text:
\section{Introduction}

In this contribution we discuss the relation between two important structures in
quantum field theory: symmetries on the one hand, and defects of various codimensions
on the other. While it is hardly necessary to emphasize the importance of symmetries,
defects in quantum field theories or, in a different formulation, quantum field theories
on stratified spaces, and their relation to symmetries are a more recent
topic of interest.

Our exposition is organized as follows. We first discuss a few general aspects of
topological codimension-1 defects, both in two-dimensional and in three-di\-men\-sional
quantum field theories. A special emphasis is put on the relation between
\emph{invertible} topological defects and symmetries.
Afterwards we consider defects for a particular class of three-dimensional topological
field theories: Dijkgraaf-Witten theories. An advantage of these models is
that they admit a ma\-the\-matically precise formulation as gauge theories, see
\cite{DW90,Fr95,Mo14} and references therein.
Apart from providing insights in a language that is close to standard field theoretical
formulations, this allows one to uncover interesting relations between categories
of (relative) bundles and recent results in representation theory.

\section{Topological defects in quantum field theories}

It has been recently realized in several different contexts that boundary conditions
and defects of various codimensions constitute important parts of the structure
of a quantum field theory. (Alternatively, one might say that it is instructive
to consider the quantum field theory not only on smooth manifolds, but also on stratified spaces.)
Codimension-1 defects, also called interfaces,
separate regions that can support different quantum field theories.
Such defects arise naturally in applications, ranging
from condensed matter systems, where they appear as interfaces between different
phases of matter, to domain walls in cosmology. The aspect we wish to emphasize in
this note is the fact that such defects provide a lot of structural insight into
quantum field theories, including in particular their symmetry structures.

Similarly as in the case of boundary conditions, there are various different types of
defects one can consider. In particular, one can impose various conservation conditions
of physical quantities on defects. An interesting subclass of codimension-1 defects
are {\em topological} defects. They are characterized by the property that the
values of correlators for configurations with defects do not change when the
location of a defect is only slightly changed, that is, changed by a
homotopy without crossing any field insertions or other defects. Examples of such
defects have been known for a long time (see e.g.\ \cite{DW82}). Consider, for
instance, the two-dimensional Ising model, defined by a ${\mathbb Z}_2$-valued
variable on the vertices of a two-dimensional CW-com\-plex.
Select a line that crosses bonds transversally, and change the coupling on each bond
that is crossed by the line from ferromagnetic to anti-ferromagnetic. In the
continuum limit this provides a topological defect line in the critical Ising model.

\subsection{Symmetries from invertible topological defects}

A particularly important subclass of topological defects are the
{\em invertible} topological defects. Owing to their topological nature, two
topological defects can be brought to coincidence, leading to a fusion product
of defects. The precise mathematical formulation of the relevant monoidal
structure depends on the dimension in which the quantum field theory is
defined. For the moment, let us consider two-dimensional quantum field theories.
All statements made in the sequel have the status of theorems \cite{FFRS04,FFRS07}
in the case of two-dimensional rational conformal field theories; the general
picture should be much more widely applicable, though. For a two-dimensional
rational conformal field theory, the topological line defects form a monoidal
category, with morphisms provided by field insertions which can change the
type of defect. In particular, there is an ``invisible defect'', which is
a monoidal unit $\mathbf 1$ in the category of defects.

An \emph{invertible} defect $D$ is characterized by the fact that there
exists another defect $D^\vee$ with the property that the fusion of $D$ and
$D^\vee$ gives the invisible defect,
  $$
  D\otimes D^\vee \cong \mathbf 1 \cong D^\vee \otimes D \,.
  $$
(Thus in particular for every theory the invisible defect is invertible.)
This behaviour leads to the relation

  $$
  \begin{picture}(210,90)
  \put(20,0)   {\includegraphics{pic_ffrs5_13CD.eps}}

  \put(13,28)  {\scriptsize$ D $}
  \put(57.6,28){\scriptsize$ D^\vee $}
  \put(82,42)  {$ = $ }
  \put(101,42) {$ \mathrm{dim}(D)^{-1} $ }
  \put(155,0){
  \put(34,20)  {\scriptsize$ D^\vee $}

  \put(33.3,65){\scriptsize$D ^\vee $}
  }
  \end{picture}
  $$

where $\mathrm{dim}(D)$ is the (quantum) dimension of the defect, which
for an invertible defect can take the values $+1$
(in unitary theories this is the only possibility) or $-1$. This relation is to be
understood as an identity of correlators when applied locally in any
configuration of fields and defect lines. With this relation, it is immediate
to deduce a connection with symmetries: one has equalities of the form
  $$
  \begin{picture}(360,57)
  \put(0,0)    {\includegraphics{pic_ffrs3_4IMN.eps}}

  \put(98,27) {$ =\, \displaystyle\frac1{\dim(D)} $ }
  \put(251,27) {$ = $ }
  \end{picture}
  $$
between correlators of different bulk fields.

Let us discuss this issue explicitly for the case of the two-dimensional critical
Ising model. There are then three primary fields, commonly referred to as the
identity field $1$, the spin field $\sigma$ and the energy field $\epsilon$. Their
conformal weights are $0$, $\frac1{16}$ and $\frac12$,
respectively. Indecomposable topological line defects of the critical Ising model
turn out to be in bijection with these fields: in particular, $1$ corresponds to
the invisible defect and $\epsilon$ to an invertible defect. (It is in fact
the latter defect that amounts to changing the couplings from ferromagnetic to
antiferromagnetic.) The action of this specific invertible defect on bulk fields
is as follows: it leaves the bulk fields corresponding to $1$ and to $\epsilon$
invariant and changes the sign of $\sigma$. Inside RCFT correlators for the
critical Ising model we thus have the equalities

  $$
  \begin{picture}(350,32)
  \put(0,2){
  \put(0,0)    {\includegraphics{pic_ffrs3_4PQ3.eps}}
  \put(13.8,5.5){\scriptsize$ 1 $ }
  \put(29,25.9) {\scriptsize$ \epsilon $ }
  \put(44,13.5) {$ = $ }
  \put(62.7,1.9){$ 1 $ }
  }
  \put(130,2){
  \put(15.2,6.9){\scriptsize$ \epsilon $ }
  \put(29.3,25.9) {\scriptsize$ \epsilon $ }
  \put(44,13.5) {$ = $ }
  \put(64.0,4.7){$ \epsilon $ }
  }
  \put(263,2){
  \put(13.8,7.4){\scriptsize$ \sigma $ }
  \put(29.7,25.9) {\scriptsize$ \epsilon $ }
  \put(44,13.5) {$ = ~~ - $ }
  \put(75.4,5.5){$ \sigma $ }
  }
  \end{picture}
  $$

As can be seen from these pictures, there is a natural action of invertible topological
line defects on field insertions: wrapping such a defect around a bulk field insertion
yields another bulk field.

A further advantage of having a realization of symmetries in terms of invertible topological
defects is seen by studying what happens when the defect is moved to a boundary with
boundary condition $M$: this process yields another boundary condition $M'$, according to

  $$
  \begin{picture}(201,88)
  \put(0,-1){
  \put(27,0)     {\includegraphics{pic_ffrs5_01EF.eps}}

  \put(19.4,16)  {\scriptsize$ D $}
  \put(55.9,18)  {\scriptsize$ M $}
  \put(119,40)   {$ = $}
  \put(141.2,18) {\scriptsize$ M' $}
  }
  \end{picture}
  $$

In a similar manner, by wrapping around various structures in an obvious way,
the group of invertible topological defects
acts as a symmetry group on {\em all} data of the field theory, including field insertions
of bulk, boundary and disorder fields, boundary conditions and types of defects.
In the framework of the TFT construction of correlators of two-dimensional
rational conformal field theories \cite{SFR06}, isomorphism classes of invertible topological
defects can be explicitly classified \cite{FFRS04,FFRS07}. In the example of the critical Ising
model one finds a symmetry group ${\mathbb Z}_2$, consisting of the invisible defect
and the $\epsilon$-defect described above, while e.g.\ for the critical three-state Potts model
one obtains a non-abelian symmetry group $S_3$.

For the present purposes, it is important to realize that topological codimension-1 defects can
also exist in more general classes of quantum field theories. Moreover, in the general case
geometric considerations suggest a natural action of such defects on field theoretic structures,
like boundary conditions and defects of various codimensions, as well.
In short: defects can wrap around field theoretic structures.

\subsection{T-dualities and Kramers-Wannier dualities from topological line defects}

Before turning to higher-dimensional field theories, we point out that topological line defects are
indeed of much wider use: they can also implement Kramers-Wannier dualities and T-dualities.
If a general topological defect wraps around a bulk field, the following situation is created:

  $$
  \begin{picture}(190,67)
  \put(0,-3){
  \put(0,0)  {\includegraphics{pic_ffrs3_1EF.eps}}

  \put(11.4,27.2){\scriptsize$ \phi $ }
  \put(39.2,58){\scriptsize$ D $ }
  \put(55,35){ $\displaystyle = ~
                \underset{{\rm defects}~D_i}{\underset{{\rm intermediate}}{\sum}}$ }
  \put(143.7,57.7){\scriptsize$ \phi $ }
  \put(131,23.3){\scriptsize$ D $ }
  \put(149.6,30.1){\scriptsize$ D_i $ }
  \put(166.9,58){\scriptsize$ D $ }
  }
  \end{picture}
  $$

In this way a bulk field $\phi$ is turned by the defect $D$ into a disorder field. To obtain
an order-disorder duality, one also needs the opposite process, turning disorder fields
into ordinary local bulk fields. It can be shown \cite{FFRS04,FFRS07} that to this
end the dual defect $D^\vee$ must be used, and that in this case one turns the disorder field
back into a bulk field if and only if the fused defect $D\,{\otimes}\, D^\vee$ is a direct sum
of invertible defects. This condition can be examined in concrete models.
It is in particular satisfied for the defect corresponding to the spin field $\sigma$
in the critical Ising model, thanks to the well known fusion rule
$\sigma \,{\otimes}\, \sigma \,{\cong}\, 1 \,{\oplus}\, \epsilon$;
this defect indeed produces the action of the Kramers-Wannier duality in the critical model.

