text:
\section{Gauge anomalies for associative algebras}\label{sec:warmup}

The mathematics in this section is not new. Its goal is to present a familiar story in a way that will generalize to VOAs. A good source for the complete story is \cite{MR2266528}.

\subsection{Azumaya algebras}
A $\bK$-algebra $A$ is \define{Azumaya} if it is finite-dimensional and simple, and its centre is exactly $\bK$. Let $A^\op$ denote the opposite algebra. Then $A$ is Azumaya if and only if its \define{enveloping algebra} $A \otimes A^\op$ is isomorphic to a matrix algebra. This is in turn equivalent to $A \otimes A^\op$ being Morita-trivial. Modules for $A \otimes A^\op$ are precisely $A$-$A$-bimodules, and so an Azumaya algebra is an algebra whose (monoidal) category of bimodules is  equivalent to $\Vect$.

\subsection{Physical interpretation}\label{subsec:azumaya-physics}
Azumaya algebras have the following interpretation in quantum physics. Consider a (one-dimensional) chain of atoms. To each atom we associate a ``local Hilbert space'' given by the underlying vector space of~$A$ (so that a basis of $A$ are the ``spins'' of the ``spin chain'');
  if the atoms are index by $i \in \{\dots, 1,2,\dots\}$, then we will write $A_i$ for the copy of $A$ at location $i$.
 The total Hilbert space is the tensor product $\bigotimes_i A_i$ of all the local Hilbert spaces. (That tensor product makes sense only for chains of finite length.)

 We now place a Hamiltonian on this Hilbert space, which will be of ``commuting projector'' type. Specifically, consider the multiplication map $m : A \otimes A \to A$. It is a map of $A$-$A$-bimodules, and by our Azumaya condition corresponds to a linear map $B \to \bK$, where we have used the identification  $\cat{Bimod}(A) \simeq \Vect$ to identify ${_A A_A} \in \cat{Bimod}(A)$ with $\bK \in \Vect$ and to identify ${_A A \otimes A_A} \in \cat{Bimod}(A)$ with some vector space $B \in \Vect$. (The vector space $B$ will be noncanonically isomorphic to the underlying vector space of $A$.) The map $B \to \bK$ is nonzero, hence a surjection, and so can be split: we can choose a bimodule map $\Delta : {_A A_A} \to {_A A \otimes A_A}$ such that $m \circ \Delta = \id_A$. Then $\Delta \circ m : A^{\otimes 2} \to A^{\otimes 2}$ is a projection (it squares to itself) onto a vector space isomorphic to $A$.
 Let $\id_i$ denote the identity on $A_i$, and for a set $I$ of indices let $\id_I = \bigotimes_{i\in I} \id_i$. Also write $(\Delta \circ m)_{i,i+1}$ for the map $\Delta \circ m$ acting on $A_i \otimes A_{i+1}$.
 The Hamiltonian we will use is:
 $$\sum_{i} \id_{<i} \otimes (\id - \Delta \circ m)_{i,i+1} \otimes \id_{>i+1}.$$

 This Hamiltonian is gapped (i.e.\ it has a finite-dimensional lowest eigenspace, and its spectrum is discrete), and its vector space of ground states  is isomorphic to $A$. Since the chain of atoms was $1$-space-dimensional, together with the Hamiltonian we have a ``gapped topological system of $(1+1)$-dimensional matter.''
 It is expected that gapped physical systems admit low-energy effective descriptions in terms of TQFTs. For the spin-chain model built from $A$, the low-energy TQFT is the one constructed from $A$ a la \cite{Schommer-Pries:thesis}.

The gapped topological phase (meaning the connected component of the space of all gapped systems under deformations that do not close the gap) constructed from a semisimple algebra $A$ depends only on the Morita equivalence class of $A$
(see e.g.\ \cite[\S2.4]{GJFa}).
To encode the algebra $A$ itself, one can remember not just the phase of $(1+1)$-dimensional matter, but also a choice of boundary condition. Namely, the spin-chain constructed above has an obvious ``free'' boundary condition, whose
boundary observables form a copy of the associative algebra $A$. The bulk observables live in the centre $\cZ(A)$, which is trivial by assumption.
Thus the bulk theory is almost trivial:
it is ``short-range entangled.''
The class of $A$ in the Brauer group $\Br(\bK)$ can be considered a ``gravitational anomaly'' for the boundary system: the boundary system fails to be an absolute $(0+1)$-dimensional QFT only because of the Brauer class of $A$.
 Some discussion of absolute versus anomalous quantum systems, in the framework of TQFTs, is available in \cite{MR3165462,me-Heisenberg}; for a condensed matter discussion focussing on gauge anomalies, see \cite[\S7]{GJFa}.

\subsection{Construction of gauge anomalies}\label{assoc-construction}
Let $A$ be Azumaya and $G \subset \Aut(A)$ a finite group of automorphisms of $A$. (One can be more general, but we will not.) We will associate to these data a \define{gauge anomaly} $\alpha \in \H^2(G; \bK^\times)$, where $G$ acts trivially on $\bK^\times$.

There are many ways to define $\alpha$. The quickest is to consider the exact sequence, valid for any algebra,
\begin{equation}
 \{1\} \to \cZ(A^\times) \to A^\times \to \Aut(A) \to \Out(A) \to \{1\} , \label{eqn-assoc}
\end{equation}
where the map $A^\times \to \Aut(A)$ picks out the inner automorphisms, and $\Out(A)$ denotes the group of outer automorphisms of $A$. Two simplifications occur when $A$ is Azumaya. First, from the definition in terms of central simple algebras, we see that $\cZ(A^\times) = \bK^\times$. Second, $\Out(A)$ is trivial by the Skolem--Noether theorem. Let us quickly prove the second claim. Any automorphism $\phi : A\to A$ determines an invertible bimodule $A_\phi \in \cat{Bimod}(A)$, which is $A$ as a left $A$-module, but for which the right $A$-action is twisted by $\phi$.
But $\cat{Bimod}(A) \simeq \Vect$, so
 there must be a bimodule isomorphism $A_\phi \isom A$. This map must take the unit $1 \in A_\phi$ to some (necessarily invertible) element $f\in A$, and compatibility with the left $A$-action says that it takes $a \mapsto af$. But compatibility with the right $A$-action then says that $a \phi(b) f= afb$ for all $a$, and so $\phi$ is inner. All together, the sequence
$$ \{1\} \to \bK^\times \to A^\times \to \Aut(A) \to \{1\}$$
is seen to be exact, and a central extension. The action $G \to \Aut(A)$ pulls back this central extension to an extension of $G$, classified by the anomaly $\alpha \in \H^2(G; \bK^\times)$.

In terms of the physical description from \S\ref{subsec:azumaya-physics}, the anomaly $\alpha$ arises as follows. The action of $G$ on $A$ determines an action of $G$ on the $(1+1)$-dimensional phase of matter built from $A$. Stack this $G$-equivariant phase of matter with a copy of the phase of matter built from $A^\op$ with trivial $G$-action. (``Stacking'' means to consider the noninteracting tensor product of the two physical systems. Physically, you place them on separate sheets separated by an infinitely strong insulating material, and then press the combined solar panel into a single layer.) Ignoring the $G$-symmetry, we find the trivial phase of matter, as it is built from the Morita-trivial algebra $A \otimes A^\op$. But the $G$-equivariance is nontrivial:
we have built a ``symmetry-protected trivial'' phase. Bosonic SPT phases in $1+1$ dimensions are classified by ordinary group cohomology.
(In the condensed matter sense, that claim is a widely believed assertion rather than a theorem; c.f.\ \cite[\S5.1]{GJFa}. It is a theorem in the TQFT sense: it is equivalent to the abstract algebraic construction given above.)

There is a final construction of the anomaly $\alpha$ which is useful to describe, since it is the easiest to generalize to the VOA setting. (We will describe it assuming the action of $G$ on $A$ is faithful; it is not hard to extend the construction to the non-faithful case.) Consider the subalgebra $A^G \subset A$ fixed by the $G$-action. Provided $G$ is finite and $\bK$ has characteristic $0$, this subalgebra remains semisimple. Its \define{commutant} is:
$$ B := \{b \in A \st [b,a] = 0 \;  \forall a \in A^G\}.$$
We claim that $B$ is a twisted group algebra for $G$. To prove this, for each $g \in G$ set:
$$ B_g := \{b \in A \st ba = g(a)b \; \forall a \in A\}.$$
Then $B_g$ is one-dimensional by the Skolem--Noether theorem (it is at most one-dimensional since $\cZ(A) \cong \bK$, and it is at least one-dimensional since we have already shown that all automorphisms of $A$ are inner) and $B_g \cap B_{g'} = \{0\}$ if $g \neq g'$ since the action is faithful. It is clear that $B_g B_{g'} = B_{gg'}$, and so $\bigoplus_{g\in G} B_g \subset A$ is a subalgebra. But the commutant of $\bigoplus_{g\in G} B_g$ in $A$ is  $A^G$, and so
$$ B = \bigoplus_{g\in G} B_g$$
by the double commutant theorem.

Finally, being a twisted group algebra, $B$ determines a cohomology class $\alpha \in \H^2(G; \bK^\times)$, easily identified with the cohomology class that classified the central extension discussed above.

\subsection{Gauging}\label{assoc-gauging}

Gauge anomalies arise as the obstruction to gauging symmetries. Consider first the case $A = \operatorname{Mat}_n(\bC)$, which corresponds via the bulk-boundary construction of \S\ref{subsec:azumaya-physics} to a quantum mechanics model with Hilbert space~$\bC^n$. A gaugeable group of symmetries is a group~$G$ acting on~$\bC^n$; then the gauged theory has as its Hilbert space the fixed subspace $(\bC^n)^G$. The anomaly must be trivialized in order to promote a group of automorphisms of $A$ to a group of automorphisms of the Hilbert space.

For associative algebras, gauging is also called ``quantum Hamiltonian reduction'';
the choice of trivialization of the anomaly corresponds to the ``quantum comoment map''.
Quantum Hamiltonian reduction can be written in various ways. Here is one.
Suppose that $G$ acts on $A$ with trivialized anomaly, meaning that we have chosen a way to lift  the homomorphism $G \to \Aut(A)$ to a homomorphism $G \to A^\times$. Then in particular the fixed subalgebra $A^G$ and the untwisted group algebras $\bK[G]$ are commutants in the Azumaya algebra $A$. This provides an isomorphism between their centres: $\cZ = \cZ(A^G) \cong \cZ(\bK[G])$, the algebra of class functions. The homomorphism $G \to \{1\}$ determines a homomorphism $\cZ(\bK[G]) \to \bK$. Specifically, $\cZ(\bK[G])$ arises as the algebra of $\bK$-valued global functions on the Deligne--Mumford stack $G/G$ corresponding to the adjoint action, and $\bK = \cO(\mathrm{pt}/G)$, and the map $\cZ(\bK[G]) \to \bK$ is restriction along the inclusion $\mathrm{pt}/G \to G/G$.
 The \define{gauging} of $A$ by $G$ is by definition the associative algebra:
\begin{equation}\label{assoc-gauging-eqn}
 A\sslash^\beta G := A^G \otimes_{\cZ} \bK.
\end{equation}
The superscript emphasizes that the gauging depends on the choice of trivialization $\beta$ of the anomaly $\alpha$.
The set of trivializations, if nonempty, is a torsor for $\H^1(G; \bK^\times)$.

