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%  slides:
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%  Categories with Inner Gauge Groups 
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%  Amnon Yekutieli
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\title[Inner Gauge Groups]
{Categories with Inner Gauge Groups}
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\author[Amnon Yekutieli]{Amnon Yekutieli}
\institute[BGU]
{Department of Mathematics\\
Ben Gurion University\\[2em]
Notes available at 
{\blue http://www.math.bgu.ac.il/$\sim$amyekut/lectures}\\[2ex]
\tiny{Written 31 May 2009}}
\date{}
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%Memorial Meeting, Nov.\ 2008}}
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\begin{document}

% %% top matter for notes %%
% \mbox{}
% \vspace{2cm}
% 
% \begin{center}
% {\large \bfseries Categories with Inner Gauge Groups}
% 
% 
% \vspace{1ex}
% Lecture Notes\\
% %\footnote[1]{written: 14 Dec 2008}
% 21 May 2009
% 
% \vspace{3ex}
% Amnon Yekutieli\\
% Ben Gurion University, ISRAEL
% 
% \texttt{\small http://www.math.bgu.ac.il/$\sim$amyekut}
% \end{center}
% 
% \vspace{2ex}
% %% ** end of top matter for notes **

 
\begin{frame}[label=title]
\titlepage
\end{frame}

\setcounter{section}{-1}
\section{Introduction}
\begin{frame}
\sect{Introduction}
\pause 

In this talk I will explain a new kind of algebraic structure, namely
a \hilite{dblue}{category with inner gauge groups}. I will give
several motivating examples.
\pause 

This structure appears naturally in the
study of twisted deformation quantization of algebraic varieties, as I will
mention at the end of the talk.
\end{frame}


\section{1. Algebroids}
\begin{frame}
\sect{1. Algebroids}
\pause

A category $\cat{A}$ is called a \hilite{rust}{linear category} if for every
pair of objects
$i, j \in \opn{Ob}(\cat{A})$
the set of morphisms
\[ \cat{A}(i, j) := \opn{Hom}_{\cat{A}}(i, j) \]
is an abelian group; 
\pause 
and for every $i, j, k \in \opn{Ob}(\cat{A})$
the composition
\[ \cat{A}(i, j) \times \cat{A}(j, k) \to \cat{A}(i, k) \]
is bilinear. 
\pause 

Note that we are treating objects of $\cat{A}$ as indices.
\end{frame} 

\begin{frame} 
For any object $i$ the set of morphisms $\cat{A}(i, i)$ is a ring. 
This gives a hint how to construct linear categories: 
\pause 

\begin{exa}
Let $A$ be a ring. Define a category $\cat{A}$ with 
$\opn{Ob}(\cat{A}) := \{ 0 \}$ and 
$\cat{A}(0, 0) := A$. 
Addition and composition come from
addition and multiplication in $A$.
Then $\cat{A}$ is a linear category.
\end{exa}
\pause 

But there are also very different kinds of linear categories too:
\pause 

\begin{exa}
Let  $\cat{C}$ be an abelian category, and let $\{ M_i \}_{i \in I}$
be a collection of objects of $\cat{C}$, indexed by some set $I$. Define a
category $\cat{A}$ as follows. The set of objects is
$\opn{Ob}(\cat{A}) := I$, and 
$\cat{A}(i, j) := \opn{Hom}_{\cat{C}}(M_i, M_j)$.
Then $\cat{A}$ is a linear category. 
\end{exa}
\end{frame} 

\begin{frame} 
\begin{dfn}
Let $\cat{A}$ be a linear category. 
\pause 

Define $\cat{A}^{\times}$ to be the
subcategory of $\cat{A}$ with 
$\opn{Ob}(\cat{A}^{\times}) :=
\opn{Ob}(\cat{A})$ and
\[ \cat{A}^{\times}(i, j) := \opn{Isom}_{\cat{A}}(i, j) , \]
the set of invertible morphisms $i \to j$.
\pause 

The category $\cat{A}^{\times}$ is a groupoid (all morphisms are
invertible), and we call it the \hilite{dblue}{gauge groupoid} of $\cat{A}$.
\end{dfn}
\pause 