Again we have a natural action on correlators:
  $$
  \begin{picture}(400,137)
  \put(0,75)   {\includegraphics{pic_ffrs3_4I.eps}}
  \put(163,75) {\includegraphics{pic_ffrs3_4J.eps}}
  \put(138,0)  {\includegraphics{pic_ffrs3_4K.eps}}
  \put(264,0)  {\includegraphics{pic_ffrs3_4L.eps}}
  \put(106,102){$\displaystyle = ~ \frac1{\dim(D)} $}
  \put(81,27)  {$\displaystyle = ~ \frac1{\dim(D)} $}
  \put(243,27) {$ = $}
  \end{picture}
  $$

\noindent
which shows how correlators involving only bulk fields are related to correlators involving
disorder fields. Specifically, through the action of the $\sigma$-defect the correlator of four
spin fields on a sphere can be seen to be equal to the correlator of four disorder fields, according to
  $$
  \begin{picture}(400,137)
  \put(0,75)   {\includegraphics{pic_ffrs3_4I.eps}}
  \put(147,75) {\includegraphics{pic_ffrs3_4J.eps}}
  \put(138,0)  {\includegraphics{pic_ffrs3_4K.eps}}
  \put(264,0)  {\includegraphics{pic_ffrs3_4L.eps}}
  \put(106,102){$\displaystyle = ~ \frac1{\sqrt2} $}
  \put(97,27)  {$\displaystyle = ~ \frac1{\sqrt2} $}
  \put(243,27) {$ = $}
  \put(4,88) {
  \put(26.7,.2)  {\scriptsize $ \sigma $}
  \put(26.9,31.4){\scriptsize $ \sigma $}
  \put(56.3,.2)  {\scriptsize $ \sigma $}
  \put(56.5,31.4){\scriptsize $ \sigma $}
  }
  \put(151,88) {
  \put(17.5,26)  {\scriptsize $ \sigma $}
  \put(26.7,.2)  {\scriptsize $ \sigma $}
  \put(26.9,31.4){\scriptsize $ \sigma $}
  \put(56.3,.2)  {\scriptsize $ \sigma $}
  \put(56.5,31.4){\scriptsize $ \sigma $}
  }
  \put(142,14) {
  \put(25.0,1)   {\scriptsize $ \mu $}
  \put(25.3,31.9){\scriptsize $ \mu $}
  \put(37,11.9)  {\scriptsize $ \epsilon $}
  \put(37,21.3)  {\scriptsize $ \epsilon $}
  \put(51.9,16)  {\scriptsize $ \sigma $}
  \put(57.3,1)   {\scriptsize $ \sigma $}
  \put(57.5,31.9){\scriptsize $ \sigma $}
  }
  \put(267,14) {
  \put(26.0,0)   {\scriptsize $ \mu $}
  \put(26.3,31)  {\scriptsize $ \mu $}
  \put(43,10.8)  {\scriptsize $ \epsilon $}
  \put(43,20.1)  {\scriptsize $ \epsilon $}
  \put(58.3,0)   {\scriptsize $ \mu $}
  \put(58.1,31)  {\scriptsize $ \mu $}
  }
  \end{picture}
  $$
and the correlator of two spin fields on a torus can be expressed as

  $$
  \begin{picture}(393,135)
  \put(0,70)    {\includegraphics{pic_ffrs3_4G.eps}}
  \put(27.1,95.4){\scriptsize $\sigma$ }
  \put(27.1,115.4){\scriptsize $\sigma$ }
  \put(90,99)  {$ = $}
  \put(113,99) {$ \displaystyle \frac12 $}
  \put(129,70) {
  \put(0,0)    {\includegraphics{pic_ffrs3_4F.eps}}
  \put(17.2,35){\scriptsize $\epsilon$ }
  \put(27.1,26){\scriptsize $\mu$ }
  \put(27.1,45.9){\scriptsize $\mu$ }
  }
  \put(216,99) {$ + $}
  \put(233,99) {$ \displaystyle \frac12 $}
  \put(251,70) {
  \put(0,0)    {\includegraphics{pic_ffrs3_4E.eps}}
  \put(17.2,35){\scriptsize $\epsilon$ }
  \put(27.1,26){\scriptsize $\mu$ }
  \put(27.1,45.9){\scriptsize $\mu$ }
  \put(44,18.2){\scriptsize $\epsilon$ }
  }
  \put(130,29) {$ + $}
  \put(147,29) {$ \displaystyle \frac12 $}
  \put(165,0) {
  \put(0,0)    {\includegraphics{pic_ffrs3_4D.eps}}
  \put(17.2,35){\scriptsize $\epsilon$ }
  \put(27.1,26){\scriptsize $\mu$ }
  \put(27.1,45.9){\scriptsize $\mu$ }
  \put(49.2,18){\scriptsize $\epsilon$ }
  }
  \put(252,29) {$ + $}
  \put(269,29) {$ \displaystyle \frac12 $}
  \put(287,0) {
  \put(0,0)    {\includegraphics{pic_ffrs3_4C.eps}}
  \put(16.8,35){\scriptsize $\epsilon$ }
  \put(27.1,26){\scriptsize $\mu$ }
  \put(27.1,45.9){\scriptsize $\mu$ }
  \put(34.3,18.1){\scriptsize $\epsilon$ }
  \put(50,32){\scriptsize $\epsilon$ }
  }
  \end{picture}
  $$

T-duality of compactified free bosons can be understood as a special type of
order-disorder duality, see Section 5.4 of \cite{FGRS07}. Indeed, the chiral
data of two T-dual full conformal field theories can be described by the same
theory, namely the one describing a $\mathbb Z_2$-orbifold of the free boson theory. The
line defect implementing the duality is then constructed
from twist fields of this orbifold theory.

\subsection{Relative field theories}

Before turning to defects in the specific class of three-dimensional Dijkgraaf-Witten theories,
we briefly explain two applications of defects. The first is in the context of {\em relative
field theories}. By definition, a $(d{-}1)$-di\-men\-sional relative field theory is a field theory
on the boundary of a $d$-di\-men\-sional field theory or, more generally, on a codimension-1 defect.
There are interesting cases in which the latter
is a topological field theory; we mention two particular situations:
\\[3pt]
(1)\,
The case that the $d$-dimensional topological field theory is even an invertible theory.
This provides a geometric setup for describing anomalous theories; see \cite{FT14}.
\\[3pt]
(2)\,
The case $d \,{=}\, 3$, with the three-dimensional topological field theory being
of Reshetikhin-Turaev type and built from a modular tensor category that is
the representation category of a vertex algebra. In this case the relative theory on a
topological surface defect is a local rational conformal field theory
whose underlying chiral theory is the one built from the vertex algebra.
This provides a geometric interpretation \cite{KS11} of the TFT construction \cite{SFR06}
of RCFT correlators.

\subsection{Quantum codes}

Another application of surface defects in three-di\-men\-sional topological field
theories are quantum codes. It is well known that the
space of qubits of the toric code on a surface $\Sigma$ can be described by
a topological field theory of Turaev-Viro type. Indeed, the vector space assigned by
the topological field theory to the surface $\Sigma$ contains
the information on a vector space together with an action of mapping class groups
which is used to construct quantum gates. Two problems are apparent: First, the
surfaces $\Sigma$ of concrete samples typically have a low genus, and thus,
according to the Verlinde formula, the TFT state space space $\tft(\Sigma)$ has
low dimension. Moreover, the representation of the mapping class group might be too
small to allow for universal gates. To circumvent these problems the idea has been
put forward \cite{BW10,BQ12,BJQ13} to consider samples with a bilayer
(or multi-layer) system and twist defects that create branch cuts. They effectively
lead to systems defined on surfaces of higher genus and thus to higher-di\-men\-sional
quantum codes with richer representations of mapping class groups.
Such systems have been analyzed mathematically \cite{FS14} by exploiting
results on permutation equivariant categories.

\section{Defects and boundary conditions in three-dimensional topological field theories}
\label{DWbound}

We now turn our attention to a particularly accessible subclass of three-di\-men\-sional
topological field theories, Dijkgraaf-Witten theories. These are topological field
theories of Turaev-Viro type, and they can be constructed explicitly in a
gau\-ge-the\-o\-retic setting with finite gauge group. We treat them as
3-2-1-extended topological field theories.

\subsection{Constructing extended Dijkgraaf-Witten theories from $G$-bun\-dles}
\label{DWsect}

As a first input datum we select a finite group $G$.
The space of field configurations on a closed oriented compact smooth
three-ma\-ni\-fold $M$ is then taken to be the groupoid $\Bun_G(M)$ of $G$-bundles on $M$.
By considering formally a path integral with vanishing action, we obtain a partition function
that evaluates to
  $$
  \tft_G(M)=\int_{\Bun_G(M)}\!\! \mathrm D A\,\, \mathrm{e}^0 = \big| \Bun_G(M)\big| \,.
  $$
Here we use the fact that the groupoid $\Bun_G(M)$ is essentially finite, so that there is a
well-defined counting measure, its groupoid cardinality. The groupoid cardinality of
an (essentially) finite groupoid $\Gamma$ is
  $$
  \big| \Gamma \big| := \sum_{\gamma\in \pi_0(\Gamma)} \frac1{|\mathrm{Aut}_\Gamma(\gamma)|} \,,
  $$
where the summation is over the set $\pi_0(\Gamma)$ of isomorphism classes of objects of
$\Gamma$, each of which is counted with the inverse of the cardinality of its automorphism
group.

Trivially, the number $\tft_G(M) \,{=}\, |\Bun_G(M)|$ defines an invariant of the
three-manifold $M$.

It is non-trivial, though, that this invariant is local in the following sense.
Let us cut the closed three-manifold into two pieces, consisting of
three-ma\-ni\-folds with boundaries that are closed oriented surfaces. Suppose that
the topological field theory can associate meaningful quantities to such manifolds
as well, which would allow us to reduce the problem of computing a three-manifold invariant
to computing invariants of simpler manifolds. To see what the topological field theory
$\tft_G$ should associate to a closed oriented two-manifold $\Sigma$, we take a
three-ma\-nifold $M$ with boundary
$\Sigma$. For any $G$-bundle $P$ on $\Sigma$, consider the groupoid $\Bun_G(M,P)$ of
$G$-bundles on $M$ that restrict to the $G$-bundle $P$ on the boundary $\partial M$. Then
$\Bun_G(M,P)$ is an essentially finite groupoid: its groupoid cardinality provides a function
  $$
  \begin{array}{rll}
  \Psi_M:\quad \Bun_G(\Sigma) &\!\! \longrightarrow \!\!& \mathbbm C\\[3pt]
  P &\!\! \longmapsto \!\!& |\Bun_G(M,P)|
  \end{array}
  $$
that depends only on the isomorphism class of the bundle $P$. To the surface $\Sigma$ itself,
we should associate the recipient for all these functions: the vector space
$\mathbbm C[\pi_0(\Bun_G(\Sigma))]$ of complex-valued functions on the set $\pi_0(\Bun_G(\Sigma))$
of isomorphism classes of $G$-bundles on $\Sigma$. We thus recover the well known
feature that a three-dimensional topological field theory assigns vector spaces to
surfaces. In the case of Dijkgraaf-Witten theories, these vector spaces are obtained by
linearization of (isomorphism classes of objects of) categories of bundles.

Closed two-manifolds can, in turn, be decomposed, e.g.\ in a pair-of-pants decomposition, by
cutting them along circles. Hence let $S$ be a closed oriented one-manifold and ask
what an extended topological field theory should associate
to $S$. Following the same type of analysis as above, we choose a surface
$\Sigma$ with boundary $S$. Fixing a $G$-bundle $P$ on the boundary $S$, we assign to it a vector
space,
  $$
  \Psi_\Sigma:\quad P \,\longmapsto\, \mathbbm C[\pi_0(\Bun_G(\Sigma,P))] \,.
  $$
The so obtained map $\Psi_\Sigma\colon \Bun_G(S)\,{\to}\, \Vect $ is a vector bundle on the space of
field configurations on $S$. Thus the extended
Dijkgraaf-Witten theory based on $G$ should associate to $S$ the $\mathbbm C$-li\-ne\-ar category
of vector bundles over the space of field configurations.
Denoting, for an essentially finite groupoid $\Gamma$, the functor
category from $\Gamma$ to a category $\mathcal C$ by $[\Gamma,\mathcal C]$,
this is the functor category
  $$
  \tft_G(S) = [\Bun_G(S),\Vect] \,.
  $$
We thus learn that a 3-2-1-extended topological field theory assigns to a one-ma\-ni\-fold
a $\mathbbm C$-linear category.
The category $\tft_G(S^1)$ can be computed explicitly: the category $\Bun_G(S^1)$
is equivalent to the action groupoid $G/\!/G$ for the adjoint action of $G$ on itself. Indeed,
after fixing a point $p \,{\in}\, S^1$, a $G$-bundle is described by its holonomy $g \,{\in}\, G$,
on which gauge transformations act by the adjoint action. Thus $\tft_G(S^1) \,{\simeq}\,
[G/\!/G,\Vect]$, and this functor category is equivalent to the category of $G$-graded vector spaces
$V \,{=}\, \bigoplus_{g\in G} V_g$ with a $G$-action such that $g(V_h) \,{\subset}\, V_{ghg^{-1}}$.
Put differently, we deal with $G$-equivariant vector bundles on the group $G$.