It can happen that $A \sslash^\beta G$ is the zero algebra. If it is not zero, then it is Azumaya.
One way to see this is to realize $A \sslash^\beta G$ as the algebra of endomorphisms of the left $A$-module $A \otimes_G \bK$ (where $G$ acts on $A$ by right multiplication via the homomorphism $\beta : G \to A^\times$).
Indeed, such a description makes clear that if $A\sslash^\beta G \neq 0$, then $A$ and $A\sslash^\beta G$ are Morita equivalent (compare \cite{MR3581593}).

\subsection{Galois action and the cohomological Brauer group} \label{assoc-cohom-Brauer}

Given a $\bK$-algebra $A$ and a field automorphism $\gamma$ of $\bK$, there is a new $\bK$-algebra $^\gamma A$ which is equal to $A$ as a ring, but with $\bK$-structure twisted by $\gamma$. If $A$ is Azumaya then so is $^\gamma A$, and if $A$ has an action of $G$ by algebra automorphisms, then so does $^\gamma A$. There are no surprises about how anomalies transform under twisting by $\gamma$: if $G$ acts on $A$ with gauge anomaly $\alpha \in \H^2(G; \bK^\times)$, then it acts on $^\gamma A$ with anomaly~$\gamma(\alpha)$. In other words, the anomaly $\alpha$ has coefficients not just in $\bK^\times$-as-an-abelian-group, but in $\bK^\times$-as-a-Galois-module.

This Galois equivariance leads to a cohomological characterization of Azumaya algebras. Suppose that $A$ and $B$ are Azumaya algebras over $\bK$ and that $\bK \subset \bL$ is a Galois extension with Galois group $\Gal(\bL/\bK)$ such that $A \otimes_\bK \bL \cong B \otimes_\bK\bL$ as $\bL$-algebras; let $f : A \otimes_\bK \bL \to B \otimes_\bK \bL$ denote a choice of isomorphism. Then $A \otimes_\bK \bL$ carries two different $\Gal(\bL/\bK)$-actions, providing the different descent data: letting $\rho$ denote the standard action of $\Gal(\bL/\bK)$ on $\bK$, the two actions are $\rho_A: \gamma \mapsto \id_A \otimes \rho(\gamma)$ and $\rho_B : \gamma \mapsto f^{-1} \circ (\id_B \otimes \rho(\gamma)) \circ f$. Neither $\rho_A(\gamma)$ nor $\rho_B(\gamma)$ is $\bL$-linear, but the composition
$$ \tilde\kappa ( \gamma) := \rho_B(\gamma) \circ \rho_A(\gamma)^{-1} $$
is. This $\tilde\kappa$ is not a homomorphism from $\Gal(\bL/\bK)$ to $\Aut(A\otimes_\bK \bL)$, but it is a twisted cocycle --- $\rho_A$ determines an action of $\Gal(\bL/\bK)$ on $\Aut(A\otimes_\bK \bL)$ --- and so determines a twisted cohomology class
$$ \kappa \in \H^1(\Gal(\bL/\bK); \Aut(A\otimes_\bK \bL)).$$
It is a basic theorem that this class, together with $A$, determines the isomorphism type of $B$.

The earlier discussion of gauge anomalies amounts to the fact that there is a universal class $\alpha^{\text{univ}} \in \H^2(\Aut(A \otimes_\bK \bL); \bL^\times)$ classifying the extension $1 \to \bL^\times \to (A \otimes_\bK \bL)^\times \to \Aut(A\otimes_\bK\bL) \to 1$. Actually, our discussion did not quite imply this, because we worked only with finite groups $G$. But the image of the cocycle $\tilde\kappa$, together with its Galois-conjugates, is a finite subset of $\Aut(A \otimes_\bK \bL)$ which is easily seen to be closed under composition, and so a subgroup, and so the earlier discussion does imply that $\alpha^{\text{univ}}$, even if it did not exist on all of $\Aut(A \otimes_\bK \bL)$, does exist after being restricted to the image of $\kappa$.

The composition of $\alpha^{\text{univ}} \circ \kappa$ is a class in the twisted cohomology $\H^2(\Gal(\bL/\bK); \bL^\times)$. Given the Morita class of $A$, this twisted cohomology class determines the Morita equivalence class of $B$.

Finally, it is a fundamental fact that for every Azumaya algebra $B$ over $\bK$, there is a Galois extension $\bK \subset \bL$ such that $B \otimes_\bK \bL \cong \operatorname{Mat}_n(\bL)$ for some $n$.
Fixing $A = \operatorname{Mat}_n(\bK)$ and taking the union over extensions $\bL \supset \bK$, one discovers:
$$ \Br(\bK) = \{\text{Azumaya algebras over $\bK$ up to Morita equivalence}\} \cong \H^2(\Gal(\bar{\bK}/\bK); (\bar{\bK})^\times).$$
The left-hand side is called the \define{Brauer group} of $\bK$ and the right-hand side is called the \define{cohomological Brauer group}. (In positive characteristic, $\bar\bK$ means the separable, rather than algebraic, closure.)

\section{Gauge anomalies for VOAs}\label{sec:VOAs}

In the remainder of this paper, we will repeat the story from \S\ref{sec:warmup} with associative algebras (the observables for $(0+1)$-dimensional quantum field theories) replaced by vertex operator algebras ($(1+1)$-dimensional quantum field theories). The question of how to gauge an action of a finite group on a VOA was first studied in \cite{MR1003430,MR1128130}. The mathematical construction of gauge anomalies is due to \cite{MR1923177}; this section consists primarily of an interpretation of that paper.

\subsection{Holomorphic VOAs} \label{voas-holo}

We will not review the definition of vertex operator algebra ---
an excellent reference is \cite{MR2023933}
--- preferring instead to indicate the meanings of words used in VOA theory by analogizing with the associative algebra case. We first point out that the notion of VOA is purely algebraic: it makes sense over any field.
We will work only over $\bC$ and its sub-fields, which  simplifies the story. (In positive characteristic, one must take care with divided powers. The  introduction of vertex algebras in \cite{MR843307} already included the positive-characteristic case.) In particular, if $V$ is a VOA over $\bK$, then so are its Galois conjugates $^\gamma V$ for $\gamma \in \Gal(\bK/\bQ)$.
In this paper VOAs are always
simple,
self-dual (i.e.\ admitting an invariant nondegenerate bilinear form),
and
$\bN$-graded by the action of $L_0$.

Given a VOA $V$, there are various categories of ``$V$-modules,'' with different regularity conditions placed on the module. We will focus on ``admissible'' modules (we will not give the definition), writing $\Rep(V)$ for the category thereof.
VOA modules correspond not to left-modules for an associative algebra $A$ but rather are analogous to $A$-$A$-bimodules: $\Rep(V)$ is the VOA analog of $\cat{Bimod}(A)$, and the Zhu algebra of $V$ (defined so that
the irreducible $V$-modules are in natural bijection with irreducible modules over the Zhu algebra; Zhu's construction in \cite{MR1317233} is technical, but see \cite{MR2568399} for a readable survey)
  is analogous to the enveloping algebra $A \otimes A^\op$.

The analog of semisimplicity for an algebra is called ``regularity'': a VOA $V$ is \define{regular} if it is $C_2$-cofinite and the category $\Rep(V)$ of admissible modules is semisimple with finitely many simples (a condition called ``rationality,'' but
 this paper avoids that term so as not to conflict with the question of definability over $\bQ$).
Suppose that we are working over $\bK = \bC$. Then regularity implies that $\Rep(V)$ is a modular tensor category
\cite{MR2468370}.
The braided monoidal structure is the VOA analog of the fact that $\cat{Bimod}(A)$ is naturally a monoidal category.

The word analogous to ``Azumaya'' is ``holomorphic'': $V$ is \define{holomorphic} if $\Rep(V) \simeq \Vect$.

\subsection{Physical interpretation} \label{subsec:voa-physics}

It is believed that $(2+1)$-dimensional topological phases of matter are classified by MTCs together with some mild extra data about the central charge. (Namely, each MTC has a central charge $c$ living in $\bQ/8\bZ$; the extra data is a lift to $\bQ$.  The root of unity $\alpha = \exp(2 \pi i c/8)$ is sometimes called the \define{(gravitational) anomaly} of the MTC.) There is no proof of this belief, largely due to the lack of a definition of ``$(2+1)$-dimensional topological phase of matter,'' but a TQFT version is proven in \cite{BDSPV}, where it is confirmed that the Reshetikhin--Turaev construction \cite{MR1091619} gives a precise classification of once-extended 3d TQFTs. If $V$ is a regular VOA over $\bC$, then the MTC $\Rep(V)$ determines a $(2+1)$-dimensional topological phase of matter, and the VOA $V$  arises as the algebra of observables for a distinguished $(1+1)$-dimensional chiral conformal boundary condition.

If $V$ is holomorphic, then the bulk $(2+1)$-dimensional phase of matter is almost trivial: it is a short range entangled (SRE) phase, encoding only the central charge of $V$.
(The central charge corresponds to the ``beyond group cohomology layer'' in \cite[\S5.2]{GJFa}.)
Consider a bulk-boundary system with $V$ determining the boundary physics and with bulk physics this SRE phase. Since the bulk phase is SRE, any choice of local boundary coordinates identifies the boundary physics with a true quantum field theory, but the theory isn't truly conformal: conformal transformations change the partition function of the theory in a simple way determined by $c$, and so $c$, or equivalently the bulks SRE phase, encodes the ``gravitational anomaly'' of the holomorphic CFT described by $V$.

\subsection{Construction of gauge anomalies} \label{voa-gauge-construction}

Let $V$ be holomorphic VOA defined over $\bC$ and let $G \subset \Aut(V)$ be a finite group of VOA automorphisms of $V$. We will associate to these data a \define{gauge anomaly} $\alpha \in \H^3(G;\bC^\times)$. We will do so by imitating the construction from \S\ref{assoc-construction}. There is no VOA analog of the exact sequence (\ref{eqn-assoc}). But we can consider the analog of the commutant subalgebras $A^G \subset A \supset B \cong \bK^\alpha[G]$.

First, it makes perfect sense in VOA theory to consider the $G$-fixed subalgebra $V^G \subset V$.
There is a version of commutant subalgebras that can be said entirely within the language of vertex algebras, called the ``coset construction'' \cite{MR778819,MR1159433}: the \define{coset} of a sub-vertex-algebra $W \subset V$ is the sub-vertex-algebra $W' \subset V$ consisting of all elements of $V$ that have trivial operator product expansion with all elements of $W$.
In our case, however, $V^G \subset V$ is a conformal embedding (i.e.\ the image of $V^G$ contains the Virasoro vector of $V$)
and so, since $V$ is assumed simple,  $(V^G)'$ is the trivial VOA.

In order to have a meaningful commutant of $V^G \subset V$, we must leave the world of VOAs, which encode the vertex operators in a chiral CFT, and also consider topological line operators. One expects on physical grounds that topological line operators for a fixed CFT form a category, because there can be topological junctions between line operators, and that they can be fused, providing the category of line operators with a monoidal structure. (A typical CFT will have infinitely many nonisomorphic simple topological line operators and so the category of line operators is not fusion. C.f.\ \cite{Andre-bicommutant}, which uses the term ``soliton'' for what we call ``topological line operator.'')