Note that for any object $i$ of $\cat{A}$ we have
\[ \cat{A}^{\times}(i, i) = \cat{A}(i, i)^{\times} , \]
the group of invertible elements in the ring $\cat{A}(i, i)$.
\end{frame} 

\begin{frame} 
A groupoid $\cat{G}$ is called \hilite{rust}{nonempty} if
$\opn{Ob}(\cat{G}) \neq \emptyset$, and \hilite{dblue}{connected} if 
$\cat{G}(i, j) \neq \emptyset$
for any pair of objects.
\pause 

\begin{dfn}
An \hilite{bgreen}{algebroid} is a linear category $\cat{A}$ whose gauge
groupoid $\cat{A}^{\times}$ is nonempty and connected.
\end{dfn}
\end{frame} 

\begin{frame} 
\begin{figure}
\includegraphics[scale=0.8, viewport = 120 580 450 750, clip = true]
{algebroid_1_plus.ps}
%\label{fig1}
\caption{An algebroid $\cat{A}$ with set of objects 
$\{ 0, 1 \}$.}
\end{figure}
\end{frame} 

\begin{frame} 
\begin{figure}
\includegraphics[scale=0.8, viewport = 120 580 450 750, clip = true]
{algebroid_2_plus.ps}
%\label{fig1}
\caption{The gauge groupoid $\cat{G} := \cat{A}^{\times}$
of the algebroid $\cat{A}$.}
\end{figure}
\end{frame} 

\begin{frame} 
Let $\cat{A}$ be an algebroid. An element
$g \in \cat{A}^{\times}(i, j)$ induces a ring isomorphism
\[ \opn{Ad}(g) : \cat{A}(i, i) \iso \cat{A}(j, j) \]
which is ``conjugation by $g$''; namely
\[ \opn{Ad}(g)(a) := g \circ a \circ g^{-1} \]
for $a \in \cat{A}(i, i)$.
\pause 

Suppose we choose a particular object
$i_0 \in \opn{Ob}(\cat{A})$ (a ``base point''), and an isomorphism
\[ g_{i_0, j} \in \cat{A}^{\times}(i_0, j) \]
for any other object $j$ (a ``path''). 
\pause 

These choices are called a \hilite{dblue}{decomposition} of $\cat{A}$.
\end{frame} 

\begin{frame} 
The fact that the decomposition is not canonical is crucial.
\pause 

We shall soon see a geometric version of an algebroid, where it is 
\hilite{rust}{impossible to decompose}.
\pause  

There is a close relationship between algebroids and Morita theory. This is
explained in Appendix A.
% \ref{sec:morita} 
\end{frame} 



\section{2. Stacks of Algebroids}
\begin{frame}
\sect{2. Stacks of Algebroids}
\label{sec:stacks}
\pause

In order to convince you that algebroids are really necessary, I will
now go to geometry. So let $X$ be a topological space.
\pause 

A category is a higher structure than a set: a category $\cat{C}$ is a set
$\opn{Ob}(\cat{C})$, together with morphisms between elements of 
$\opn{Ob}(\cat{C})$.
\pause 

The geometric version of a set is a sheaf of sets. Such a sheaf $\mcal{S}$
assigns a set $\mcal{S}(U)$ to any open set $U \subset X$, and a restriction
function
\[ \opn{rest}_{V / U} : \mcal{S}(U) \to \mcal{S}(V) \]
when $V \subset U$. These restriction functions must satisfy certain axioms.
\end{frame} 

\begin{frame} 
The geometric version of a category is a \hilite{rust}{stack}. 
\pause 

A stack $\gerbe{C}$ assigns a category $\gerbe{C}(U)$ to any open set $U$, and 
a restriction functor
\[ \opn{rest}_{V / U} : \gerbe{C}(U) \to \gerbe{C}(V) \]
to an inclusion $V \subset U$ of open sets. 
\pause 