This category is actually nothing but the {\em Drinfeld center} of the fusion category
$G$-$\Vect$ of $G$-graded vector spaces. For later reference, we recall the definition of
the Drinfeld center $\calz(\cala)$ of a monoidal category $\cala$: an object of $\calz(\cala)$
is a pair $(U,c_{U,-})$ consisting of an object $U$ of $\cala$ and a functorial family of
isomorphisms $c_{U,X}\colon U \,{\otimes}\, X \,{\to}\, X \,{\otimes}\, U$,
called a half-braiding, subject to some further coherence properties; a morphism from
$(U,c_{U,-})$ to $(V,c_{V,-})$ is a morphism $f\colon U \,{\to}\, V$ in $\cala$ such that
$(\mathrm{id}_- \,{\otimes}\, f) \,{\circ}\, c_{U,-} \,{=}\, c_{V,-} \,{\circ}\, (f \,{\otimes}\,
\mathrm{id}_-)$. The half-braidings endow the Drinfeld center $\calz(\cala)$ with the structure
of a {\em braided} monoidal category. Dropping the half-braiding gives a forgetful functor
  $$\begin{array}{rll}
  \calz(\cala) &\!\! \to \!\!& \cala\\[2pt]
  (U,c_{U-}) &\!\! \mapsto \!\!& U
  \end{array}$$
which has a natural monoidal structure. If a category $\calc$ is already braided, then the
braiding provides a braided monoidal functor $\calc \,{\to}\, \calz(\calc)$. The opposite
braiding endows the underlying category with another structure of a braided category,
denoted by $\calc^{\mathrm{rev}}$, and similarly gives a braided monoidal functor
$\calc^{\mathrm{rev}} \,{\to}\, \calz(\calc)$. These two functors combine into a braided
monoidal functor
  \begin{equation}
  F:\quad \calc\boxtimes\calc^{\mathrm{rev}}\to \calz(\calc)\
  \label{cschweigert:cantriv}
  \end{equation}
from the Deligne product of $\calc$ and $\calc^{\mathrm{rev}}$ to the center.
A braided category is called \emph{non-degenerate} iff this functor $F$ is a braided
equivalence. A non-degenerate ribbon category is called a \emph{modular} category.

The preceding considerations for Dijkgraaf-Witten theories lead us naturally to
the general definition
of 3-2-1-extended topological field theories as a symmetric monoidal 2-functor
  $$
  \tft:\quad \cobord_{3,2,1} \to 2\mbox-\Vect(\mathbbm C) \,.
  $$
This definition naturally generalizes Atiyah's classical definition of (non-extended)
topological field theory.
In the rest of this subsection we elaborate on this 2-func\-tor. In the definition,
$2$-$\Vect(\mathbbm C)$ is the symmetric monoidal bicategory
of finitely semisimple $\mathbbm C$-linear abelian categories, with the Deligne product
as a mo\-no\-i\-dal structure. The bicategory $\cobord_{,3,2,1}$ is
an extended cobordism category whose objects are closed oriented one-manifolds $S$,
whose 1-morphisms $S \,{\to}\, S'$ are two-ma\-ni\-folds $\Sigma$ with boundary together
with a decomposition $\partial\Sigma \,{\xrightarrow\simeq}\, S'\,{\sqcup}\, {-}S$, and
whose 2-morphisms are three-manifolds with corners, up to diffeomorphism, again with an
appropriate decomposition of the boundary.

According to the definition, the 2-functor $\tft$ assigns to the oriented circle
${\mathbb S}^1$ a finitely semisimple $\mathbbm C$-linear category
${\mathcal C} \,{:=}\, \tft({\mathbb S}^1)$. The trinion (pair of pants), regarded as a
cobordism ${\mathbb S}^1 \,{\sqcup}\, {\mathbb S}^1 \,{\to}\, {\mathbb S}^1$ gives a functor
$\otimes\colon\, {\mathcal C}\,{\boxtimes}\, {\mathcal C} \,{\to}\, {\mathcal C}$. Finally,
suitable three-manifolds with corners provide natural isomorphisms that
endow this functor with an associativity constraint and the category ${\mathcal C}$ with
the structure of a braiding. The constraints on these natural transformations, expressed
by pentagon and hexagon diagrams, can be deduced from homotopies between three-manifolds
with corners. In this way the category ${\mathcal C} \,{=}\, \tft({\mathbb S}^1)$ is
expected to be endowed with the structure of a modular tensor category
(see \cite{BDSV14} for some recent progress). The Reshetikhin-Turaev
construction can be seen as a converse, constructing a 3-2-1-extended topological
field theory from a modular tensor category.

\subsection{Topological field theories with boundaries and defects}\label{sec:defbound}

Incorporating defects and boundaries into a topological field theory amounts to consider
an enlarged bicategory $\cobord^\partial_{3,2,1}$ of cobordisms.
As a warm-up, we recall the situation for two-dimensional topological field theories
with boundaries, also known as open/closed topological field theory

 \cite{La01,AN06,MS06,LP08}
Beyond disjoint unions of oriented circles, the category $\cobord^\partial_{2,1}$ has intervals
$\mathbb I_{a_1,a_2}$, with boundary conditions $a_1,a_2$ attached to the end points,
as additional objects. Morphisms are now surfaces with boundaries consisting of segments
which are are either ``cut-and-paste  boundaries'' implementing locality or true physical boundaries.

A two-dimensional open/closed topological field theory is a symmetric monoidal functor
$\tft\colon \cobord^\partial_{2,1}\to\Vect$. It assigns a vector space
  $$
  A_a := \tft(\mathbb I_{aa})
  $$
to an interval with boundary condition $a$ at either end point.
For any boundary condition $a$, a disk with three marked intervals on the boundary provides
a cobordism $\mathbb I_{aa} \,{\sqcup}\, \mathbb I_{aa} \,{\to}\, \mathbb I_{aa}$; applying
the functor $\tft$ thus yields a linear map $A_a \,{\otimes}\, A_a \,{\to}\, A_a$.
Further analysis \cite{MS06,LP08} shows that this map furnishes a (not necessarily commutative)
associative product on $A_a$, and that $A_a$ is even endowed
with the structure of a symmetric Frobenius algebra. A similar argument shows that for any
other boundary condition $b$, the cobordism $\mathbb I_{aa}\sqcup \mathbb I_{ab} \,{\to}\, \mathbb I_{ab}$
endows the vector space $M_b \,{:=}\, \tft(\mathbb I_{ab})$ with a linear map
$A_a \,{\otimes}\, M_b \,{\to}\, M_b$ and thereby acquires the structure of an $A_a$-module.

In this way one arrives at the following structure: boundary conditions form a $\mathbbm C$-linear
category that is equivalent to the category $A_a\text{-mod}$ of $A_a$-modules for any given
boundary condition $a$. To the interval $\mathbb I_{b_1,b_2}$ the topological field theory assigns
  \begin{equation}
  \tft(\mathbb I_{b_1,b_2})= \mathrm{Hom}_{A_a\text{-mod}}(b_1,b_2) \,,
  \label{cschweigert:interval}
  \end{equation}
the vector space of morphisms of $A_a$-modules. In particular, for any boundary condition
$b$, we find $\tft(\mathbb I_{b,b}) \,{=}\, \mathrm{End}_{A_a\text{-mod}}(b)$, which is an algebra that
is Morita equivalent to the algebra $A_a$. If $A_a$ is semisimple, then the commutative Frobenius algebra
$C \,{:=}\, \tft(S^1)$ that is assigned to the circle turns out to be the center of the algebra $A_a$:
  $$
  C= \tft(S^1)= Z(\tft(\mathbb I_{aa}) )= Z(A_a) \,.
  $$

For three-dimensional topological field theories, we will find a higher-categorical analogue
of these structures. In a complete treatment one should start by giving a precise definition
for the extended category $\cobord^\partial_{3,2,1}$ of cobordisms;
several definitions have been proposed for this enlarged category, see e.g.\ \cite[Sect.\ 4.3]{Lu09}.
Here we refrain from presenting a formal definition, and instead concentrate on some of the physical
requirements that such a category should satisfy. Again we start from oriented three-manifolds, but this
time we also allow them to have  oriented
boundaries that are actual physical boundaries to which boundary
conditions need to be assigned, rather than the cut-boundaries that were already present in
Section \ref{DWsect}, which are instead a device for incorporating some aspects of locality into
the theory. We also admit codimension-1 submanifolds, called surface defects. Such a defect separates
two three-dimensional regions each of which supports a topological field theory, and the
theories associated to the two regions may be different. Labels must be assigned to
such defects, too. In this way, we get in particular invariants of three-manifold with embedded
surfaces. In the present setting of 3-2-1-extended topological field theories,
it is natural to go one step further and allow for
codimension-2 defects as well. In particular, there will be generalized Wilson lines
separating two surface defects from each other, and likewise boundary Wilson lines that are
confined to a boundary component.

In a next step one imposes locality by cutting such a three-manifold to produce three-ma\-nifolds
with cut-boundaries. It is natural to impose the condition that the cut-surface intersects all
structure in the three-manifold, e.g.\ surface defects and boundaries, transversally. Boundary manifolds
now carry a decoration consisting of lines labeled by surface defects and boundaries labeled
by boundary conditions. Imposing further locality, as in Section \ref{DWsect}, requires to
cut such decorated surfaces transversally. This yields new types of labeled one-manifolds, which are to
be taken as the objects in the bicategory $\cobord^\partial_{3,2,1}$. Let us describe these objects
in detail: There are oriented intervals and circles, with finitely many marked points in the interior
that divide them into subintervals. Each subinterval is to be marked with a topological field theory.
Points adjacent to two subintervals are to be marked by a surface defect, boundary points by
a boundary condition. The possible boundary conditions and types of surface defects have to be
determined by a field-theoretic analysis.
For the case of Dijkgraaf-Witten theories we present this analysis in the next subsection.

\subsection{A model independent analysis of defects and
boundaries}\label{sec:tftb}

We now turn to a general analysis of boundary conditions and defects in three-dimensional
topological field theories of Reshetikhin-Turaev type.
The labels for the bulk Wilson lines in such a theory are supplied by some modular tensor category $\calc$.
The boundary conditions of the theory
form a bicategory: its objects are boundary conditions, its 1-morphisms are labels for
boundary Wilson lines, and its 2-morphisms label point-like insertions on boundary Wilson lines.
Our goal is to describe this bicategory in terms related to the modular tensor category $\calc$.

We want the boundary Wilson lines for a given boundary condition to be topological as well.
As a consequence there is a fusion of boundary Wilson lines and we have invariance under two-dimensional
isotopies. This leads to the requirement that for any boundary condition $a$ the
boundary Wilson lines and their insertions form a $\mathbbm C$-linear, finitely semisimple, rigid monoidal
category $\calw_a$. The category $\calw_a$ is thus a fusion category; since boundary Wilson lines are
confined to the two-dimensional boundary,
$\calw_a$ is, however, not endowed with the structure of a braiding.