For each $g\in G$, consider the category $\cC_g$ of topological line operators $X$
 such that, for each vertex operator $v \in V$,
 $$
 \begin{tikzpicture}[baseline=(v.base)]
   \draw[ultra thick] (0,0) -- (0,2); \path (.25,.25) node {$X$}; \path (1,1) node (v) {$\bullet \,v$};
   \draw[ultra thin] (-2,0) -- (2,0) -- (2,2) -- (-2,2) -- cycle;
 \end{tikzpicture}
 \quad=\quad
 \begin{tikzpicture}[baseline=(v.base)]
   \draw[ultra thick] (0,0) -- (0,2); \path (.25,.25) node {$X$}; \path (-1,1) node (v) {$\bullet \,g(v)$};
   \draw[ultra thin] (-2,0) -- (2,0) -- (2,2) -- (-2,2) -- cycle;
 \end{tikzpicture}
  $$
 in the sense that if you take the operator-valued holomorphic function ``insert the vertex operator $v$ at a point to the right of the line operator $X$'' and analytically continue it over the $X$-line, you get the operator-valued holomorphic function ``insert the vertex operator $g(v)$ at a point to the left of the line operator $X$.''

There is a natural functor from this category $\cC_g$ to what in the VOA literature is called the category of ``$g$-twisted $V$-modules.''
Indeed, a typical topological line operator $X$ in a CFT does not admit any topological ``endings,'' i.e.\ there are usually no topological junctions between $X$ and the trivial line operator. But there is a vector space of chiral conformal endings of $X$. (Compare: The topological endings of the trivial line operator are the vertex operators in $V$ of conformal dimension $0$, which is to say just the multiples of the vacuum. The chiral conformal endings of the trivial line are all of $V$.) When $X$ is a line operator satisfying $Xv = g(v)X$, the space of chiral conformal endings for $X$ is a $g$-twisted module.

The functor $\cC_g \to \{\text{$g$-twisted $V$-modules}\}$, $X \mapsto \{\text{endings of $X$}\}$, is expected to be an equivalence, and so we will conflate the two categories. The only way it could fail to be an equivalence is if there are line operators in $\cC_g$ which admit no endings at all. Such line operators can exist in a TQFT (assuming the phrase ``TQFT'' implicitly restricts just to topological endings of topological line operators) but should not exist for a CFT described by a VOA.

When $V$ is holomorphic, the category $\cC_g$ is (noncanonically) equivalent to $\Vect$ \cite[Theorem 2]{MR1794264}.
Consider the direct sum of categories $\cC = \bigoplus \cC_g$. Under the physical expectation that line operators can be fused, $\cC$ will be a $G$-pointed fusion category.
The nontrivial data of a $G$-pointed fusion category is its associator; up to equivalence of $G$-pointed fusion categories, that data is a cohomology class $\alpha \in \H^3(G;\bC^\times)$ \cite[Example 2.3.8 and Proposition 2.6.1]{EGNO}. By construction $V^G$ is the commutant VOA of $\cC \subset V$, since it consists of those vertex operators that commute with the line operators in $\cC$,
and one expects a ``double commutant theorem'' realizing $\cC = \bigoplus \cC_g$ as the commutant of $V^G$.

Actually, working purely in VOAs, it is not known that one can fuse arbitrary twisted modules. We will therefore impose the following widely-believed assumption:

\mytheorem[Regularity Assumption]{Let $V$ be a holomorphic VOA and $G \subset \Aut(V)$ a finite group. Then the fixed sub-VOA $V^G$ is regular.}

\noindent The main result of \cite{CarnahanMiyamoto} implies that this Regularity Assumption holds if $G$ is solvable.
With the Regularity Assumption in place, the results of \cite{MR1923177} fully justify the story told in this section: in particular, the category $\cC$ is fusion, and is the commutant of $V^G$, and so the gauge anomaly $\alpha \in \H^3(G;\bC^\times)$ is defined.

\subsection{Gauging}\label{voa-gauging}

To justify the name ``gauge anomaly,'' we must construct, for each trivialization~$\beta$ of~$\alpha$, a \define{gauging} $V\sslash^\beta G$ of $V$. In the physicist's style we will first give a recipe for $V\sslash^\beta G$. In \S\ref{voa-linalg} we will explain the question that this recipe answers.
The recipe will imitate the quantum Hamiltonian reduction formula~(\ref{assoc-gauging-eqn})
from \S\ref{assoc-gauging}: for Azumaya algebras, the gauging was $A \sslash^\beta G := A^G \otimes_{\cZ} \cO(\mathrm{pt}/G)$, where $\cZ \cong \cZ(A^G) \cong \cZ(B)$ and where $\beta$ identified the twisted group algebra $B = \bigoplus B_g \cong \bK^\alpha[G]$ with the untwisted group algebra $\bK[G]$.

In the VOA case, the gauge anomaly $\alpha$ arises as the associator on the $G$-pointed fusion category $\cC = \bigoplus_g \cC_g$. The trivialization is therefore the data of an equivalence $\beta : \cC \simeq \Vect[G]$, where $\Vect[G]$ means the $G$-pointed fusion category of $G$-graded vector spaces with trivial associator. This is directly analogous with the story about $B \cong \bK^\alpha[G]
\overset \beta
\cong \bK[G]$.

We need an analog for the centre $\cZ(A^G)$. The \define{centre} of any algebra is its commutant as a subalgebra of itself. For $W$ a regular vertex algebra, we could take the coset of $W \subset W$, but we would get a trivial answer. Instead, as in \S\ref{voa-gauge-construction}, we will take its categorical commutant, i.e.\ the category of the line operators on $W$ that commute with all vertex operators in $W$. The category of such line operators is precisely $\Rep(W)$: each line operator has a $W$-module worth of chiral conformal endings (and we take for granted the assumption that every line operator has endings, so that the functor from line operators to modules is an equivalence).

A second justification for setting $\cZ(W) := \Rep(W)$ comes from comparing \S\ref{subsec:azumaya-physics} and \S\ref{subsec:voa-physics}. The centre $\cZ(B)$ of a semisimple algebra $B$ arises as the algebra of operators for the $(1+1)$-dimensional topological order constructed from $B$, whereas $\Rep(W)$ are the operators for the $(2+1)$-dimensional topological order constructed from the regular VOA $W$.

The isomorphism $\cZ(A^G) \cong \cZ(B)$ used in \S\ref{assoc-gauging} has an analog: conditional on the Regularity Assumption, it is shown in \cite{MR1923177} that there is a canonical equivalence $\cZ(V^G) := \Rep(V^G) \simeq \cZ(\cC)$, where the right-hand side denotes the Drinfeld centre of the fusion category $\cC$.

Moreover, analogous to the map $\cZ(\bK[G]) = \cO(G/G) \to \bK = \cO(\mathrm{pt}/G)$, there is a canonical map $\cZ(\Vect[G]) \to \Rep(G)$, coming from realizing $\cZ(\Vect[G])$ and $\Rep(G)$ as categories of sheaves on $G/G$ and $\mathrm{pt}/G$ respectively. The map $\cZ(\Vect[G]) \to \Rep(G)$ extends to an equivalence $\cZ(\Vect[G]) \simeq \cZ(\Rep(G))$ of MTCs \cite[Example 3.15]{MR3077244}.
One can recover the trivialization $\beta$ from the composition $\cZ := \cZ(\cC) \overset\beta\simeq \cZ(\Vect[G]) \to \Rep(G)$.

All together, we can analogize  equation~(\ref{assoc-gauging-eqn}) to
\begin{equation} \label{voa-gauging-eqn}
  V\sslash^\beta G := V^G \otimes_\cZ \Rep(G).
\end{equation}
The last step is to explain how to tensor a VOA with a fusion category. The short answer is that $V\sslash^\beta G$ is a conformal extension of $V^G$ determined by $\cZ \to \Rep(G)$: the map $V^G \mono V\sslash^\beta G$ is analogous to the map $A^G \epi A \sslash^\beta G := A^G \otimes_\cZ \bK$; in categories, there is no real distinction between quotients and extensions (e.g.\ extensions of algebras often lead to quotients of module categories).

The long answer is that the data of the map $\cZ \to \Rep(G)$ is equivalent to the data of an ``\'etale algebra object'' $E \in \cZ$, i.e.\ a separable braided-commutative algebra which contains only one copy of the unit object of $\cZ$. The equivalence identifies $\Rep(G)$ as the category of $E$-module-objects in $\cZ$, and identifies $E$ as the $\cZ$-object worth of ``internal endomorphisms'' of the unit in $\Rep(G)$.
But \'etale algebra objects in $\Rep(V^G)$ are precisely conformal extensions of $V^G$.
All together, equation~(\ref{voa-gauging-eqn}) says that $V\sslash^\beta G$ is the conformal extension of $V^G$ corresponding to the \'etale algebra $E \in \Rep(V^G)$ corresponding to the map $\Rep(V^G) \overset\beta\simeq \cZ(\Vect[G]) \to \Rep(G)$.

The \'etale algebra object $E \in \cZ$ is ``lagrangian'' in the sense that it realizes $\cZ$ as the Drinfeld centre of $\Rep(G)$. It follows that $V \sslash^\beta G$ is holomorphic.

\subsection{Linear algebraic description}\label{voa-linalg}

The previous sections provided a recipe that took in a holomorphic VOA $V$ and a finite group $G\subset \Aut(V)$ and a trivialization $\beta$ of the anomaly and produced (assuming the Regularity Assumption) a VOA $V \sslash^\beta G$ that we called the ``gauging'' of $V$ by $G$. We did not explain abstractly what ``gauging'' means, nor did we prove that one needs precisely a trivialization of $\beta$. We will do so now. Specifically, we will arrive at the answer by explaining how $V \sslash^\beta G$ decomposes as a module over its subalgebra $V^G$. We call this a ``linear algebraic description'' of $V \sslash^\beta G$ since we will not try to describe the operator product.

The first step is to describe the category $\Rep(V^G)$ in which $V \sslash^\beta G$ lives. Suppose that $G$ acts with anomaly $\alpha \in \H^3(G;\bC^\times)$, which we don't yet assume to be trivializable; then $\Rep(V^G) \simeq \cZ(\Vect^\alpha[G])$.
Slightly abusively, let us write $G/G$ for the set of conjugacy classes in $G$, and for each $g \in G/G$ choose a representative also called $g$.  Let $C(g) \subset G$ denote its centralizer. There is a \define{slant product} map $\iota_g : \H^3(G;\bC^\times) \to \H^2(C(g);\bC^\times)$, given on cocycle representatives by
$$ (\iota_g\alpha)(x,y) = \alpha(g,x,y) - \alpha(x,g,y) + \alpha(x,y,g). $$
(Here and throughout, we write the group law on  $\H^\bullet(G;\bC^\times)$ additively.)

Write $\Rep^{\iota_g\alpha}(C(g))$ for the $\iota_g\alpha$-block in the category of projective $C(g)$-modules.
Then, as a linear category,
$$ \cZ(\Vect^\alpha[G]) \simeq \bigoplus_{g\in G/G} \Rep^{\iota_g\alpha}(C(g)).$$
For each $g \in G/G$, let $V_g \in \cC_g$ denote (a choice for) the simple $g$-twisted $V$-module. Then $V_g$ is naturally a $\iota_g\alpha$-projective $C(g)$-module, i.e.\ $V_g \in  \Rep^{\iota_g\alpha}(C(g))$.
In fact, under the equivalence $\Rep(V^G) \simeq \cZ(\Vect^\alpha[G])$, the simple objects of $\Rep^{\iota_g\alpha}(C(g))$ are identified with the $V^G$-submodules of $V_g$.

