundaries and domain walls} Commun. Math. Phys. 313 (2012) 351-373 (\href{http://arxiv.org/abs/1104.5047}{arXiv:1104.5047})

\end{itemize}
[[!redirects defect]] [[!redirects defects]]

[[!redirects field theory with singularities]]

[[!redirects defect field theory]] [[!redirects defect field theories]]

[[!redirects defect QFT]] [[!redirects defect QFTs]]



\end{document}