To derive constraints on the category $\calw_a$, we postulate that there should exist a process
of moving a bulk Wilson line to the boundary, and that this should be as smooth as possible.
(In other words, we exclude possible boundary conditions for which such a smooth process does
not exist.) This provides a functor
$F_a\colon \calc \,{\to}\, \calw_a$. Next we impose the condition that the following two processes
are equivalent: fusing two bulk Wilson lines and then moving them into the boundary, and moving
the two bulk Wilson lines individually to the boundary and afterwards fusing them in the boundary;
schematically,
  $$
  \begin{picture}(350,88)
  \put(0,3)     {
  \put(0,0)     {\includegraphics{pic_fusV_sd22AB.eps}}
  \put(15,71)     {\scriptsize $ U $}
  \put(56.5,46.3) {\scriptsize $ U{\otimes}V $}
  \put(54,26)     {\begin{turn}{15} \scriptsize $ F_a(U{\otimes}V) $\end{turn}}
  \put(82.7,78)   {\scriptsize $ V $}
  \put(216.5,0){
  \put(-1,27.7)   {\scriptsize $ F_a(U) $}
  \put(15,71)     {\scriptsize $ U $}
  \put(48.3,26.3) {\begin{turn}{80} \scriptsize $ F^{}_{\!a}(U)\,{\otimes}\, F^{}_{\!a}(V) $\end{turn}}
  \put(82.7,78)   {\scriptsize $ V $}
  \put(82,27.7)   {\scriptsize $ F_a(V) $}
  } }
  \end{picture}
  $$
In formulas, what we postulate is the existence of coherent isomorphisms
  $$
  F_a(U{\otimes_\calc} V) \cong F_a(U) \,{\otimes_{\calw_a}}\, F_a(V)
  $$
for all $U,V \,{\in}\,\calc$. This means that the functor $F_a$ has the structure of a
\emph{monoidal} func\-tor.

Similarly, the following two processes should be equivalent:
  $$
  \begin{picture}(350,87)
  \put(0,3)     {
  \put(0,0)     {\includegraphics{pic_fusV_sd24AB.eps}}
  \put(2.9,34.1)  {\begin{turn}{-20}$\scriptstyle F_a(U)$\end{turn}}
  \put(61.9,70.8) {\scriptsize $ U $}
  \put(59,28.2)   {\scriptsize $ X $}

  \put(254.5,71)  {\scriptsize $ U $}
  \put(266.1,28.2){\scriptsize $ X $}

  \put(304.6,26.2){\begin{turn}{20}$\scriptstyle F_a(U)$\end{turn}}
  }
  \end{picture}
  $$
Here a bulk Wilson line labeled by an object $U$ of $\calc$ is moved to either side of a boundary Wilson line
labeled by an object $X$ of $\calw_a$. Accordingly we postulate the existence of coherent isomorphisms
  $$
  F_a(U) \,{\otimes_{\calw_a}}\, X \cong X \,{\otimes_{\calw_a}}\, F_a(U) \,.
  $$
This constitutes an additional structure on a monoidal functor from a braided category
to a fusion category, called the structure of a \emph{central}
functor. It can be alternatively described as the choice of a lift $\widetilde F_a$ of $F_a$
to the center of $\calw_a$, i.e.\ as the existence of a braided monoidal functor $\widetilde F$
such that the diagram
  $$\xymatrix{
  & \calz(\calw_a) \ar[d]\\
  \calc \ar^{F_a}[r] \ar@{-->}^{\widetilde F_a}[ur]& \calw_a
  }$$
commutes, with the vertical arrow the forgetful functor described in Section \ref{DWsect}.
It is now  an evident naturality requirement to ask that $\widetilde F$ is an \emph{equivalence}
of braided categories: the only reason for the existence of a consistent prescription for
moving a specific boundary Wilson line past any arbitrary boundary Wilson line
should be that this boundary Wilson line actually comes from
a Wilson line in the bulk via the process described by the functor $F_a$.

We thus learn in particular that in order to admit boundary conditions of the type just introduced,
a modular tensor category $\calc$ must be a Drinfeld center. This constitutes a new
phenomenon in three dimensions, for which there is no
counterpart in two dimensions: while any commutative algebra
(arising, as explained in Section \ref{sec:defbound}, as the value of a two-dimensional topological field
theory on a circle) is the center of some algebra (namely, of itself), it is {\em not\/}
true that any braided monoidal fusion category (arising as the value of a
three-dimensional topological field theory on a circle) is the Drinfeld center of a fusion category.

In this context, the following notions are relevant \cite{DMNO13,DNO13}: Two modular
tensor categories $\calc_1$ and $\calc_2$ are said to be {\em Witt equivalent\/} iff there
exist fusion categories $\cala_1$ and $\cala_2$ and a braided equivalence
  $$
  \calc_1\boxtimes \calz(\cala_1) \xrightarrow{~~\simeq~~} \calc_2\boxtimes\calz(\cala_2) \,.
  $$
The Deligne product induces on the set of equivalence
classes a natural group structure; this group is called the Witt group
of modular tensor categories. It contains, as a subgroup, the Witt group of finite abelian
groups $A$ that are endowed with a quadratic form $q\colon A \,{\to}\, {\mathbb C}^\times$.
(Such a form determines a modular tensor category, a fact that is relevant
for abelian Chern-Simons theories.) The obstruction to the existence of boundary conditions
for a three-dimensional topological field theory based on a modular tensor category
$\calc$ can thus be restated as the condition that the class of $\calc$ in the Witt group must
vanish, $[\calc] \,{=}\, 0$.

Suppose now we have already found one boundary condition $a$, i.e.\ one Witt-tri\-vi\-alization
      $\widetilde F_a\colon \calc \,{\xrightarrow{\,\simeq\,}}\, \calz(\calw_a)$. If $b$ is any other boundary

condition, denote by $\calw_{a,b}$ the category of boundary Wilson lines separating a piece of
boundary in boundary condition $a$ from a piece in boundary condition $b$. Fusion of Wilson lines
inside the piece in boundary condition $a$ with a Wilson line separating $a$ and $b$ then
yields a functor
  $$
  \calw_a \,{\times}\, \calw_{a,b} \longrightarrow \calw_{a,b} \,.
  $$
Hereby the category $\calw_{a,b}$ acquires the structure of a module category over the monoidal category
$\calw_a$. This amounts to a categorification of the situation in two-dimensional topological
field theory analyzed in Section \ref{sec:defbound}. The bicategory of module categories
over $\calw_a$ is therefore a natural candidate for the bicategory of boundary conditions.

In much the same way as a left module $M$ over a ring $R$ is a right module over the
ring $\mathrm{End}_R(M)$, the $\calw_a$-module category $\calw_{a,b}$ is also a right
module category over the monoidal category ${\mathcal E\!nd}_{\calw_a}(\calw_{a,b})$
of module endofunctors of $\calw_{a,b}$. But at the same
time it is also a right module category over the fusion category $\calw_b$ of Wilson lines inside
the boundary with boundary condition $b$. Naturality thus suggests
an equivalence $\calw_b\,{\simeq}\, {\mathcal E\!nd}_{\calw_a}(\calw_{a,b})$. This can only hold true
if the fusion category ${\mathcal End}_{\calw_a}(\calw_{a,b})$ is a Witt trivialization of $\calc$ as
well. Now indeed it is known \cite{Sc01} that for any module category there is a
canonical braided equivalence
  $$
  \calz(\calw_a) \simeq \calz({\mathcal E\!nd}_{\calw_a}(\calw_{a,b})) \,,
  $$
so that we do have a canonical Witt trivialization for ${\mathcal E\!nd}_{\calw_a}(\calw_{a,b})$ as well.

More generally, given two boundary conditions described by $\calw_a$-module category $\calm_b$ and
$\calm_c$, the category of boundary Wilson lines is given by the functor category
${\mathcal F\!un}_{\calw_a}(\calm_b,\calm_c)$ of module functors. This can be regarded as a categorification
of Equation (\ref{cschweigert:interval}). At this point a word of warning is in order: via pullback along
the forgetful functor
  $$
  \calc \xrightarrow{~\simeq~} \calz(\calw_a) \xrightarrow{~\phantom{\simeq}~}  \calw_a \,,
  $$
any $\calw_a$-module category has a natural structure of a $\calc$-module category;
however, not every $\calc$-module category describes a boundary condition for the
Reshetikhin-Turaev theory based on $\calc$. As an illustration consider Kitaev's toric
code, which is the Dijkgraaf-Witten theory based on $\mathbb{Z}_2$. In that case there are
$6$ inequivalent

    indecomposable
$\calc$-module categories over the Drinfeld center
$\calc\,{=}\,\calz(\Vect(\mathbb Z_2))$ of the category $\Vect(\mathbb Z_2)$ of
$\mathbb Z_2$-graded complex vector spaces, but only 2

   indecomposable
module categories over the fusion category $\Vect(\mathbb Z_2)$. These two
$\Vect(\mathbb Z_2)$-module categories correspond to the two known \cite{brki}

   elementary
boundary conditions for the toric code.

The results obtained above are easily extended to surface defects that separate
three-dimensional topological field theories of Reshetikhin-Turaev type based on two
modular tensor categories $\calc_1$ and $\calc_2$. Again, one starts with a fusion category
$\calw_d$ of defect Wilson lines that are confined to the
defect surface. We now get two monoidal functors
  $$
  \calc_1\xrightarrow{~~} \calw_d\qquad\text{and}\qquad
  \calc_2^{\mathrm{rev}} \xrightarrow{~~} \calw_d

  $$
which combine into a single functor
  $$
  F_d:\quad \calc_1 \,{\boxtimes}\, \calc_2^{\mathrm{rev}} \xrightarrow{~~~} \calw_d \,.
  $$
Again this functor $F_d$ has the structure of a central functor, yielding a functor
  $$
  \widetilde F_d:\quad \calc_1 \,{\boxtimes}\, \calc_2^{\mathrm{rev}} \xrightarrow{~~\simeq~~} \calz(\calw_d)
  $$
that is again argued to be, by naturality, a braided equivalence. This leads to the following
obstruction: surface defects of the type studied here only exist if the classes of the
two modular tensor categories in the Witt group are equal, $[\calc_1] \,{=}\, [\calc_2]$.

Our discussion is, deliberately, not at the highest possible conceptual level.
In analogy with the fact that
Morita equivalent algebras can be naturally discussed in a bicategorical setting with
bimodules as 1-morphisms, in the present situation there is an underlying 3-categorical structure.
Also, we did not
discuss fusion of topological surface defects. Indeed, surface defects form a monoidal bicategory.
The monoidal unit is called the {\em transparent surface defect}, which separates a modular
tensor category $\calc$ from itself. The underlying Witt trivialization is based
on $\calc$, seen as a fusion category, and the braided equivalence
  $$
  \calc\boxtimes \calc^{\mathrm{rev}} \xrightarrow{~~\simeq~~} \calz(\calc)
  $$
which follows by Equation (\ref{cschweigert:cantriv})
from the fact that $\calc$ is modular. Ordinary Wilson lines are then
recovered as defect Wilson lines separating the transparent defect from itself.

\subsection{Defects and boundaries in Dijkgraaf-Witten theories}\label{sec:DW}

We now describe an explicit gauge-theoretic construction \cite{FSV14} of boundary conditions and
surface defects for Dijkgraaf-Witten theories. A key idea is to keep the two-step
procedure outlined in Section \ref{DWsect}: first we assign to manifolds of various dimension
groupoids of (ge\-nera\-lized) bundles, and then we linearize those categories.