Suppose now that we choose a trivialization $\beta$ of $\alpha$. Then $\iota_g\beta$ (represented by the cocycle $(\iota_g\beta)(x) = \beta(g,x) - \beta(x,g)$) is a trivialization of $\iota_g\alpha$, i.e.\ it determines a one-dimensional $\iota_g\alpha$-projective $C(g)$-module that we will call $\bC_{\iota_g\beta} \in \Rep^{\iota_g\alpha}(C(g))$. Given $M \in \Rep^{\iota_g\alpha}(C(g))$, let $M^{\iota_g\beta} := \hom(\bC_{\iota_g\beta}, M)$. It is the space of ``fixed points'' of the $C(g)$-action if we use $\iota_g\beta$ to identify $\Rep^{\iota_g\alpha}(C(g))$ with $\Rep(C(g))$.

Then, as a representation of $V^G$, we have
\begin{equation} \label{voa-linalg-eqn}
 V \sslash^\beta G \cong \bigoplus_{g \in G/G} V_g^{\iota_g\beta}.
\end{equation}
One can show this by inspecting the lagrangian \'etale algebra object in $\Rep(\Vect[G])$ corresponding to the fusion category $\Rep(G)$.

An immediate consequence of equation~(\ref{voa-linalg-eqn}) is that the only common $V^G$-submodule of $V$ and $V \sslash^\beta G$ is $V^G = V_e^{\iota_e\alpha}$ itself (where of course $e\in G$ denotes the unit). Based on this, we take:

\mytheorem[Definition]{Let $V$ be a holomorphic VOA and $G \subset \Aut(V)$ a finite group of automorphisms such that the Regularity Assumption holds. A \define{gauging of $V$ by $G$} is any holomorphic VOA $V\sslash G$ extending the fixed subalgebra $V^G$ such that the only common $V^G$-submodule shared by $V$ and $V\sslash G$ is $V^G$ itself. The action of $G$ on $V$ is \define{nonanomalous} if a gauging exists.}

We have seen that if the anomaly admits a trivialization $\beta$, then $V \sslash^\beta G$ is a gauging in this sense.
Conversely, by studying lagrangian algebras in $\cZ(\Vect^\alpha[G])$, one can show that the action of $G$ on $V$ is nonanomalous if and only if the anomaly is trivializable, and that gaugings are in bijection with trivializations.
\vspace*{-3pt}

\section{Main result}\label{sec:mainthm}

Most of \S\ref{sec:VOAs} requires working over the complex numbers $\bC$ for a simple reason: there is no purely algebraic way to talk about ``line operators,'' because there is no purely algebraic way to talk about real submanifolds of $\bC$. Said another way, the MTC structure on $\Rep(W)$, for $W$ a regular VOA, requires {a priori} that we work over $\bC$: algebraically, one can contemplate intertwining operators parameterized by configurations of points on $\bP^1$; only after taking $\bC$-points can one identify configuration spaces on $\bP^1$ with classifying spaces for braid groups. Indeed, the construction of the MTC structure on $\Rep(W)$ given in \cite{MR2468370} repeatedly chooses parameters living on real intervals in $\bC$.

Thus the  definition of  the anomaly $\alpha$ of an action of $G$ on a holomorphic VOA $V$ requires that $V$ be defined over $\bC$. But suppose that $\gamma$ is a field automorphism of $\bC$. Then there is a Galois-conjugate VOA $^\gamma V$ equal to $V$ as a VOA over $\bQ$ but with the $\bC$-structure twisted by $\gamma$. It is also holomorphic, since its linear category of representations is defined algebraically, and also has a $G$-action, hence its own anomaly.
Our goal in this section is to prove:

\mytheorem[Main Theorem]{
Let $V$ be a holomorphic VOA over $\bC$ and $G \subset \Aut(V)$ a finite group of automorphisms, and let $\gamma$ be a field automorphism of $\bC$ and $^\gamma V$ the Galois-conjugate VOA. Conditional on the Regularity Assumption, let $\alpha \in \H^3(G;\bC^\times)$ denote the anomaly of the action of $G$ on $V$. Then the anomaly of the action of $G$ on $^\gamma V$ is $\gamma^2(\alpha)$.
}\vspace*{-7pt}

\subsection{Recollection on the cyclotomic Galois group} \label{galois-recollections}

Let $\bmu \subset \bC^\times$ denote the group of roots of unity. We will write $\bmu$ additively; it is isomorphic to $\bQ/\bZ$ via the exponential map $x \mapsto \exp(2\pi i x)$. If $G$ is a finite group, then the map $\H^\bullet(G;\bmu) \to \H^\bullet(G;\bC^\times)$ is an isomorphism for $\bullet \geq 1$.

If $\gamma \in \Aut(\bC)$ and $x \in \bmu$ has order $N$, then $\gamma(x) \in \bmu$ also has order $N$. It follows that $\gamma$ acts on $\bmu$ by multiplication by some $n \in \widehat{\bZ}^\times$, where $\widehat{\bZ}$ denotes the profinite completion of $\bZ$ and $\widehat{\bZ}^\times$ is its group of units.
The map $\gamma \mapsto n : \Aut(\bC) \to \widehat{\bZ}^\times$ is a surjection of $\Aut(\bC)$ onto the cyclotomic Galois group $\Gal(\bQ^{\mathrm{cyc}}/\bQ) \cong \Aut(\bmu) \cong \widehat{\bZ}^\times$.

\subsection{Nonanomalous actions} \label{nonanom}

There are two purely algebraic parts of \S\ref{sec:VOAs}. First, the definitions in \S\ref{voas-holo} of the words ``holomorphic'' and ``regular'' refer only to the structure of the linear category of representations, and so if $V$ is holomorphic, then $^\gamma V$ is as well. Second, the definition of a ``gauging'' of $V$ by $G$ given in \S\ref{voa-linalg} is purely algebraic.
To see this, first note that $(^\gamma V)^G = {^\gamma (V^G)}$, and so the expression $^\gamma V^G$ is unambiguous.
Second, suppose that  $V\sslash G$ is a gauging of $V$ by $G$, i.e.\ a holomorphic extension of $V^G$ which, as a $V^G$-module, shares only the submodule $V^G$ with~$V$. Then $^\gamma(V\sslash G)$ is a gauging of $^\gamma V$ by $G$, since the only common $^\gamma V^G$-submodule shared by $^\gamma V$ and $^\gamma(V\sslash G)$ is $^\gamma V^G$. In summary:

\mytheorem[Lemma]{If the action of $G$ on $V$ is nonanomalous, then so is the action of $G$ on $^\gamma V$. \qed}

\pagebreak[0]
It is not hard to prove that if $G$ acts on $V_1$ with anomaly $\alpha_1$ and on $V_2$ with anomaly $\alpha_2$, then it acts on $V_1 \otimes V_2$ with anomaly $\alpha_1 + \alpha_2$, where we have written the group law on $\H^3(G;\bC^\times)$ additively.

\mytheorem[Corollary]{Suppose that $G$ acts on $V_1$ with anomaly $\alpha$ and on $V_2$ with anomaly $-\alpha$, and suppose that the Main Theorem holds for $V = V_1$. Then it also holds for $V = V_2$. \qed}

\subsection{A construction of Evans and Gannon} \label{egconst}

Conditional on the Regularity Assumption, the paper \cite{EvansGannon} constructs, for each finite group $G$ and each class $\alpha \in \H^3(G;\bmu)$, a VOA $V$ with a faithful $G$-action with anomaly $\alpha$. Given the corollary in \S\ref{nonanom}, it suffices to prove the Main Theorem for the VOAs constructed by Evans and Gannon. This section reviews their construction.

Let $G$ be a finite group. Then there is a finite abelian extension $1 \to A \to E \to G \to 1$ such that the restriction map $\H^3(G;\bmu) \to \H^3(E;\bmu)$ is the zero map. This fact has been rediscovered a number of times, most recently in \cite{PhysRevX.8.031048}.
The Lyndon--Hochschild--Serre spectral sequence enhances the usual 5-term inflation-restriction exact sequence (for $\bmu$-cohomology) to one of the form:
\begin{multline*}
  1 \to \H^1(G;\bmu) \to \H^1(E;\bmu) \to H^1(A;\bmu) \to \H^2(G;\bmu) \to \H^2(E; \bmu) \\ \to \Lambda(G;A) \overset\delta\to \H^3(G; \bmu) \overset0\to \H^3(E;\bmu) \to \dots
\end{multline*}
where $\Lambda(G;A)$ maps naturally to $\H^2(A;\bmu)$, with kernel $\H^1(G; \H^1(A;\bmu)$.
It follows that for each $\alpha \in \H^3(G;\bmu)$, we can choose $\beta \in \Lambda(G;A)$ with $\delta\beta = \alpha$.

The strategy of \cite{EvansGannon}, then, is to start with a holomorphic VOA $W$ with a faithful action by $E$ with trivialized anomaly. (They prove that the permutation action on $W = (V^\natural)^{\otimes |E|}$, where $V^\natural$ denotes the moonshine VOA, works. One can always arrange a nonanomalous $E$-action by starting with some holomorphic VOA $W$ on which $E$ acts with anomaly of order $N$ and then taking $W' = W^{\otimes N}$. It could happen that $W'$ is nonanomalous but not canonically so: the anomaly may not have a distinguished trivialization. But all choices of trivialization of the anomaly for $W'$ will lead to the same trivialization of the anomaly for $W'' = (W')^{\otimes |\H^2(E;\bmu)|}$.)
 In particular the induced $A$-action on $W$ is nonanomalous, and so gaugings of $W$ by $A$ are parameterized by classes in $\H^2(A;\bmu)$. The central result underlying the construction is:

\mytheorem[Proposition (\cite{EvansGannon})]{
Conditional on the Regularity Assumption,
$W \sslash^\beta A$ carries a faithful $G$-action with anomaly $\alpha = \delta\beta$. \qed
}

In the Proposition, the VOA $W \sslash^\beta A$ depends only on the image of $\beta \in \Lambda(G;A)$ under the map $\Lambda(G;A) \to \H^2(A;\bmu)$, but the information in $\beta$ in the kernel of this map is used to determine the action of $G$. (The group $\H^1(A;\bmu)$ acts on any orbifold $W \sslash A$, and so an action of $G$ on $W$ can be adjusted by an element of $\H^1(G; \H^1(A;\bmu)) = \hom(G, \H^1(A;\bmu))$; compare the ``electromagenetic duality'' from \cite{FT18} or the ``T-duality'' from \cite{MR3916985}.)