These restriction functors must
satisfy ``higher sheaf axioms''.
%See Figure \ref{fig:stack_1_a}.
\end{frame} 

\begin{frame} 
\begin{figure}
\includegraphics[scale=0.8, viewport = 100 475 500 735, clip = true]
{stack_1_plus.ps}
\caption{A stack $\gerbe{C}$ on $X$}
\label{fig:stack_1}
\end{figure}
\end{frame} 

\begin{frame} 
\begin{figure}
\includegraphics[scale=0.8, viewport = 100 475 500 735, clip = true]
{stack_1_a_plus.ps}
\caption{A stack $\gerbe{C}$ on $X$}
\label{fig:stack_1_a}
\end{figure}
\end{frame} 

\begin{frame} 
\begin{figure}
\includegraphics[scale=0.8, viewport = 100 475 500 735, clip = true]
{stack_1_b_plus.ps}
\caption{A stack $\gerbe{C}$ on $X$}
\label{fig:stack_1_b}
\end{figure}
\end{frame} 

\begin{frame} 
\begin{figure}
\includegraphics[scale=0.8, viewport = 100 475 500 735, clip = true]
{stack_1_c_plus.ps}
\caption{A stack $\gerbe{C}$ on $X$}
\label{fig:stack_1_c}
\end{figure}
\end{frame} 

\begin{frame} 
We usually write 
\[ i|_V := \opn{rest}_{V / U}(i) \]
for $i \in \opn{Ob}(\gerbe{C}(U))$.
\pause 

A \hilite{dblue}{stack of groupoids} is a stack $\gerbe{G}$ such that for every
$U$ the
category $\gerbe{G}(U)$ is a groupoid. 
\pause 

Here is a very important notion for this talk.
\pause 

\begin{dfn}
A \hilite{rust}{gerbe} is is a stack of groupoids $\gerbe{G}$ which is
\hilite{bgreen}{locally nonempty} and \hilite{bgreen}{locally connected}. 
\end{dfn}

%Locally nonempty means that for every point $x \in X$ there is an open set $U$,
%$x \in U$, such that $\opn{Ob}(\gerbe{G}(U)) \neq \emptyset$.
%See Figure ??.
\end{frame} 

\begin{frame}
%\begin{figure}
\centering
\includegraphics[scale=0.8, viewport = 100 475 500 735, clip = true]
{stack_2_a_plus.ps}
%\caption{A gerbe $\gerbe{G}$ on $X$}
%\label{fig:stack_2_a}
%\end{figure}

\parbox[t]{\textwidth}{\hilite{bgreen}{Locally nonempty} means that for every
point $x \in X$\\ 
\mbox{} }
\end{frame} 

\begin{frame} 
%\begin{figure}
\centering
\includegraphics[scale=0.8, viewport = 100 475 500 735, clip = true]
{stack_2_b_plus.ps}
%\caption{A gerbe $\gerbe{G}$ on $X$}
%\label{fig:stack_2_b}
%\end{figure}

\parbox[t]{\textwidth}
{there is an open set $U$, $x \in U$, such that 
$\opn{Ob}(\gerbe{G}(U)) \neq \emptyset$. 
\\ \mbox{} }
\end{frame} 

%\begin{frame} 
% Locally connected means that for any pair of objects
%  $i, j \in \opn{Ob}(\gerbe{G}(U))$,
% and for any point $x \in U$, there is an open neighborhood $V$,
% $x \in V \subset U$, such that $i|_V \cong j|_V$ in the groupoid
% $\gerbe{G}(V)$.
%See Figure ??
%\end{frame} 

\begin{frame} 
%\begin{figure}
\centering
\includegraphics[scale=0.8, viewport = 100 475 500 735, clip = true]
{stack_3_a_plus.ps}
%\caption{A gerbe $\gerbe{G}$ on $X$}
%\label{fig:stack_3_a}
%\end{figure}