Our ansatz is inspired by the notion of a {\em relative bundle}: Given a morphism
$j\colon Y \,{\to}\, X$ of smooth manifolds and a group homomorphism $\iota\colon H \,{\to}\, G$,
the category of relative bundles has as objects triples consisting of a $G$-bundle $P_G \,{\to}\, X$,
an $H$-bundle $P_H \,{\to}\, Y$, and an isomorphism
  $$
  \alpha:\quad \mathrm{Ind}_H^G (P_H) \,\xrightarrow{\,\,\cong\,\,}\, j^* P_G
  $$
of $G$-bundles on $Y$. A morphism is a pair consisting of a morphism
$P_G \,{\xrightarrow{\,\varphi_G^{}\,}}\, P_G'$ of $G$-bundles on $X$
and a morphism $P_H \,{\xrightarrow{\,\varphi_H^{}\,}}\, P_H'$ on $Y$
of $H$-bundles, obeying the obvious compatibility constraint
  $$\xymatrix{
    \mathrm{Ind}_H^G (P_H^{})\ar^{~~~\alpha}[r]\ar_{\mathrm{Ind}_H^G(\varphi_H^{})} [d]
    & j^*P_G^{}\ar^{j^*\!\varphi_G^{}}[d]\\
  \mathrm{Ind}_H^G (P_H')\ar^{~~~\alpha'}[r] &j^*P_G'
  }$$
with the isomorphisms $\alpha$ and $\alpha'$.

In our construction, relative bundles are the appropriate tool for describing boundaries. In the
case of a surface defect $\Sigma \,{\hookrightarrow}\, M$ the situation is somewhat more subtle.
Suppose that $\Sigma$ divides the three-manifold $M$ into two disjoint connected components $M_\pm$.
We consider the situation that Dijk\-graaf-Witten theories for two, possibly different, finite
groups $G_\pm$ are associated to the components $M_+$ and $M_-$. Denote by
$\overline M_\pm \,{:=}\, M_\pm \,{\cup}\, \Sigma$ the closure of $M_\pm$ in $M$, and by
$j_\pm\colon \Sigma \,{\hookrightarrow}\, \overline M_\pm$ the corresponding embeddings.
We then consider, for a given group homomorphism $H\to G_+ \,{\times}\, G_-$ as field
configurations the groupoid whose objects consist of two $G_\pm$-bundles $P_\pm \,{\to}\, M_\pm$,
an $H$-bundle $P_H \,{\to}\, \Sigma$ and an isomorphism
  $$
  \beta :\quad \mathrm{Ind}_H^G(P_H) \,\xrightarrow{\,\,\cong\,\,}\,  j_+^*P_+^{}\times j_-^*P_-^{}
  $$
of $G_+{\times} G_-$-bundles on $\Sigma$.

We have to add yet one further datum to Dijkgraaf-Witten theories. Namely, typically, the
linearization of groupoids is twisted by a 2-cocycle on the relevant groupoid of
generalized bundles. This requires the choice of additional data on the finite groups involved.
We first discuss the situation without boundaries and without surface defects, for a
Dijkgraaf-Witten theory based on a finite group $G$. Then
the additional datum is a three-cocycle $\omega \,{\in}\, Z^3(G,\mathbbm C^\times)$ on the group.
A more geometrically inclined reader is invited to think about this cocycle as a three-cocycle
on the stack $\Bun_G$ of $G$-bundles, and thus as a (Chern-Simons) 2-gerbe on $\Bun_G$.
Such a three-cocycle leads to a holonomy on closed three-manifolds and thus furnishes a
topological Lagrangian, yielding a three-dimensional topological field theory $\tft_{G,\omega}$.
To determine the category $\tft_{G,\omega}({\mathbb S}^1)$, we need a 2-cocycle $\tau(\omega)$
on the groupoid $\Bun_G(\mathbb{S}^1) \,{\simeq}\, G{/}\!{/} G$, the action groupoid for the
adjoint action of $G$ on itself. The transgressed 2-cocycle
$\tau(\omega)$ can be obtained \cite{Wi08} from the group cocycle $\omega$ by
evaluating $\omega$ on a suitable triangulated three-ma\-ni\-fold. This indeed produces
the well-known 2-cocycle \cite{DPR90} for Dijkgraaf-Witten theories.

We now illustrate the generalization of this prescription to boundaries and defects in an
example. Let $\mathbb I$ be an interval with a marked point in its interior, corresponding to
a surface defect. To each of the two subintervals ${\mathbb I}_{1,2}$ we a assign
a finite group $G_{1,2}$ and three-cocycles
$\omega_{1,2} \,{\in}\, Z^3(G_{1,2},\mathbbm C^\times)$, of which we think of as
topological bulk Lagrangians. To the end point adjacent to the first interval,
we assign a group homomorphism $\iota_1\colon\, H_1 \,{\to}\, G_1$ and a boundary Lagrangian
as follows. We think about the three-cocycle $\omega_1$ on $G_1$ as a Chern-Simons 2-gerbe on
$\Bun_{G_1}$. The group homomorphism $\iota_1$ induces a morphism
$\Bun_{H_1} \,{\to}\, \Bun_{G_1}$ of 2-ger\-bes. A similar situation, one categorical dimension lower,
is familiar from the study of D-branes, see e.g.\ \cite{FNSW10}: for D-branes, one has a gerbe on
the bulk manifold and a morphism of gerbes on the boundary, from the trivial gerbe
to the restriction of the bulk gerbe. This is an instance of the general principle that
in gauge theories, holonomies of manifolds with boundary, and thus topological actions
on such manifolds, can only be defined if a trivialization on the boundary is chosen.

In the situation at hand, what is to be trivialized is the restriction of the Chern-Simons
2-gerbe to $\Bun_{H_1}$. Thus in terms of group cochains, we are looking for a
2-cochain $\theta_1 \,{\in}\, C^2(H_1,\mathbbm C^\times)$ that represents a morphism
from the trivial 2-gerbe on $\Bun_H$ to the 2-gerbe described by the restriction of the
three-cocycle $\omega_1$ on $G_1$ to $\Bun_{H_1}$. Accordingly we
impose the condition ${\mathrm d}\theta_1 \,{=}\, \iota_1^*(\omega_1)$. We think about $\theta_1$
as a topological boundary Lagrangian. The situation at the other end point is, in complete
analogy, described by a group homomorphism $\iota_2\colon H_2 \,{\to}\, G_2$ and a 2-cochain
$\theta_2 \,{\in}\, C^2(H_2,\mathbbm C^\times)$ such that
${\mathrm d}\theta_2 \,{=}\, \iota_2^*(\omega_2)$.

In a similar vein, the situation for the surface defect is a higher-categorical analogue
of bibranes \cite{FSW08}, which describe the target space physics of topological defects
in two-dimensional conformal field theory:
We now have a group homomorphism $\iota_{12}\colon H_{12} \,{\to}\, G_1\,{\times}\, G_2$
and a 2-cochain $\theta_{12} \,{\in}\, C^2(H_{12},\mathbbm C^\times)$ satisfying
${\mathrm d}\theta_{12} \,{=}\, \tilde\iota_{2}^*(\omega_2) \,{\cdot}\, \tilde\iota_{1}^*(\omega_1)^{-1}$,
with the group homomorphisms $\tilde\iota_i\colon H_{12} \,{\xrightarrow{\,\,\iota_{12}\,}}\,
G_1 \,{\times}\, G_2 \,{\xrightarrow{\,\,p_i\,}}\, G_i$ obtained from $\iota_{12}$ and the
canonical projections.
Indecomposable boundary conditions and defects correspond to group homomorphisms
$\iota_{1,2}$, respectively $\iota_{12}$, that are injective, i.e.\ are subgroup embeddings.

In \cite{FSV14}, an explicit prescription was given  for transgressing these cochains to
a 2-cocycle on the relevant category of (generalized) relative bundles. This prescription
is a generalization of the one given in \cite{Wi08} for Dijkgraaf-Witten theories without
boundaries or defects. It makes the categories associated to one-manifolds with
defects and boundaries quite explicitly computable.

On the other hand, there are also general results for boundary conditions and defects in
three-dimensional topological field theories of Reshetikhin-Turaev type. Suppose the
theory is based on a modular tensor category $\mathcal C$. The model independent analysis
of Section \ref{sec:tftb} reveals that topological boundary conditions only exist
if the category $\mathcal C$ is braided equivalent to the Drinfeld center of a fusion
category $\mathcal A$. For Dijkgraaf-Witten theories based on a finite group $G$ and
a three-cocycle $\omega \,{\in}\, Z^3(G,\mathbbm C^\times)$, this condition is satisfied
automatically: the category $\mathcal C \,{=}\,\, \mathcal C_{G,\omega}$ is the
Drinfeld center of the category $\Vect(G)^\omega$ of $G$-graded vector spaces
with associator twisted by the three-cocycle $\omega$.

Recall from the general analysis in Section  \ref{sec:tftb}
that boundary conditions are in bijection
with module categories over the fusion category $\mathcal A$. For the fusion category
$\Vect(G)^\omega$ relevant for Dijkgraaf-Witten theories, indecomposable module categories are,
in turn, known \cite{Os03} to be in bijection with subgroups
$\iota\colon H \,{\hookrightarrow}\, G$ and 2-cochains $\theta \,{\in}\, C^2(H,\mathbbm C^\times)$
such that ${\mathrm d}\theta \,{=}\, \iota^*\omega$. This matches exactly the gauge-theoretic
description of types of boundary conditions. According to the general results \cite{FSV13},
the topological field theory should assign to an interval with Dijkgraaf-Witten theory
based on $(G,\omega)$ and boundaries with boundary conditions $(H_1,\theta_1)$ and
$(H_2,\theta_2)$ the category
  $$
  \mathrm{Fun}_{{\Vect(G)}^\omega}({\mathcal M}(H_1,\theta_1), {\mathcal M}(H_2,\theta_2))
  $$
of module functors from the indecomposable module category ${\mathcal M}(H_1,\theta_1)$
over the fusion category $\Vect(G)^\omega_{}$ to the indecomposable module category
${\mathcal M}(H_2,\theta_2)$. A non-trivial calculation \cite{FSV14} shows that
this category coincides, as a finitely semisimple abelian category, with
the category obtained by linearizing categories of generalized relative bundles.

A similar calculation can be performed for the circle with surface defects as point-like
insertions. This requires a higher-categorical version of the notion of a trace, namely
a trace 2-functor on bimodule categories with values in $\mathbbm C$-linear categories.
Such a coherent trace can indeed be constructed \cite[Sect.\,3]{fss} and gives rise
to results that are in accordance with
gauge theoretic calculations in Dijkgraaf-Witten theories \cite[Sect.\,5]{fss}.

\subsection{Symmetries from invertible topological surface defects}\label{sub:syminv}

We now turn to the relation between symmetries of three-dimensional topological field
theories of Turaev-Viro type and invertible surface defects. For such a theory, the modular
tensor category $\mathcal C$ of bulk Wilson lines is the Drinfeld center of a fusion category
$\mathcal A$, i.e.\ ${\mathcal C} \,{=}\, {\mathcal Z}({\mathcal A})$.

According the paradigm discussed in Section 1, symmetries should correspond to invertible topological
surface defects, and their action should be described by wrapping those defects. By the
analysis presented in the preceding subsection, invertible surface defects are described
by invertible bimodule categories over the fusion category $\mathcal A$. These form a
monoidal bicategory; restricting to only invertible 1-morphisms and 2-morphisms in this bicategory,
one obtains a categorical 2-group, the Brauer-Picard 2-group \cite{ENOM10}.
Its isomorphism classes form a finite group $\BrPic(\mathcal A)$; in the sequel
work with this group, rather than the underlying 2-group.