\subsection{Completion of the proof}

Thus to prove the Main Theorem, it suffices to study the case $V = W\sslash^\beta A$ from \S\ref{egconst}. Specifically, we know that $G$ acts on $W\sslash^\beta A$ with anomaly $\alpha = \delta \beta$. Suppose that $\gamma \in \Aut(\bC)$ acts on $\bmu$ by multiplication by $n \in \widehat{\bZ}^\times$. (As in the previous sections, we will write $\bmu$ additively.) Then we wish to show that $G$ acts on $^\gamma(W \sslash^\beta A)$ with anomaly $n^2\alpha$.
Since $\delta : \Lambda(G;A) \to \H^3(G;\bmu)$ is linear,
it suffices to prove:

\mytheorem[Claim]{$^\gamma(W \sslash^\beta A) \cong {^\gamma W} \sslash^{n^2\beta} A$.}

To prove the claim, we begin with the linear algebraic description of $W \sslash^\beta A$ from \S\ref{voa-linalg}. Using that $A$ is an abelian group, equation~(\ref{voa-linalg-eqn}) says that as a $W^A$-module,
$$ W \sslash^\beta A \cong \bigoplus_{a \in A} W_a^{\iota_a\beta}. $$
and so
$$ ^\gamma(W \sslash^\beta A) \cong \bigoplus_{a\in A} {^\gamma(W_a^{\iota_a\beta})}.$$

We must now understand the Galois conjugate $W^A$-module $^\gamma(W_a^{\iota_a\beta})$.
By assumption $\gamma$ acts by multiplication by $n \in \widehat{\bZ}^\times$ on $\bmu$, and so
$$ ^\gamma(W_a^{\iota_a\beta}) \cong (^\gamma(W_a))^{\iota_a n\beta}.$$

What is $^\gamma(W_a)$? It is not $(^\gamma W)_a$. Indeed, the definition of ``$a$-twisted $W$-module'' refers explicitly to the eigenvalues of $a$.
These eigenvalues (or rather their logarithms, so that we can work additively)
 get multiplied by $n \in \widehat{\bZ}^\times$ under Galois conjugation by $\gamma$. To compensate for this one must divide $a$ by $n$. The result is:
$$ ^\gamma(W_a) \cong (^\gamma W)_{n^{-1}a}.$$

Taking the direct sum over $a\in A$, we find:
\begin{equation}\label{eqn-aclaim}
 ^\gamma(W \sslash^\beta A) \cong \bigoplus_{a\in A} {^\gamma(W_a^{\iota_a\beta})} \cong \bigoplus_{a \in A} (^\gamma W)_{n^{-1}a}^{\iota_a n\beta} \cong \bigoplus_{a' \in A} (^\gamma W)_{a'}^{\iota_{na'} n\beta} \cong \bigoplus_{a' \in A} (^\gamma W)_{a'}^{n^2\iota_{a'} \beta}.\end{equation}
Here we have reindexed $a' = n^{-1}a$. We also used that $\iota_{nx} \beta = n\iota_x \beta$. To see this, note that, for fixed $y$, $\iota_x\beta(y) = \beta(x,y) - \beta(y,x) = -\iota_y\beta(x)$ is a 1-cocycle  in the $x$ variable.

Equation~(\ref{eqn-aclaim}) establishes the Claim at the linear algebraic level, i.e.\ as modules over $^\gamma W^A$. But, since $A$ is an abelian group, \'etale algebras in $\cZ(\Vect[A]) \simeq \Rep(^\gamma W^A)$ are determined by their underlying modules. Thus the linear algebraic isomorphism implies a VOA isomorphism.
Finally, the VOA isomorphism is compatible with the $G$-action.
This completes the proof of the Claim and hence of the Main Theorem.

\section{Questions and conjectures}\label{sec:conclusions}

\subsection{Moonshine for every group} \label{moonshine-speculation}

Since $\Aut(\bmu) \cong \widehat{\bZ}^\times$ is abelian, its action on $\bmu$ can be ``twisted'' by any power. Given $i\in \bZ$, we will write $\bmu^{\otimes i}$ for the abelian group $\bmu$ made into a $\widehat{\bZ}^\times$-, and hence $\Aut(\bC)$-, module via the action $n \triangleright x := n^i x$.
The Main Theorem can then be summarized as saying that as an $\Aut(\bC)$-module, gauge anomalies of VOAs live in $\H^3(-;\bmu^{\otimes 2})$.

\mytheorem[Corollary of the Main Theorem]{
Suppose that the action of $G$ on $V$ preserves a $\bK$-form for some field $\bK \subset \bC$. Then the anomaly of the action lives in
$
  \H^0\bigl(\Gal(\bar{\bK}/\bK); \H^3\bigl(G; \bmu^{\otimes 2}\bigr)\bigr).
$
}

Consider the case $\bK=\bQ$. The ``defining property of $24$'' from \cite{MR554399} --- that $n^2 \equiv 1 \mod 24$ for $n \in \widehat{\bZ}^\times$ --- implies that for any group $G$, $\H^0(\Gal(\bar{\bQ}/\bQ); \H^\bullet(G; \bmu^{\otimes 2}))$ is entirely $24$-torsion. Let $V^\natural$ denote the moonshine VOA. The monster group $\bM$ acts on $V^\natural$ preserving a $\bQ$-form~\cite{MR2928458}. Taking for granted the Regularity Assumption, let $\omega^\natural$ denote the anomaly of this action. In unpublished work \cite{Mason-Monster}, Mason computed the image of $\omega^\natural$ under the total slant product map
$$ \prod_{g\in \bM} \iota_g : \H^3(\bM;\bmu) \to \prod_{g \in \bM} \H^2(C(g); \bmu)$$
and showed that the image has exact order $24$. Together with the above corollary, we find:

\mytheorem[Corollary]{The anomaly $\omega^\natural$ of the action of $\bM$ on $V^\natural$ has exact order $24$.}

The exact order of $\omega^\natural$ was first computed in \cite{MR3916985} using a different argument;
that paper modelled CFTs in terms of conformal nets (rather than VOAs), for which the Regularity Assumption is known \cite{MR1806798} to hold.

The construction from \cite{EvansGannon}, reviewed in \S\ref{egconst}, establishes that there is a weak form of ``moonshine'' for every finite group $G$ and every anomaly $\alpha \in \H^3(G;\bmu)$. The resulting VOA
 is by no means canonical, and has astronomically large central charge (of order $|G| \times |\H^3(G;\bmu)|$). Various authors have speculated versions of the following (I heard it from I.\ Frenkel), while knowing that, as stated, it is surely too naive:

\mytheorem[Conjecture]{
For each quasisimple group $G$, there is a distinguished ``moonshine'' VOA representation $V_G$ of $G$. For the Monster group, $V_\bM = V^\natural$, and for compact Lie groups, $V_G$ is the level-one WZW model. The VOAs $V_G$
can be used in the classification of finite simple groups analogously to the way Lie algebras are used in the classification of simple Lie groups.
}

The words ``level-one'' in the phrase ``level-one WZW model'' of a simple Lie group $G$ refers to the fact that the anomaly for these models is a generator of the infinite cyclic group $\H^4(BG;\bZ)$. Theorem 8.6 of \cite{MR3990846} comes close to showing that $\omega^\natural$ generates $\H^3(\bM;\bmu) \cong \H^4(B\bM;\bZ)$. For all but finitely many finite simple groups $G$ of Lie type, $\H^3(G;\bmu)$ is cyclic, and the calculations in \cite{MR3990846} suggest that cyclicity also holds for ``most'' of the sporadic simple groups. Thus the above Conjecture may be refined by speculating that one way $V_G$ should be distinguished is that its anomaly should generate $\H^3(G;\bmu)$.

The calculations in \cite{MR3990846} also show that for ``most'' sporadic simple groups $G$, $\H^3(G;\bmu)$ is entirely 24-torsion. (The known exceptions are the Janko groups $J_1$, $J_2$, and $J_3$.) Given our Main Theorem, this increases the odds that the distinguished representation $V_G$ posited by the Conjecture might be defined over $\bQ$. For comparison, according to unpublished work of J.~Grodal~\cite{Grodal}, if $G$ is a quasisimple group of Lie type defined over $\bF_q$, then $\H^3(G;\bmu) \approx \bZ/(q^2-1)$. (The ``$\approx$'' sign indicates that there are some systematic adjustments needed to accommodate
twisted forms and
Schur multipliers of $G$, and that there are finitely many groups $G$ for which further corrections are necessary.) Note that, letting  $\bmu \subset \bar{\bF}_q^\times$ now denote the group of roots of unity in positive characteristic,
 $$\bZ/(q^2-1) \cong \H^0(\Gal(\bar{\bF}_q/\bF_q); \bmu^{\otimes 2}).$$
This is consistent with the construction of \cite{MR3456026} of a VOA over $\bF_q$ for each group of Lie type defined over $\bF_q$.
Missing is a good theory of ``gauge anomalies'' in positive characteristic.

\subsection{A vertex Brauer group}

Recall the story of the cohomological Brauer group outlined in~\S\ref{assoc-cohom-Brauer}. When a group $G$ acts on an Azumaya algebra $A$ over $\bar{\bK}$, the anomaly lives not just in $\H^2(G;\bar{\bK}^\times)$ thought of as an abstract group: under the $\Gal(\bar{\bK}/\bK)$-action on $\{$Azumaya algebras over $\bar{\bK}\}$, the anomaly transforms according to the action on (the coefficients of) $\H^2(G;\bar{\bK}^\times)$. This control over the Galois action implies that there is an anomaly-type map for Galois-twisted actions, which in the case of descent data for $\Gal(\bar{\bK}/\bK)$ provides a map
$$ \{\text{Azumaya algebras over $\bK$}\} \to \H^2\bigl(\Gal(\bar{\bK}/\bK); \bar{\bK}^\times\bigr).$$
Moreover, this map provides an isomorphism of the right-hand side with the Brauer group  $\Br(\bK)$.

Our Main Theorem suggests that there may be a similar story for VOAs. For $G$ a finite group, our Main Theorem identifies anomalies of actions of $G$ on holomorphic VOAs as living in $\H^3(G;\bmu^{\otimes 2})$. This suggests:

\mytheorem[Conjecture]{
For each field $\bK$, there is a \define{vertex Brauer group} of Morita equivalence classes of holomorphic VOAs over $\bK$ modulo tensoring with $E_{8,1}$.
When $\bK$ is of characteristic $0$, then the vertex Brauer group is isomorphic to the Galois cohomology group $\H^3(\Gal(\bar{\bK}/\bK); \bmu^{\otimes 2})$.
}
We restrict to charactersitic zero for two reasons. First, there is so far no theory of 't Hooft anomalies of actions on VOAs in positive characterstic. Second, in the Azumaya case, the map
$$ \H^2\bigl(\Gal(\bar{\bK}/\bK);\bmu\bigr) \to \H^2\bigl(\Gal(\bar{\bK}/\bK);\bar{\bK}^\times\bigr) \cong \Br(\bK)$$
is an isomorphism only in characteristic $0$; in characteristic $p >0$ it is an isomorphism onto the prime-to-$p$ part of the right-hand side (see e.g.\ \cite[Chapter VI, Corollary 6.3.6]{MR2392026}).

We now outline, in parallel with \S\ref{assoc-cohom-Brauer}, how one could hope to prove the Conjecture.
Suppose that $V$ and $W$ are holomorphic VOAs over $\bK$ such that $V \otimes_\bK \bar{\bK} \cong W \otimes_\bK \bar{\bK}$. If $V$ is fixed, then the data of the $\bK$-form $W$ is equivalent to the data of a $1$-cocycle with twisted coefficients
$$\kappa \in \H^1\bigl(\Gal(\bar{\bK}/\bK); \Aut_{\bar{\bK}}(V \otimes_\bK \bar{\bK})\bigr).$$
Suppose that the  't Hooft anomalies of finite group actions on $V \otimes_\bK \bar{\bK}$ are the restriction of a universal anomaly living on $\Aut_{\bar{\bK}}(V \otimes_\bK \bar{\bK})$; according to our Main Theorem, this universal anomaly should live in
$$ \alpha \in \H^3\bigl( \Aut_{\bar{\bK}}(V \otimes_\bK \bar{\bK}); \bmu^{\otimes 2}\bigr).$$
Then the composition $\alpha \circ \kappa$ would be a twisted cohomology class
$$ \alpha \circ \kappa \in \H^3\bigl( \Gal(\bar{\bK}/\bK); \bmu^{\otimes 2}\bigr)$$
that depends only on the $\bK$-form $W$ of $V$.

Unfortunately, we lack direct access to this universal class $\alpha$: with current understanding of VOAs, we can only define anomalies for actions of finite groups, but $\Aut_{\bar{\bK}}(V \otimes_\bK \bar{\bK})$ is typically an infinite (in fact, affine algebraic) group. One could salvage this as follows. Suppose that there is a finite-degree Galois extension $\bK \subset \bL$ such that $V \otimes_\bK \bL \cong W \otimes_\bK \bL$. Then, still fixing $V$, the data of the $\bK$-form $W$ is given by a twisted cohomology class $\kappa_\bL \in \H^1(\Gal(\bL/\bK); \Aut_\bL(V \otimes_\bK\bL))$. Fix a cocycle for $\kappa_\bL$ and let $G \subset \Aut_\bL(V \otimes_\bK\bL)$ denote the union of the image of $\kappa_\bL$ together with its Galois conjugates. Then $G$ is a finite group, and so does have a well-defined anomaly $\alpha_\bL \in \H^0(\Gal(\bar{\bK}/\bL);\H^3(G; \bmu^{\otimes 2}))$.
Our proof of the Main Theorem does not provide sufficient control over the Galois action on cocycles --- at too many steps we worked instead simply with cohomology classes --- but it is reasonable to expect that there is a valid composition
$$ \alpha_\bL \circ \kappa_\bL \in \H^3\bigl( \Gal(\bL/\bK); \H^0\bigl(\Gal(\bar{\bL}/\bL); \bmu^{\otimes 2}\bigr)\bigr) \subset \H^3\bigl( \Gal(\bar{\bK}/\bK); \bmu^{\otimes 2}\bigr),$$
providing the class $\alpha \circ \kappa$ above.

Thus we expect that $\bK$-forms of a fixed holomorphic VOA lead to classes in the putative ``cohomological vertex Brauer group'' $\H^3(\Gal(\bar{\bK}/\bK); \bmu^{\otimes 2})$. In the associative  case, the next step is to classify Azumaya algebras over $\bar{\bK}$: they are all matrix algebras. In the vertex case, it is hopeless to try to classify holomorphic VOAs up to isomorphism, since in particular the set of even unimodular lattices injects into the set of holomorphic VOAs. (In small central charge, there is an expected classification \cite{MR1213740}, but it is not elementary.)

Instead, we should recognize the Azumaya classification as being about Morita equivalence classes: over $\bar{\bK}$, all Azumaya algebras are Morita equivalent. We can hope that a similar statement holds for VOAs.
There is no consensus definition of ``Morita equivalence'' of VOAs. We will take the following definition:

\mytheorem[Definition]{Suppose that $V_1$ and $V_2$ are regular VOAs which are both conformal extensions of the same regular VOA $W \subset V_1,V_2$ (in particular,  all three of $V_1$, $V_2$, and $W$ have the same central charge). Then $V_1$ and $V_2$ are \define{Morita equivalent} if the corresponding \'etale algebras in $\Rep(W)$ are $E_2$-Morita equivalent.}
Implicit in the definition is the fact that, given a braided monoidal category $\cC$, there is a ``Morita $3$-category'' $\cat{Alg}_2(\cC)$ whose objects are braided-commutative algebra objects in $\cC$; compare \cite[\S8]{JFS}. In the case of a braided fusion category $\cC$, an $E_2$-Morita equivalence of \'etale algebras $A$ and $B$ is the same as a $\cC$-balanced Morita equivalence of fusion categories $\cat{Mod}(A) \simeq \cat{Mod}(B)$. It follows in particular that $\Rep(V_1) \simeq \Rep(V_2)$ as MTCs.

Suppose that $V_1$ and $V_2$ are holomorphic VOAs. Then, using that two fusion categories are Morita equivalent if and only if they have equivalent centres \cite[Theorem 3.1]{MR2735754}, one finds that $V_1$ and $V_2$ are Morita equivalent as soon as they share a common regular subalgebra.

The analog of the fact that all Azumaya algebras over $\bC$ are Morita trivial would then be the conjecture that the central charge is the only Morita invariant of a holomorphic VOA. Equivalently:

\mytheorem[Conjecture]{
Suppose $V_1$ and $V_2$ are two holomorphic VOAs over $\bC$ of the same central charge. Then there is a regular VOA $W$ of the same central charge embedding into both $V_1$ and $V_2$.
}
It is not at all clear whether to believe this conjecture, because we don't have enough understanding of the set of all VOAs of large central charge: the Conjecture holds for the few holomorphic VOAs that are known simply because the only  available methods to construct holomorphic VOAs are all of the form ``pass to a regular subalgebra, then extend to a holomorphic algebra in a different way'' and so always produce Morita-equivalent VOAs.
(For instance, the Conjecture holds for all holomorphic VOAs of central charge $c\leq24$ by \cite{vEMS,MS19}. Holomorphic VOAs of larger central charge have not been well studied; the Conjecture applies, by construction, to those from \cite{largeC}.)
For lattice VOAs, the Conjecture follows from a $\bZ$-version of the Graham--Schmidt process, which produces, for any two unimodular lattices of the same rank, a common finite-index sublattice, and which provides the foundation of Kneser's neighbourhood method.
Physically, the Conjecture would follow from the expectation that every full conformal field theory with $c_L = c_R$ admits a gapped boundary condition.

Finally, if this Conjecture holds, then we could take the VOAs $E_{8,1}^{\otimes n}$ as our standard Morita-class representatives, where ``$E_{8,1}$'' means the level-one WZW theory for $E_8$ equipped with its ``Chevalley'' $\bQ$-form constructed in \cite{MR2928458}. This is why ``modulo tensoring with $E_{8,1}$'' appears in the statement of the Conjecture at the beginning of this subsection.

\subsection{K-theoretic interpretation of anomalies}

The existence of a vertex Brauer group isomorphic to $\H^3(\Gal(\bar{\bK}/\bK);\bmu^{\otimes 2})$ would imply the existence of a ``refined anomaly'' whenever the action of $G$ on a holomorphic vertex algebra $V$ preserves a $\bK$-form of $V$. Specifically, it would provide an anomaly map
\begin{equation}\label{eqn-veryrefinedanomaly}
 \{\text{holomorphic VOAs over $\bK$ with $G$-action}\} \to \H^3\bigl(G \times \Gal(\bar{\bK}/\bK); \bmu^{\otimes 2}\bigr).
\end{equation}
The ordinary gauge anomaly and the vertex Brauer class would be the restrictions to $\H^3(G;\bmu^{\otimes2})$ and $\H^3(\Gal(\bar{\bK}/\bK); \bmu^{\otimes 2})$ respectively.

The group $\H^3(G \times \Gal(\bar{\bK}/\bK); \bmu^{\otimes 2})$ is built, via a spectral sequence, from components of the form $\H^{3-i}(G; \H^{i}(\Gal(\bar{\bK}/\bK;\bmu^{\otimes 2})))$ for $0\leq i \leq 3$. Suppose that $\bK$ is a number field (i.e.\ a finite-degree extension of $\bQ$). Then the proposed vertex Brauer group is not itself particularly interesting: it follows from a theorem of Tate and Poitou \cite{MR0175892,PoitouLille} (see \cite[\S II.6.3, Theorem B]{MR1466966}) that $\H^3(\Gal(\bar{\bK}/\bK); \bmu^{\otimes 2})$ is isomorphic to $(\bZ/2)^N$, where $N$ is the number of field embeddings $\bK \mono \bR$. But the Galois cohomology groups $\H^i(\Gal(\bar{\bK}/\bK);\bmu^{\otimes 2})$ for $i\leq 2$ are quite interesting.
A theorem of Tate in the number-field case \cite{MR0422212,MR0429837}, and of Merkurjev and Suslin in general \cite{MR675529}, provides
  a close relationship between $\H^2(\Gal(\bar{\bK}/\bK);\bmu^{\otimes 2})$ and the second K-group $\rK_2(\bK)$:
\begin{equation}\label{MSthm}
 \H^2(\Gal(\bar{\bK}/\bK); \bmu^{\otimes 2}_n) \cong \rK_2(\bK) \otimes \bZ/n\bZ,\end{equation}
where $\bmu_n^{\otimes 2}$ denotes the $n$-torsion subgroup of $\bmu^{\otimes 2}$.
(See \cite[Chapter 8]{MR2266528} and \cite[\S VI.4]{MR2392026}.)

Suppose that $G$ is a finite group acting on a holomorphic VOA $V$ over $\bK$.
 The refined anomaly~(\ref{eqn-veryrefinedanomaly}) would map to a class in $\H^1(G; \H^2(\Gal(\bar{\bK}/\bK);\bmu^{\otimes 2}))$. The isomorphism (\ref{MSthm}) then suggests:

\mytheorem[Conjecture]{
Each action of a finite group $G$  on a holomorphic VOA $V$ over $\bK$ determines
 a homomorphism $G \to \rK_2(\bK)$.
}

  Let $n$ denote the {exponent} of $G$, i.e.\ $g^n = 1$ for all $g \in G$. Then a map $G \to \rK_2(\bK)$ will land in the $n$-torsion subgroup of $\rK_2(\bK)$. Suppose that $\bK$ contains a primitive $n$th root of unity~$\zeta$. The choice of $\zeta$ determines an isomorphism between the $n$-torsion subgroup of $\rK_2(\bK)$ and the $n$-torsion subgroup of  $\mathrm{Br}(\bK)$, because it identifies both with the untwisted cohomology group $\H^2(\Gal(\bar{\bK}/\bK); \bZ/n)$. The choice of $\zeta$ is also what is needed to define algebraically the category $\cC_g$ of $g$-twisted $V$-modules.  Over an algebraically closed field, $\cC_g$ is (noncanonically) equivalent to $\Vect$, but such an equivalence need not survive Galois descent: rather, $\cC_g$ will be equivalent to the category of modules of some Azumaya algebra $A_g$.
  The map $g \mapsto [A_g] \in \Br(\bK)$
    presumably agrees with the map $G \to \rK_2(\bK)$ predicted in the Conjecture.

\subsection{Galois and Grothendieck--Teichmuller groups and modular data}

Let $W$ be a regular VOA over $\bC$. We emphasized at the start of \S\ref{sec:mainthm} that the MTC structure on $\Rep(W)$ depends, {a priori}, on the $\bC$-structure of $W$.
To control this {a priori} $\bC$-dependence one should study the following question. Choose $\gamma \in \Aut(\bC)$. As a $\bC$-linear category, $\Rep(^\gamma W) \simeq {^\gamma \Rep(W)}$, since the notion of ``representation'' is defined algebraic. How do the MTC structures compare?

\mytheorem[Conjecture]{Fix a semisimple $\bC$-linear category $\cC$ with finitely many simples objects. The set of MTC structures on $\cC$ is permuted by a canonical action of $\Aut(\bC)$.}

Up to issues of completion, this action of $\Aut(\bC)$  should come from the injection of $\Gal(\bar{\bQ}/\bQ)$ into the Grothendieck--Teichmuller group, which is, again up to issues of completion, the automorphism group of the $E_2$ operad. MTCs are a particular type of (categorical) algebra for the $E_2$ operad.