Applying an extended topological field theory to a two-manifold with boundary
yields a functor. Take this two-manifold to be a
cylinder with a surface defect of type $D$ wrapping the non-contractible cycle:

  $$
  \begin{picture}(225,70)
  \put(20,1)  {\includegraphics{pic_fpsv_11A.eps}}
  \put(133.6,32) {\boldmath{$D$}}
  \end{picture}
  $$

This yields an endofunctor $F_D\colon\, {\mathcal C} \,{\to}\, {\mathcal C}$. A detailed
analysis \cite{FPSV14} shows
that if the defect $D$ is invertible, then the functor $F_D$ has the structure of a braided monoidal
autoequivalence. In the case of Dijkgraaf-Witten theories, the modular tensor
category of bulk Wilson lines is ${\mathcal C} \,{=}\, {\mathcal Z}(\Vect(G))$,
and an invertible (and thus indecomposable) defect is given by a subgroup
$\iota\colon H \,{\hookrightarrow}\, G\,{\times}\, G$ and a 2-cochain
$\theta \,{\in}\, C^2(H,\mathbbm C^\times)$. The functor $F_D \,{\equiv}\, F_{(H,\theta)}$
can then be explicitly obtained by linearizing the span of action groupoids
  $$ \xymatrix{
  &H/\!/H\ar^{\hat\pi_2}[dr]\ar_{\hat \pi_1}[dl]&\\
  G/\!/G && G/\!/G
  }$$
for adjoint actions which describe the relevant categories of
(relative) bundles. Here $\hat \pi_i$ is the functor on groupoids that is induced from the
group homomorphism $H\,{\xrightarrow{\,\iota\,}}\, G \,{\times}\, G\,{\xrightarrow{\,p_i\,}}\, G$.
One can then compute a pull-push functor on the linearizations to arrive at the functor
  $$
  F_{(H,\theta)}:\quad {\mathcal Z}(\Vect(G))=[G/\!/G,\Vect]
  \longrightarrow [G/\!/G,\Vect] \,,
  $$
which comes with a natural monoidal structure and which is braided. This construction
provides an explicit group homomorphism
  $$
  \BrPic(\mathcal A) \longrightarrow \mathrm{brdEq} ({\mathcal Z}(\Vect(G))) \,
  $$
from the Brauer-Picard group to the group $\mathrm{brdEq} ({\mathcal Z}(\Vect(G)))$
of braided autoequivalences of the Drinfeld center of $\Vect(G)$. A detailed comparison
\cite{FPSV14} shows that this group homomorphism is exactly the one considered in
\cite{ENOM10}; this embeds the construction of that paper naturally
into the framework of topological field theories with defects. It is also shown in \cite{ENOM10}
that the group homomorphism is actually an \emph{iso}morphism. According to the considerations
above, this representation theoretic result
can be reinterpreted as the statement that symmetries of topological field theories of
Turaev-Viro type can already be detected from their action on the category of bulk Wilson lines.

\subsection{Symmetries for abelian Dijkgraaf-Witten theories}

We now restrict our attention to the class of Dijkgraaf-Witten theories with vanishing
three-cocycle and based on a finite abelian group $A$. In this case the Brauer-Picard group
is explicitly known \cite{ENOM10}:
  $$
  \BrPic(\Vect(A)) \cong \mathrm{O}_q(A\oplus A^*)\,,
  $$
where $A^*$ is the character group of $A$, $q\colon A\,{\oplus}\, A^*\,{\to}\,\mathbbm C^\times$
is the natural quadratic form that is determined by $q(g,\chi) \,{=}\, \chi(g)$ for $g\,{\in}\,A$
and $\chi \,{\in}\, A^*$, and $\mathrm{O}_q(A{\oplus} A^*)$ is the subgroup of
those automorphisms of the finite group $A\,{\oplus}\, A^*$ that preserve this quadratic form $q$.

There are three obvious types of symmetries the Dijkgraaf-Witten theory should possess:

\\[4pt]
(1)\, \emph{Symmetries of} $\Bun_A$.
\\[2pt]
As explained in Section \ref{DWbound}, we can think about the stack $\Bun_A$ as a target space
for Dijkgraaf-Witten theories. Accordingly, symmetries of $\Bun_A$, i.e.\ group automorphisms of
the abelian group $A$, can be expected to be kinematical symmetries of the Dijkgraaf-Witten theory.
\\
Indeed, for $\varphi \,{\in}\, \mathrm{Aut}(A)$ the graph
{\sl graph}$(\varphi)\,{\subset}\, A\,{\oplus}\, A$ is a subgroup; together with the trivial
2-cochain $\theta \,{=}\, 1\,{\in}\, C^2(${\sl graph}$(\varphi),\mathbbm C^\times)$
it describes an invertible surface defect. Following the prescription given in Section
\ref{sub:syminv}, we compute the corresponding braided autoequivalence to be the element
  $$
  \varphi\oplus (\varphi^*)^{-1}:\quad A\oplus A^* \to A\oplus A^*
  $$
of $\mathrm{O}_q(A{\oplus} A^*)$.

\\[4pt]
(2)\, \emph{Automorphisms of the Chern-Si\-mons $2$-gerbe on} $\Bun_A$.
\\[2pt]
Recall that the three-cocycle $\omega\,{\in}\, C^3(A,\mathbbm C^\times)$ has the
interpretation of a Chern-Simons 2-gerbe on $\Bun_A$. In the case at hand this 2-gerbe
is trivial. Nevertheless its automorphisms are not trivial;
rather, they are given by 1-gerbes on $\Bun_A$.
Their isomorphism classs are given by the cohomology group $H^2(A,\mathbbm C^\times)$.
By transgression \cite{Wi08}, any cohomology class gives an alternating bihomomorphism
$\beta\colon A \,{\times}\, A \,{\to}\, \mathbbm C^\times$, which in physical terms
can be interpreted as a ``B-field''. (Indeed, for a
finite abelian group, transgression provides a group isomorphism between the
cohomology group $H^2(A,\mathbbm C^\times)$ and the group of alternating bihomomorphisms.)
\\
Again we can explicitly identify an invertible topological surface defect: the one
given by the diagonal subgroup $A_{\mathrm{diag}}\,{\subset}\, A\,{\oplus}\, A$ together with
a representative of a cohomology class in $H^2(A,\mathbbm C^\times)$, which we identify
with the associated alternating bihomomorphism $\beta$. The corresponding braided
equivalence can be computed to act as
  $$
  \begin{array}{rll}
  A\oplus A^* &\!\!\longrightarrow \!\!& A\oplus A^* \\[4pt]

  (g,\chi)&\!\! \longmapsto \!\!& (g,\chi+\beta(g,-)) \,.
  \end{array}
  $$
Thus this type of automorphism shifts the character by the contraction of the B-field
$\beta$ with the group element.
\\[4pt]

(3)\, \emph{Electric-magnetic dualities.}
\\[2pt]
It is finally expected that electric-magnetic dualities form a class
of symmetries of Dijk\-graaf-Witten theories (see e.g.\ \cite{BCKA13} for
a discussion of such symmetries for Turaev-Viro theories based on finite-dimensional
semisimple Hopf algebras).
\\
To present a simple example, assume that $A$ is a cyclic group and fix a
group isomorphism $\delta\colon A\,{\to}\, A^*$. Then there is a natural braided
equivalence
  $$
  \begin{array}{rll}
  A\oplus A^* &\!\!\longrightarrow \!\!& A\oplus A^*\\[4pt]
  (g,\chi)&\!\! \longmapsto \!\!& (\delta^{-1}(\chi),\delta(g)) \,.
  \end{array}
  $$
This can again be described by an invertible surface defect, namely the one given
by the diagonal subgroup $A_{\mathrm{diag}}\,{\subset}\, A\,{\oplus}\, A$ and
the alternating bihomomorphism $\beta$ defined by
  $$
  \beta(g_1,g_2) := \frac{\delta(g_1)(g_2)}{\delta(g_2)(g_1)}
  \quad\text{ for }~ g_1,g_2\,{\in}\, A\,.
  $$

A careful investigation \cite{FPSV14} of the structure of the finite
group $\mathrm{O}_q(A\oplus A^*)$ shows that in fact the three types of symmetries
listed above generate this group, i.e.\ they already provide \emph{all}
symmetries of the theory.

\section{Conclusions}

Here is a brief summary of crucial insights presented in this contribution:

\begin{itemize}
\item
Topological defects are important structures in quantum field theories.
\\[-8pt]~

\item
By general principles they describe symmetries and dualities of quantum field theories
in such a way that the action of these symmetries and dualities on all different
structures of a quantum field theory becomes apparent. This includes the action on
field insertions as well as the action on boundary conditions and defects of various
codimensions.
\\[-8pt]~

\item
Topological codimension-1 defects and boundaries provide a natural setting for
studying relative field theories. This includes in particular anomalous field theories and
the TFT construction of RCFT correlators.
\\[-8pt]~

\item
Applications of topological defects virtually concern all fields of physics
to which quantum field theory is applied.
\end{itemize}

\noindent
There are several classes of quantum field theories in which topological defects can be
treated quite explicitly. Besides two-dimensional rational conformal field theories, these
include three-dimensional topological field theories of Reshetikhin-Turaev type.
We have shown that a natural class of defects and boundary conditions in such theories
is obstructed, with an obstruction taking values in the Witt group of non-degenerate braided
fusion categories. We have also
demonstrated, for the subclass of Dijkgraaf-Witten theories, how defects and boundary conditions
can be formulated in a gauge-theoretic setting using (generalizations of) relative bundles.
This provides a natural embedding of structures of categorified representation theory, in
particular of the theory of module and bimodule categories over monoidal categories, into the
more comprehensive setting of (extended) topological field theory.

\bibliographystyle{amsalpha}

redpajama_set_name: RedPajamaArXiv


text:
Terrorist Attack on Somalia
This past Saturday, an explosion occurred in the capital city of Mogadishu. The attack killed more than 300 people and injured more than 300.
Two truck bomb explosions were reported to be Al-Qaeda linked. Officials have reported that this is the most massive, deadliest attack to happen in Somalia. President Mohamed Abdullahi declared a 3 day mourning period for the ravaged country.
Somalia government has stated that the event occurred at the hands of al-Shabab extremist group. This jihadist group is known to target areas of high profile of the capital. The group has promised to carry out more attacks as a result of announced bilateral military efforts with the Trump administration against the group. Many are outraged because of the lack of media coverage of this massive attack. Emily Dauth a senior at harborfields says " A lot of horrible things happen every day throughout the world but a lot of attention is focused on things closer to home. Even more recently the news has been focusing on all the hurricanes and tropical storms that have been happening, but not addressing issues such as this one."
Why has terrorism in Somalia gone unnoticed? Many have said they felt as though it's unjust that Las Vegas shooting gets a bizarre amount of coverage and support, while the incident in Somalia is going unnoticed. Senior Lauren Banks stated, "I just think it's not really fair that a lot of people have died and no one knows or is mourning them."
The US stated that, "Such cowardly attacks reinvigorate the commitment of the United States to assist our Somali and African Union partners to combat the scourge of terrorism to promote stability and prosperity for the Somali people and their regional neighbors."
Many people are left with nothing, and Somalia need for help is so extensive that President Abdullahi has issued a plea for blood donations to aid the wounded.
Fortunately, International aid is on course to help Somalia. According to the Miami Herald, "A United States military plane landed in Somalia's capital Tuesday with medical and humanitarian aid supplies. Dozens of critically injured have been airlifted to Turkey for treatment. A medical team from Djibouti arrived Monday to evacuate the wounded. Kenya will evacuate 31 injured Somalis from Mogadishu for specialized treatment in Nairobi and send 11 tons of medicine and supplies to Somalia, Kenya announced Tuesday."
All in all, death tolls are rising and the country of Somalia is in desperate need of aid to help them through their most devastating attack yet. Since the attack, thounands of Somali's have marched through the capitcal city of Mogadishu in opposition to the the terrorist attack.
Sarah-Elizabeth Leveque October 19, 2017
No More Vaping in the Bathroom!
Las Vegas Massacre
Sarah-Elizabeth Leveque October 5, 2017

redpajama_set_name: RedPajamaCommonCrawl


Le Toy Van camper van - this wooden toy camper van is so adorable! Designed just like the traditional VW camper van, it comes with a detachable surf board that sits on the roof, internal kitchenette and padded mattress. The roof comes off for easy access and imaginative play.


Dark Justice Series – A 'dark' satire on the state of race relations in the US – News and Views!
(hosted on funnyordie.com) Che Holloway (Rochester, NY – Right) stars as Police Officer Johnson, the only African-American Police Officer in a predominantly white town.
Can we as a nation LAUGH about the current racial tension in America?
Well, according to Mike Gerbino (Director) and Che Holloway (Actor), both upstate natives of New York, we should be doing exactly that as a means to explore and address the complex dynamic that is race relations in the US.
Dark Justice is a six part webseries that focuses on a small-town police station becoming a conduit for challenging stereotypes surrounding African Americans, and also white privilege that obstructs constructive criticism of the majority group in society. Che Holloway, a young (25 years old) up and coming actor who splits his time between New York City and Rochester, sat down with me to talk about the series and his passion for acting ….
Saj: The web series deals with a lot of issues to do with racial tension. How did you get to play a part in the production?
Che: It was interesting … the writer/director Mike Gerbino we had a mutual friend Alan Williams who told me about it … As a young black male, the script really resonated with me.
Saj: So, in total, how long did it take to produce?
Che: I wanna say about 2 weeks total … in reality the production was split into parts due to taking breaks … we filmed the first episode a year before the rest of them, just to test the waters.
Saj: And you are originally from Rochester, right? You split your time between NYC and Rochester during filming, how did that work?
Che: Yeah, well I am a native of Rochester …. I started my acting career at AMDA (The American Musical and Drama Academy in New York City) so I moved there for a year, and stayed afterwards, working acting jobs. Then I decided to come home. It is pretty cool, I went out there to survive, and then I came back.
Saj: So, what sparked your interest in acting?
Che: Well, this is going to sound really funny but when I was around 10, I was watching Harry Potter, and I was thinking y'know these kids are around my age, and I could do that. So that's what started the 'fire'. And a couple of years after that I went to School of the Arts in Rochester, graduated, from there I did a Geva theatre Summer Academy, and I went to AMDA. My first national gig was on the Oprah Winfrey Network, on the season 2 finale of Unfaithful.
Saj: Going back to the Dark Justice series, explain a little about the storyline. Why is comedy a good way to approach this subject matter? And how was it for you as a person of color performing some of the material?
Che: It was hard personally in the sense that I worried about what the backlash could be. In the end I decided life was too short, people will always have opinions good or bad, so why not just deliver. The series makes you think, it isn't that far from reality, a little exaggerated sure, but definitely close to reality at times. We are hoping that with enough interest, we can put it on something like Hulu or Netflix.
Saj: Thinking about the way that justice is served, I was watching Dateline the other night (the Charlie Tan case that made national news), and I had a thought that if he hadn't been an individual from a well-off community, and instead had been african-american, this may have played out differently ….
Che: Oh yeah, I agree with that wholeheartedly.
Saj: And have you ever personally experienced stereotyping?
Che: Well a couple of years ago, I was in a play, To Kill a Mockingbird, in a small town near here (which I won't name). As I left the back, there was a man on a motorbike who just looked at me and said "you need to get out of here, you don't belong here". I didn't know what to do, I just thought, here I am far away from home, in his neck of the woods, so if I were to say or do anything I would be seen as the aggressor, even though I wasn't. I'm more cerebral, I think about things before I act. So, I took a chill pill, told him "have some gum" (had to get one in) and then left. It was a little awkward, you see those kinds of things happen on TV, and don't expect them to happen in real life.
Saj: So … you mentioned you are an alum of School of the Arts. There are obviously a few famous individuals who also graduated from there (Taye Diggs and Seymour Dustin Hoffman to name but two). Do you feel that pressure of having a lot to live up to?
Che: Laughs oh yes, I have big boots to fill. I got to meet Taye Diggs (former alumni of School of the Arts) while he was performing in Hedwig and the Angry Inch, and we took a selfie of course (laughs). He is a really good guy.
Saj: So what else do you have in the pipeline?
Che: I have a project called Elysian fields, which is kind of like a combination of Othello and Django Unchained. And this summer we are going to be shooting season 2 of Dark Justice.
Saj: Sounds like you are keeping busy!
Eastleigh: What are the implications for this swing-seat for the UK and Europe?


Progress have launched a campaign Prime Time to try to turn the talk about using primaries to select Labour candidates into action.
As part of this, Will Straw has written a new paper, extending the argument for primaries which he and Nick Anstead made in their Fabian pamphlet 'The Change We Need'.
The paper seeks to address criticisms including that made by Stuart White here on Next Left.
Jessica Asato writes on LabourList, addressing the most common objections to primaries. Long-standing advocate of primaries Anthony Painter has blogged on the campaign too.
I have also written a short commentary for Progress. At present, there are hopes and fears about what primaries would mean. I think we should hold some pilots and experiments to test these, so we can have a more informed debate after the next election.
It would make sense to use the opportunity of the increased number of late retirements to experiment with some late selections in this Parliament.
If Labour is considering primaries, then there is an especially strong case for a primary to select the next London Mayoral candidate. David Lammy, having argued for primaries in a Fabian lecture over a year ago, wrote in the Standard on Friday advocating primaries for the London Mayoral selections ahead of 2012.
Imagine if the parties held a primary election in every London borough. Candidates would have to campaign across the capital, building momentum, borough by borough, community by community. No longer could politicians get away with just campaigning in their heartlands. No longer would Labour voters in Hillingdon or Havering feel ignored by those at the top, since their votes would count for just as much as a Labour voter in Hackney. And yes, it would work the other way round, too, with Tory voters of Newham having their say in their selection.
The Guardian's Dave Hill saw this as a Lammy bid to run but also wrote on the merits of London primaries as perhaps the best way to engage in and resolve the party's debate about its London strategy.
As open a selection as possible in the 2012 Mayoral cycle could both maximise Labour's chances of retaining City Hall and be one of the most important ways Labour could start to engage with the emerging 'movement politics'.
In selecting a candidate for a directly elected position - where there is preferential and so cross-party voting - a broad range of London Labour opinion ought to be able to agree to the principle of an open contest. Indeed, if Ken Livingstone does confirm that he will run again, he would begin with considerable advantages in profile and name recognition. But I would imagine that those from the other wing of the party who are advocating primaries would have to say 'let the best man or woman win', rather than adopting an instrumental approach depending on whether they expected to do well.
But I think it would be a particularly good idea to build support on this now before the debate is dominated by which specific possible candidates might gain from such an approach - and to avoid the shambles and stitch-ups of Labour's London selection in 2000.
But perhaps primary momentum would carry us even further than that. Tom Harris says the 'inevitable consequence' of primaries would lead political parties to use a primary to elect their leaders too. He intended this partly as a warning as to whether the argument for primaries stacks up, while noting parties might find the attractions in the idea too.
Andrew Sparrow has written about this for the Guardian. He picks up on a comment which I made on Harris' thread, I expect the Conservatives would have little difficulty in replacing the current members' run-off of the final two candidates with an open vote among those who register to participate. I predict they will hold a leadership election on that model within the next 10 years. Labour would find that rather more difficult, because of its electoral college and federal structure.
There are certainly barriers to change - and questions which the primaries campaign needs to answer. But they are asking some important questions of their own too.
I support closed primaries, where party members can vote for a PPC.
But open primaries would just make elections into such a long and arduous process where government is seen as this far away and dangerous place, that we could end up with an American style anti-government culture, which would make ending child poverty and funding public services much harder.
I've had a quick read of Will's new pamphlet. I'm still unconvinced that Will has taken the full measure of the criticisms of 'open primaries' as illiberal and bad for democracy.
In the section of his paper addressing criticisms, he doesn't directly address the criticism that open primaries threaten freedom of association by opening up candidate selection to the general public (who may or may not share the values of the party). This concern is addressed implicitly later in the paper where Will suggests a 'semi-open primary' as a possible model: would-be participants are required to sign up to a statement of party values before they can vote in the primary. This would go some way to address the concern. But Will also leaves open the possibility of a purely open primary which takes us back to the original problem - we would just have to hope that those voting in the primary share enough of a party's values not to undermine it as an effective community of shared belief.
Will does directly address the criticism that open primaries risk being bad for democracy by pushing candidate selection towards the preferences of the median voter. His argument is that it only pushes selection closer to the median voter among supporters of the party.
Two points on that. First, if the primary is purely open, then it isn't true that it will only push selection closer to the median voter among a given party's supporters. The selection, by definition, is open to all voters.
Second, if party supporters tend to have views closer to the median voter in the population at large than party members, then it will still be the case that opening up selection to party supporters, even under a relatively closed primary system, will tend to push candidate selection closer to the views of the median voter in the population as a whole.
This, of course, has an upside. But it has a downside, too. In the long-run it will tend to reduce the scope and depth of democratic deliberation as candidates all get drawn from, and reflect, the centre ground.
Take one example: inheritance tax. This is an issue where we need a social democratic party to get out there and make the case against the current grain of public opinion. It may well be - I stand to be corrected - that a majority of Labour supporters currently share the widespread public opposition to inheritance tax. It should be the job of party members to work to win over party supporters, as part of the job of winning over the wider public in general. But wouldn't even a relatively closed primary system work against this? Wouldn't the pressure be on to reflect back existing public attitudes rather than to challenge them where they need challenging?
Thanks very much for engaging on the substance of my paper. Both your points about freedom of association and democracy are interesting but only valid, I believe, in the unique case of a free-for-all open primary where anyone can vote. This is only practiced in the US state of Wisconsin and I oppose it.
There is a danger of getting bogged down in the definitions since my preferred option can be referred to as either semi-open or semi-closed and these terms mean different things to different people. Instead, let me set out what I see as the problem with free-for-all open primaries and the rules that I would put in place to address them.
In a primary system where anyone can vote regardless of their values, the result could become diluted (ie approach the median voter) and become open to sabotage. I think you can avoid this with (i) a primary election day for all parties where voters have to choose which party to vote for OR (ii) a Labour-only primary election day where people who want to vote have to register first. The leap of logic I make here is that members of the public just won't bother to register if they don't share Labour's values. The feared "entryism" has not happened in the US nor did it happen in Totnes.
Your point on IHT is fascinating and raises a good question about how we avoid candidates playing to the gallery in primary elections. Progress are advocating Labour members' choosing the short list as the Tories did in Totnes. I didn't cover this in my paper but can see that it could be used to weed out candidates who oppose IHT and other key Labour policies.
I am really unconvinced by the worry about the median voters.
If there were a Labour primary in Hayes and Harlington, John McDonnell would walk it - if there were a London wide primary Ken would win, if there were a Birkenhead primary, Frank Field might well get beat by a local leftie.
Swing State Project has some excellent work comparing representatives on a left-right scale to the electoral make-up of their district, and it certainly isn't the case that you only get progressives winning primaries in progressive districts - organisation seems to matter a lot more than ideology.
Thanks for these comments, Will, which do go further to allay some of my anxieties (and maybe I would have spared you the trouble if I'd read the paper less quickly). I think the idea of party members controlling the short list takes some sting out of both criticisms.
Also, to engage in a bit of self-criticism, it occurs to me that in one sense the freedom of association criticism is misplaced - or, rather, it might not be quite right to state the criticism in these terms.
Are we talking about the state requiring parties to hold primaries, or parties volunteering to do so? If the latter, then, strictly speaking, the primary is an exercise of freedom of association, not a curtailment of it. It might still be criticised, if the design is wrong, as a strange, self-denying exrecise of freedom of association ('Come and pick our candidate for us - we are not worthy!'), but its not a restriction of freedom of association by the state.
I still worry that in an open primary - even when all parties hold them on the same day and you can only participate in one - there is a possibility of candidate selection being affected by non-supporters. But you are right that the extent of this problem is an empirical question and if the empirics suggest its not usually a big deal, then maybe...its not a big deal.
Thanks for comments and interesting discussion.
Stuart, In replying to yr initial post agin primaries, I thought that the freedom of association argument could work that way around, ie for the opportunity to choose to open the process up, if chosen by the party rather than mandated by the state.
It seems to me that the question of whether primaries would lead to a greater diversity or rather more homogenity of candidates (in terms of their views, and in terms of the diversity of candidates in other ways) is an unknown, as we have very little evidence as to how voter preferences in Britain would operate if a wide electorate had a choice between candidates of the same party.
Sunder: I'm sure your valid point about freedom of association filtered into my head, got filed away, and finally came back into my consciousness this lunchtime!
On your last point (and also Don's point), yes, I agree it is an unknown. I should have stated my criticism in terms of what could happen rather than in terms of what will happen.
I am persuaded that its worth experimenting with primaries and seeing what happens.