Each MTC $\cC$ determines a projective representation of $\operatorname{SL}(2,\bZ)$ called the \define{modular data} of $\cC$. We will refer to this representation as $\int_{T^2}\cC$; the notation follows the ``factorization algebra'' notation used for example in \cite{WalkerTQFTs,ScheimbauerThesis}. The representation $\int_{T^2}\cC$ has many nice properties. Among them are (i)
$\int_{T^2}\cC$ has a basis indexed by the simple objects in $\cC$ and (ii) the representation $\int_{T^2}\cC$ is defined over the cyclotomic field $\bQ^{\mathrm{cyc}}$.
Our Main Theorem strongly suggests that, in the case when $\cC = \Rep(W)$ for a regular VOA $W$,

\mytheorem[Conjecture]{At the level of modular data, we have $\int_{T^2} \Rep(^\gamma W) \cong \gamma^2\bigl( \int_{T^2} \Rep(W)\bigr)$.}

This conjecture is enticingly close to the following result of \cite{MR1266785,MR3435813}. Let $\cC$ be an arbitrary MTC, and choose $\gamma \in \Gal(\bQ^{\mathrm{cyc}}/\bQ) \cong \widehat{\bZ}^\times$. Then there is a permutation $\sigma$ of the simple objects of $\cC$, and hence of the basis of $\int_{T^2}\cC$, which intertwines $\int_{T^2}\cC$ with $\gamma^2(\int_{T^2}\cC)$.
Thus the conjecture predicts that $\int_{T^2}\Rep(W)$ and $\int_{T^2}\Rep(^\gamma W)$ are isomorphic as projective $\mathrm{SL}(2,\bZ)$-modules.

\subsection{Cyclotomicity of MTCs}

Suppose one has some arbitrary module $X$ of $\Gal(\bar{\bQ}/\bQ)$ such that the formula $\gamma \triangleright x := \gamma^2(x)$ also defines an action of $\Gal(\bar{\bQ}/\bQ)$ on $X$. Then the action on $X$ factors through $\Gal(\bQ^{\mathrm{cyc}}/\bQ)$. Indeed, if $(\gamma_1\gamma_2)^2 = \gamma_1^2\gamma_2^2$ then $\gamma_1\gamma_2=\gamma_2\gamma_1$, and the abelianization of $\Gal(\bar{\bQ}/\bQ)$ is precisely $\Gal(\bQ^{\mathrm{cyc}}/\bQ)$.

The repeated occurrences of a squared Galois action suggest that perhaps the entire theory of MTCs is cyclotomic. For comparison, finite group theory is cyclotomic --- all characters are cyclotomic, and all representations split over $\bQ^{\mathrm{cyc}}$ --- and MTCs are a version of ``finite quantum groups.'' The following conjectures have been contemplated by various authors, e.g.\ \cite{MR2922607,EvansGannon}.

\mytheorem[Conjectures]{\begin{enumerate}
  \item If $\cC$ is an MTC, then $\cC$ has a (canonical?) form over $\bQ^{\mathrm{cyc}}$.
  \item If $\cC$ is an MTC, then $\cC$ arises as $\Rep(W)$ for some (canonical?) regular VOA $W$.
  \item If $W$ is a regular VOA, then $\Rep(W)$ has a (canonical?) form over $\bQ^{\mathrm{cyc}}$.
\end{enumerate} \vspace*{-9pt}}

redpajama_set_name: RedPajamaArXiv


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Lehigh Valley Health Network's campus integration plan in Schuylkill County consolidating acute, medical/surgical services to the East Campus is continuing. That consolidation is expected to be completed by the end of 2017.
Part of this construction is enhancement of the Diagnostic Imaging/Radiology Department at the East Norwegian Street hospital. That department expansion includes the additions of nuclear medicine, digital mammography, a new interventional radiology suite and a new patient registration/reception area. Additional, but more minor renovations are taking place on the hospital's fourth floor.
The current plan keeps Lehigh Valley Hospital–Schuylkill Family Birth and Newborn Center, Lehigh Valley Health Network's Maternity Unit in Schuylkill County, on the South Campus as well as Adult and Adolescent Behavioral Health Units. A decision about the location of maternity services is still being evaluated.
An Express/Urgent Center is being planned for the South Jackson Street facility as well.


text:
package org.xlcloud.service.transformer;

import static org.junit.Assert.assertEquals;
import static org.junit.Assert.assertTrue;

import java.util.List;

import org.apache.commons.lang.builder.ToStringBuilder;
import org.apache.commons.lang.builder.ToStringStyle;
import org.apache.log4j.Logger;
import org.junit.Assert;
import org.junit.Before;
import org.junit.Test;

import org.xlcloud.service.Session.Endpoints.Endpoint;
import org.xlcloud.service.VirtualCluster;
import org.xlcloud.service.model.VirtualClusterModel;

/**
 * JUnit test suite for testing dozer transformations between
 * {@link VirtualCluster} and {@link VirtualClusterModel}.
 *
 * @author Maciej Osyda, AMG.net
 */
public class VirtualClusterTransformerTest extends TransformerTestSupport {

    private static final Logger LOG = Logger.getLogger(VirtualClusterTransformerTest.class);

    private Transformer<VirtualClusterModel, VirtualCluster> vcTransformer;

    @Before
    public void initTransformer() {
        this.vcTransformer = new DozerTransformer<VirtualClusterModel, VirtualCluster>(VirtualClusterModel.class, VirtualCluster.class);
    }

    @Test
    public void testTwoWayTransform() {
        VirtualCluster dto = randomVCluster(Mode.FULL);

        LOG.info("dto: " + ToStringBuilder.reflectionToString(dto, ToStringStyle.MULTI_LINE_STYLE));
        VirtualClusterModel modelFromDto = vcTransformer.transformFromDto(dto);
        LOG.info("model: " + ToStringBuilder.reflectionToString(modelFromDto, ToStringStyle.MULTI_LINE_STYLE));
        VirtualCluster dtoFromModel = vcTransformer.transformFromModel(modelFromDto);
        LOG.info("dtoFromModel: " + ToStringBuilder.reflectionToString(dtoFromModel, ToStringStyle.MULTI_LINE_STYLE));

        assertEquals(dto.getId(), dtoFromModel.getId());
        assertEquals(dto.getName(), dtoFromModel.getName());
        assertEquals(dto.getProjectId(), dtoFromModel.getProjectId());
        assertEquals(dto.getVirtualClusterDefinitionId(), dtoFromModel.getVirtualClusterDefinitionId());

        Assert.assertTrue(dto.getSessions().getSession().size() > 0);
        //we don't store sessions in database
        Assert.assertTrue(dtoFromModel.getSessions() == null);

        assertParameters(dto.getParameters(), dtoFromModel.getParameters());
    }

    private void assertEndpoints(List<Endpoint> endpoints, List<Endpoint> endpoints2) {
        assertEquals(endpoints.size(), endpoints2.size());
        for (Endpoint e1 : endpoints) {
            boolean serviceFound = false;
            for (Endpoint e2 : endpoints2) {
                if (e1.getAddress() != null && e1.getAddress().equals(e2.getAddress())){
                    serviceFound = true;
                    Assert.assertNotNull(e1.getEndpointName());
                    assertEquals(e1.getEndpointName(), e1.getEndpointName());
                }
            }
            assertTrue(serviceFound);
        }
    }

    private void assertParameters(VirtualCluster.Parameters parameters, VirtualCluster.Parameters parameters2) {
        assertEquals(parameters.getParameter().size(), parameters2.getParameter().size());
        for (VirtualCluster.Parameters.Parameter s1 : parameters.getParameter()) {
            boolean serviceFound = false;
            for (VirtualCluster.Parameters.Parameter s2 : parameters2.getParameter()) {
                if (s1.getName() != null && s1.getName().equals(s2.getName())){
                    serviceFound = true;
                    Assert.assertNotNull(s1.getValue());
                    assertEquals(s1.getValue(), s1.getValue());
                }
            }
            assertTrue(serviceFound);
        }
    }
}

redpajama_set_name: RedPajamaGithub


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The wind's whispers were hard to ignore; I shuffled another inch towards the edge.


text:
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Author K. Slaughter Found Liable for Employment Taxes, But Not Penalties, in Tax Court Opinion
Attorney's Fees in Interpleader Action? Not If It Impairs Recovery of Federal Tax Lien.
Published by at
Can the party filing an interpleader in California recover the costs and fees of filing the interpleader? No…. not if it impairs the recovery of the United States on its federal tax lien.
California Code of Civil Procedure section 386.6, generally, authorizes the court to make an award of attorney's fees and costs to an interpleading plaintiff. Federal interpleader actions are authorized by 28 U.S.C. § 1335 and Rule 22 of the Federal Rules of Civil Procedure where persons holding funds to property upon which conflicting claims are being made by others may deposit the funds with the District Court and require the other parties to litigate their interests. Libby v. City Nat'l Bank, 592 F.2d 504, 507 (1978). A federal court has the power to award fees through those rules and 28 U.S.C. § 2361, which empowers courts to "make all appropriate orders to enforce its judgment." "An award of attorney fees and costs to a stakeholder, however, is not automatic. The decision rests within the discretion of the court." Underwriters Group, Inc. v. Clear Creek Indep. Sch. Dist., G-05-334, 2006 U.S. Dist. LEXIS 47907 (D. Tex. June 30, 2006).
However, as the Ninth Circuit held in Abex Corp. v. Ski's Enterprises, Inc., 748 F.2d 513, 516 (9th Cir. 1984), courts have clearly held… that the existence of prior federal tax liens gives the government a statutory priority over the interpleader plaintiff's ability to diminish the fund by an award of fees." [Internal citations omitted.]
The priority of federal tax liens is governed by Section 6323 of the Internal Revenue Code, 26 U.S.C. Certain subsections in Section 6323 provide for some interests to be "superpriorities," entitled to priority over the federal tax lien regardless of whether the interest was perfected before the tax lien – for example, mechanic's liens under $5,000. Paragraph (a)(8) of Section 6323 provides for an exception for attorneys' liens where the judgment was procured through the attorneys' work, so long as the lien was not to procure a judgment against the United States, providing:
With respect to a judgment or other amount in settlement of a claim or of a cause of action, as against an attorney who, under local law, holds a lien upon or a contract enforcible against such judgment or amount, to the extent of his reasonable compensation for obtaining such judgment or procuring such settlement, except that this paragraph shall not apply to any judgment or amount in settlement of a claim or of a cause of action against the United States to the extent that the United States offsets such judgment or amount against any liability of the taxpayer to the United States.
26 U.S.C. § 6323(e)(3) provides for the recovery of attorney's fees by a lien holder when the fees are "actually incurred in collecting or enforcing [a lien] obligation" superior to the lien(s) of the United States (emphasis added).
However, neither of these exceptions apply to an interpleading plaintiff who is not a lienholder. Thus, an innocent third party may be forced to bear the cost of filing an interpleader action. The Seventh Circuit, in Campagna-Turano Bakery, Inc. v. United States, 632 F.2d 39, 44 (7th Cir. 1980), recognized the apparent inequity in this result in some cases, but found that its hands were statutorily tied:
Campagna argues persuasively that the result we reach is inequitable. Although not legally required to bring an interpleader action, it was forced to do so in order to avoid liability exceeding the amount of the debt owed to Yamo. Now, it is unable to recover costs and attorneys' fees resulting from the initiation of that action. Perhaps § 6323 should contain an exception for plaintiffs in Campagna's situation. But it is for Congress, not this court, to write that exception.
Despite the potential lack of reimbursement for bringing the interpleader action, incentive to file exists nonetheless because, under Section 2361 of Title 28, U.S.C., the court may "discharge the plaintiff from further liability" where the conditions satisfy the requirements of the Federal Interpleader Act. See also Metropolitan Life Ins. Co. v. Davis, Civil No. JFM-10-2785, 2011 WL 2148714, at *6 (D. Md. May 31, 2011).
See other blog posts on this topic at https://taxattorneyoc.com/blog/2019/03/12/trustees-and-executors-should-worry-when-taxes-are-owed-the-federal-insolvency-statute/ and https://taxattorneyoc.com/blog/2019/05/05/enforcement-of-money-judgments-against-property-in-jurisdiction-of-california-probate-court-the-interaction-between-cal-code-civ-proc-sections-709-030-695-010-and-697-310/.
Posted on 06/01/2019 by Daniel W. Layton.
Tax Attorney OC – Daniel Layton
23 Corporate Plaza Drive, Suite 100,
taxattorneyoc.com/
The use of this website does not constitute legal advice or an attorney-client relationship. It is for informational purposes only. Daniel Layton makes no representation or warranty regarding the materials on this website and assumes no liability for the use or interpretation of information contained herein.
© 2014-2022 Daniel Layton- Tax Attorney Newport Beach and Tustin, Orange County, CA - All Rights Reserved.