text:
Home » Players » Steve Smith
WATCH: Australian fans in uproar as TV umpire denies Steve Smith a stunning catch – AUS vs SA, 3rd Test
Posted On / January 9, 2023 / by Anirudh Singh
The third umpire ruled out the catch taken by Steve Smith in the second slip.
The third Test between Australia and South Africa ended in a draw.
Third umpire ruled out the catch taken by Steve Smith (Photo Source: Twitter)
Australia superstar Steve Smith almost had another screamer to add to his long list of catches, producing a brilliant effort during the third and last Test against South Africa in Sydney, only to be denied by the TV umpire.
After beating Dean Elgar multiple times during the opening overs of South Africa's first innings, Australia seamer Josh Hazlewood found the edge of the Proteas skipper's bat on the last ball of his second over.
Fielding at the second slip, Smith dived forward to his right towards the dipping ball and momentarily looked as if he had taken a clean catch. His teammates celebrated in unison as if Smith had taken the catch of the series – which he would have – but the umpires weren't convinced at all. They sent the decision upstairs to the third umpire with a soft signal of 'not out'.
"I don't think Steve Smith's totally convinced, but on the angles and replays we've seen already, this looks like it's carried," said cricketer-turned-commentator Ricky Ponting on air.
Meanwhile, TV umpire Richard Kettleborough continued to have a look at the replays from different angles.
"(The ball) has actually carried into his hand on the full," added Ponting.
"That has not bounced before it hit his hand, but you heard the third umpire say, 'his fingers have split'. So, they're trying to check with his fingers being split if any part of that ball has actually hit the grass after it's gone into his hand on the full.
"The ball has not bounced before it's got to him; it's gone into his hands on the full. And now the judgement that the third umpire has to make now … does the ball then go down onto the ground and up?
"It's very, very hard to tell, even from the angle that we've got."
After several minutes, Kettelborough eventually gave the decision in batsman's favour, much to the disappointment of the local crowd.
Not out!
The third umpire decided this effort from Steve Smith touched the ground and Dean Elgar lives on #AUSvSA pic.twitter.com/fy8bI6CAS6
— cricket.com.au (@cricketcomau) January 7, 2023
Some Australian fans wanted to let the decision slip away from their memory. "The legend @bhogleharsha said: you always see what you want to see in these situations, it's always 50:50. Move on," a fan wrote.
After declaring their innings at 475/4 with the help of Usman Khawaja's unbeaten 195 and Smith's 104, Australia bundled out South Africa on 255 in the first innings and kept them under the pump in the second essay.
The visitors score read 106/2 when players of both sides shook hands and accepted the draw. Australia sealed the series 2-0 with comprehensive victories in Brisbane and Melbourne.
TAGS: Steve Smith
CATEGORY: Steve Smith, Video


text:
Q: Using JSTL, jsp, Jquery to create nested comment (Java servlet) I created the JavaBean for Comments and CommentDAO, and I successfully send them to jsp file. However, I would like to create a nested structure by adding padding left to the element but I couldn't add the comments that are grandchild of the parent comment.
Here is the code:
public class Comments {
    private int commentID;
    private String comment;
    private int parentID;
    private int userID;
    private int articleID;
    private Boolean hide;
    private int commentDepth;
    private int childCount;

    public void setCommentDepth(int commentDepth) {
        this.commentDepth = commentDepth;
    }

    public int getCommentDepth() {
        return commentDepth;
    }

    public void setChildCount(int childCount) {
        this.childCount = childCount;
    }

    public int getChildCount() {
        return childCount;
    }

    public void setHide(Boolean hide){ this.hide=hide; }

    public Boolean getHide(){ return hide; }

    public void setArticleID(int articleID) {
        this.articleID = articleID;
    }

    public int getArticleID() {
        return articleID;
    }

    public void setUserID(int userID) {
        this.userID = userID;
    }

    public int getUserID() {
        return userID;
    }

    public void setCommentID(int commentID) {
        this.commentID = commentID;
    }

    public int getCommentID() {
        return commentID;
    }

    public void setComment(String comment) {
        this.comment = comment;
    }

    public String getComment() {
        return comment;
    }

    public void setParentID(int parentID) {
        this.parentID = parentID;
    }

    public int getParentID() {
        return parentID;
    }
}

Change comments to ThreadedComments which contains childCount and commentDepth
public static List<Comments> toThreadedComments(List<Comments> comments){

        //The resulting array of threaded comments
        List<Comments> threaded = new ArrayList<>();

        //An array used to hold processed comments which should be removed at the end of the cycle
        List<Comments> removeComments = new ArrayList<Comments>();

        //get the root comments first (comments with no parent)
        for(int i = 0; i < comments.size(); i++){
            Comments c = comments.get(i);
            if(c.getParentID() == 0){
                c.setCommentDepth(0); //A property of Comment to hold its depth
                c.setChildCount(0); //A property of Comment to hold its child count
                threaded.add(c);
                removeComments.add(c);
            }
        }

        if(removeComments.size() > 0){
            //clear processed comments
            comments.removeAll(removeComments);
            removeComments.clear();
        }

        int depth = 0;
        //get the child comments up to a max depth of 10
        while(comments.size() > 0 && depth <= 10){
            depth++;
            for(int j = 0; j< comments.size(); j++){
                Comments child = comments.get(j);
                //check root comments for match
                for(int i = 0; i < threaded.size(); i++){
                    Comments parent = threaded.get(i);
                    if(parent.getCommentID() == child.getParentID()){
                        parent.setChildCount(parent.getChildCount()+1);
                        child.setCommentDepth(depth+parent.getCommentDepth());
                        threaded.add(i+parent.getChildCount(),child);
                        removeComments.add(child);
                        continue;
                    }
                }
            }
            if(removeComments.size() > 0){
                //clear processed comments
                comments.removeAll(removeComments);
                removeComments.clear();
            }
        }

        return threaded;
    }

function to get allArticleComments:
 public static List<Comments> getArticleComment(int articleID) {
        List<Comments> articleComments = new ArrayList<>();
****connection to database and return all comments with the same ArticleID.

}

Jstl to print the nested comments: (I cant figure out how to use recursive function using javascript)
<c:if test="${threadedComments !=null}">
  <c:forEach items="${threadedComments}" var="comments">
    <c:if test="${comments.childCount > 0}">

${comments.commentID}: ${comments.comment}

        <c:forEach items="${threadedComments}" var="nestedcomments">
          <c:if test="${nestedcomments.parentID == comments.commentID}">

${nestedcomments.commentID}:${nestedcomments.comment}

          </c:if>
        </c:forEach>
    </c:if>
  </c:forEach>
</c:if>

send JavaBean from servlet:
List<Comments> allComments = CommentDAO.getArticleComment(articleID);
                List<Comments> threadedComments = CommentDAO.toThreadedComments(allComments);

                request.setAttribute("threadedComments" , threadedComments);

redpajama_set_name: RedPajamaStackExchange


text:
I'm Thirsty is a full review and alcohol history podcast. Dave and John sit down to taste the latest out there while telling everyone willing to listen all about the history of the alcohol that they review. Though they might not have a certificate to consider themselves experts, these two know a thing or two about drinking. So sit back, relax, and enjoy a nice beverage as Dave and John introduce you to the wonderful world of spirits.
A monthly podcast.
Episode 12: Fruit Inspired Beer
Dave and John are back to discuss the important things in life - beer! In this episode, the two taste test fruit inspired beers.
Episode 11: We are Stout Men
This month the boys are back to give you the info on stouts and review three very popular ones. They also add in a new beer that was just released but isn't a stout (Trust us, you'll want to hear this review). Check out what they say about what they...
By John Bruske | December 2, 2015 | 0 comments |
Episode 10: IPA's and Pale Ales
Dave and John are back! In this double length episode (had to make up for not recording in September), Dave and John finally move the show to the "I'm Thirsty" portion. They give a wonderful introduction into the world of beer for beginners and continue on by taste sampling some...
By John Bruske | October 26, 2015 | 0 comments |
Episode 9: Back to School Comes Full Circle
The boys are at it again. They went back in time to 1986 to watch a classic comedy called Back to School starring Rodney Dangerfield, and a cast of amazing people. They paired it with New Holland Brewery's Full Circle. A delicious micro brew from the west side of Michigan....
By John Bruske | August 23, 2015 | 0 comments |
Episode 8: Get Off the Sandlot, Melonhead
The boys are back at it again, this time sitting on a porch on a warm summer evening talking about an American classic and having some beer. John and Dave will discuss their movie selection process and then get into their discussion of the amazing 1993 classic, "The Sandlot." Of...
By John Bruske | July 26, 2015 | 0 comments |
Episode 7: Steamy Windows and Steamy Glasses – Revenge of the Nerds
The boys are back with beer in hand to watch the 1984 classic, "Revenge of the Nerds." A movie about nerds and their quest for acceptance and love. And as always John and Dave pair it with a beverage. Now Dave found it hard to find the exact beer he...
By John Bruske | June 28, 2015 | 0 comments |
Episode 6: It's a Cool World but this isn't Your Father's Root Beer
This month, your friends John and Dave get together to watch Cool World (1992), and pair it with a delicious Root Beer flavored beer. The two give their feelings on the movie and also on this beer. Want to know what they are? Listen and find out.
By John Bruske | May 25, 2015 | 0 comments |
Episode 5: Across the Universe with a Black and Tan in My Hand
In this episode, Dave flies solo and calls an audible to the schedule. He decided to hold off on doing the animated film and tried to cheer himself up by watching one of his go to movies. He decided to watch Across the Universe that is based around The Beatles...
By John Bruske | April 29, 2015 | 0 comments |
Episode 4: Grimm Love and Grapefruit Shandy
The 'And Scene' boys are back with their latest movie, that comes as a suggestion from Justin Corbett from the Comical Podcast, Grimm Love. And as always these two pair it with a beer they think you should check out. So this episode they went with a Leinenkugel Grapefruit Shandy....
By John Bruske | March 29, 2015 | 0 comments |
Episode 3: She's Out of My League, Lucky Girl
In this episode the beer drinking guys that watch and review movies do a 180 and surprise the women in their lives with a chick flick for Valentine's Day. These two did a movie draft to see what movie they would watch and Dave goes out and picks up a...