redpajama_set_name: RedPajamaCommonCrawl


I can see a similar tidal wave on the horizon with an Apple tablet. At first, users will sneak them into the office as companion devices. Before long, they'll insist that IT support them directly by providing expensive docking cradles and proprietary keyboard/mice combinations — all so that they can ditch their carefully tested, enterprise-ready, company-provided desktop or laptop in favor of their shiny new toy.
And you are so lazy, that you don't want to face up to your years of failure to make brave decisions and select secondary vendors such as Linux, and will continue to blindly dig the grave you have already made yourself.
2009 O'Reilly Ebook Revenue up 104% – Tools of Change for Publishing – O'Reilly posts that their e-book revenue is up 108% for the year as their printed book sales drops. I love Safari and typically use it for a couple of hours a day. Since they have released the HTML Beta interface it is much better.
Google announces the gradual end of support for IE6 in it's applications.
We are surely in the second innings on the fifth day with only the tail enders to play.
Link-state routing protocols such as OSPF and IS-IS elect a designated router on every link in order to perform flooding of topology information. In wireless ad-hoc networks, there is different notion of a link, packets can and do go out the same interface; hence, a different approach is needed in order to optimize the flooding process.
Justa reminder that OSPF and ISIS are well known implementations of the SPF routing algorithm.
For the capacity planning one, have a look at http://www.perfdynamics.com/.
The guy has written a few books on the topic that help answer questions like that.


At our 2018 Australian Payments Summit we heard from John Phillips, of 460degrees, a passionate advocate working to raise awareness and execution of Self-Sovereign Identity (SSI).
We recently sat down with John to discuss SSI, and how he believes it could benefit both the payments industry and financial sector.
Can you describe Self-Sovereign Identity and how you got involved?
The best description I have heard came from a talk by fellow SSI evangelist, Drummond Reed at Evernym who defines SSI as a "lifetime portable identity for any person, organisation, or thing that does not depend on any centralised authority and can never be taken away".
I'd seen many poorly managed identity and access management solutions in my career across education, finance and e-commerce; the idea of doing this well resonated very strongly with me. I can think of very few things that would be better to be known for, than leaving a positive impact on a global scale for digital identity.
What was it about Self-Sovereign Identity that resonated so strongly?
The aim of the global SSI movement is to make the fundamental building blocks for digital identity freely available. This is achieved using the growing publicly available catalogue of open source software and standards. Companies can then build their own services and business models on top of these building blocks.
Where do you think the greatest benefits for our industry can be realised?
Existing mechanisms to establish sufficient trust are often localised to a single organisation and are not transferable, portable or extensible. Moreover, they don't generally preserve privacy, minimise disclosure or reflect ownership responsibilities. SSI aims to change that on a global scale.
Once we agree on common, open standards, for the basic building block level, there is the opportunity to take away the unnecessary friction, privacy erosion, and lack of interoperability that exists within the current fragmented systems.
I see three key areas where SSI can assist the payments industry and financial sector.
Managing Risk – Risk can be minimised for all participants while continuing to meet regulatory and service provision requirements across a range of compliance areas such as KYC, Anti-Money Laundering and Counter-Terrorism Financing.
Managing Privacy – Providing the appropriate level of privacy to parties. We are seeing this overseas with the introduction of the EU General Data Protection Regulation (GDPR), and in Australia through the implementation of the Consumer Data Right (CDR).
Managing Trust – Providing a transparent service that all participants "trust", that is, trust in who is involved and that the process will deliver the outcomes they expect.
Where do you think the Self-Sovereign Identity movement is headed?
Several SSI implementations have already begun to appear around the world. For example, in the commercial sector, CULedger is used as a global utility to provide Know Your Customer (KYC) checks. Government bodies have also started implementing SSI solutions: My Data in Finland, the British Columbia Verifiable Organisational Network in Canada, digital birth certificates in Illinois in the USA, and in certain vertical sectors, like health, in the UK with Truu ID, a trusted digital identity for doctors.
These are pockets, but it once they can connect globally, we can achieve the network effect. Identity, like payments, is a network business and similarly requires the collaboration of many organisations and jurisdictions to function effectively.
We already have the potential to use a global utility for identity – through the Sovrin Network. You can also implement SSI in a single, standalone sense.
When SSI really becomes powerful is when you can connect these pockets of implementation – that's when the true potential of portability and interoperability is realised!
Thanks to John for spending the time with us today!
If you'd like to comment on the future of Digital Identity in Australia, the Australian Payments Council is currently consulting and accepting submissions on its 2019 Payments Plan. Since 2015 the Council has had a strong focus on developing a Digital Identity Trust Framework.


Riders at SEA can request Uber when leaving the airport. Learn how to receive these trip requests and cash in on the action.
Per SEA regulations, driver-partners must adhere to the terms below. Failure to do so may result in the receipt of a citation or even restriction of access to airport trips.
Please note that Uber will not pay for any citations that you may incur. If you receive a citation, the Port will allow a 10 day period for appeals if you believe the citation was unjust. If you fail to appeal or pay the citation invoice, these will be sent to Uber to pay the Port, however you will receive email communication and the corresponding amount will be deducted from your earnings.
Commercial partners may fulfill requests for uberBLACK and SUV and may opt-in to fulfill requests for SELECT and uberXL.
We've instituted a special first vehicle in is the first one out (FIFO) system at Seattle-Tacoma International Airport. The FIFO system only works within the designated, geofenced area and driver-partners waiting elsewhere will not receive trip requests through the Uber driver app.
Driver-partners must have a valid Sea-Tac Airport permit (Gatekeeper).
Driver-partners must have a valid business license.
Driver-partners must have valid commercial vehicle insurance.
Driver-partners may not wait longer than 10 minutes in the loading areas.
Driver-partners must keep headlights on at all times.
Driver-partners must not exceed the posted speed limit of 10mph.
At SEA, Uber's technology uses a first-in-first-out (FIFO) queue. This means that, in order to receive a trip request, driver-partners need to enter a designated waiting area, located within a "FIFO zone". Driver-partners will then be placed in a virtual queue where the first driver-partner to enter will be the first to receive a trip request.
At SEA, the FIFO zone covers all airport grounds, as shown in the map below. Commercial driver-partners are welcome to wait at our new holding lot located at 2902 S. 171st Street SeaTac, WA 98188.
Once you accept a trip request, the pickup location at SEA is located on the lower level Arrivals drive (see map below).
Based on the operating agreement with SEA, riders can be dropped off at the upper level Departures drive. If you'd like to receive a trip request afterwards, you'll need to keep the Uber driver app on to maintain your place in the FIFO queue.
Re-Match allows all partners who just completed an airport dropoff to temporarily receive a pickup request without having to wait in the FIFO queue.
To qualify for Re-Match, partners must complete a dropoff at the airport. If you're in the terminal and do not receive a request after a few moments, it means there was no pickup available to rematch and you need to leave the terminal drives.
If you don't receive a Re-Match trip, you can still wait in the FIFO queue for an airport request. Please do not loiter in the terminal, as idling at the curb can still result in a citation.
If you do receive a Re-Match trip, follow the route and use the south entrance to reach the pickup location on the Arrivals drive. You may want to call your rider to let them know that you are on your way!
No. Partners will only be eligible for Re-Match after dropping off at the airport.
Does Re-Match increase my wait time in the FIFO queue?
During times when there are many drop offs the opportunity for Re-Match is higher. At these times the wait times in the FIFO queue may increase. However, Re-Match usually reduces the number of cars in the FIFO queue such that overall wait times generally remain about the same as before.
What is the fastest way to go from Departures to the pickup location?
SeaTac Airport has requested that drivers use the Airport Expressway and the return to terminal loops to reach the pickup locations. Please do not exit on 170th street as those intersections can become very congested.
I'd like to know more about Re-Match.
Checking the arrivals schedule at Seattle-Tacoma International Airport can help you maximize your earnings with information about when flights will be landing at the airport and when it will be busiest. See SEA ARRIVALS SCHEDULE.


text:
Crazy Rhythms
The Good Earth
Louie Louie Collection
The History of Northwest Rock, Vol. 3
Über The Feelies
Usually, the first thing out of everyone's mouth about the Feelies has to do with the guitars -- those frenetic, frazzled, jittery, jagged guitars. But by too often playing the guitar card, critics have downplayed the band's unconventional use of percussion. On the band's legendary debut, all four members lent a hand to give each song an irresistible backbeat decked out in nervous tics and shivers, all of it washed over by a flood of burbling, buzzing, chiming guitars. Vocals fall somewhere in between Jonathan Richman's nasal drip and Iggy Pop's zombie carnie spiel. But what impresses most about Crazy Rhythms is still the beat! Many of the songs start out as percussive cat-and-mouse games that break into a scamper once the guitars show up. This record will send you, willingly or not, into a convulsive nerd mating dance. In the landscape of modern rock, it towers majestic in the middle of nowhere since nothing comes close to sounding like it, not even subsequent albums by the band. The Good Earth, their second album, has a more rustic quality that makes it feel like a drive across the Great Plains on a fistful of Mini-Thins. Only Life retained the rusticity of the second album, but was covered with the track marks of the Velvet Underground. All three of these records are incomparable, important albums, but it's the first that will get under your skin the quickest and stay there the longest. Chad Driscoll
Pylon, R.E.M., The Bongos, The Go-Betweens, The Mekons, The Scene Is Now
Jangle Pop
Usually, the first thing out of everyone's mouth about the Feelies has to do with the guitars -- those frenetic, frazzled, jittery, jagged guitars. But by too often playing the guitar card, critics have downplayed the band's unconventional use of percussion. On the band's legendary debut, all four members lent a hand to give each song an irresistible backbeat decked out in nervous tics and shivers, all of it washed over by a flood of burbling, buzzing, chiming guitars. Vocals fall somewhere in between Jonathan Richman's nasal drip and Iggy Pop's zombie carnie spiel. But what impresses most about Crazy Rhythms is still the beat! Many of the songs start out as percussive cat-and-mouse games that break into a scamper once the guitars show up. This record will send you, willingly or not, into a convulsive nerd mating dance. In the landscape of modern rock, it towers majestic in the middle of nowhere since nothing comes close to sounding like it, not even subsequent albums by the band. The Good Earth, their second album, has a more rustic quality that makes it feel like a drive across the Great Plains on a fistful of Mini-Thins. Only Life retained the rusticity of the second album, but was covered with the track marks of the Velvet Underground. All three of these records are incomparable, important albums, but it's the first that will get under your skin the quickest and stay there the longest.
Chad Driscoll
Alt/Punk