\parbox[t]{\textwidth}{
\hilite{bgreen}{Locally connected} means that for any pair of objects
$i, j \in \opn{Ob}(\gerbe{G}(U))$, \\
and for any point $x \in U$,}
\end{frame} 

\begin{frame} 
%\begin{figure}
\centering
\includegraphics[scale=0.8, viewport = 100 475 500 735, clip = true]
{stack_3_b_plus.ps}
%\caption{A gerbe $\gerbe{G}$ on $X$}
%\label{fig:stack_3_b}
%\end{figure}

\parbox[t]{\textwidth}{
there is an open neighborhood $V$, 
$x \in V \subset U$, such that \\
$i|_V \cong j|_V$ in the groupoid $\gerbe{G}(V)$.}
\end{frame} 

\begin{frame} 
A \hilite{rust}{stack of linear categories} is a stack $\gerbe{A}$ such that
each of the
categories $\gerbe{A}(U)$ is linear, and the restriction functors 
$\gerbe{A}(U) \to \gerbe{A}(V)$ are linear.
\pause 

A stack of linear categories $\gerbe{A}$ gives rise to a stack of groupoids 
$\gerbe{A}^{\times}$, in the obvious way, namely
\[ \gerbe{A}^{\times}(U) := \gerbe{A}(U)^{\times} \]
for any open set $U$. 
\pause 

We call $\gerbe{A}^{\times}$ the gauge stack of 
$\gerbe{A}$.
\end{frame} 

\begin{frame} 
The next definition is due to Kontsevich \cite{Ko}.
\pause 

\begin{dfn}
A \hilite{dblue}{stack of algebroids} on $X$ is a stack of linear categories
$\gerbe{A}$, whose gauge stack $\gerbe{A}^{\times}$ is a gerbe.
\end{dfn}
\pause 

The most interesting stacks of algebroids are the following kind:
\pause 

\begin{dfn}
Let $\gerbe{A}$ be a stack of algebroids on $X$. We say that 
$\gerbe{A}$ is \hilite{rust}{really twisted} if it has no global objects;
namely the set
$\opn{Ob}(\gerbe{G}(X))$ is empty.
\end{dfn}
\pause 

In Appendix B
%\ref{sec:cech}  
there is an example of a really twisted stack of
algebroids on an algebraic variety $X$, constructed from a nonzero class in
$\mrm{H}^2(X, \mcal{O}_X)$. 
\end{frame} 



\section{3. Categories with Inner Gauge Groups}
\begin{frame}
\sect{3. Categories with Inner Gauge Groups}
\pause

Let's return to the discrete (i.e.\ non-geometric) situation. 
\pause 

Suppose $\cat{A}$ is an algebroid.
We shall try to see what information is needed to recover the
algebroid $\cat{A}$ from the gauge groupoid $\cat{G} = \cat{A}^{\times}$.
\pause 

There are two reasons for doing that. The first is that it might be
easier, in the geometric setup, to study the stack of groupoids than
the stack of algebroids, since it is a less complicated creature.
(I am using the word ``creature'' since ``object'' is over-used
here.) 
\pause 

The other reason is that we need to find analogues of the notion of
algebroid in other contexts -- replacing the category of rings by some other
category, say $\cat{P}$, with suitable structure. 
\end{frame} 

\begin{frame} 
So let $\cat{P}$ be a category. To make things easier to grasp, let's
pretend $\cat{P}$ is some category of algebras. 
\pause 

\begin{dfn} 
An \hilite{dblue}{inner gauge group structure} on $\cat{P}$ is the assignment
of the following data to each algebra $A \in \cat{P}$:
\pause 
\begin{itemize}
\item A group $\opn{IG}(A)$,  called the \hilite{bgreen}{group of inner gauge
transformations}.
\pause 
\item A group homomorphism
\[ \opn{ig} : \opn{IG}(A) \to \opn{Aut}_{\cat{P}}(A) . \]
\end{itemize}
\pause 
These assignments must be functorial in $A$. 
\pause 

We say that $\cat{P}$ is a \hilite{rust}{category with inner gauge groups}.
\end{dfn}
\pause 

A more formal definition can be found in Appendix C. %\ref{sec:whole}.
\end{frame} 

\begin{frame} 
\begin{exa}
Take $\cat{P} := \cat{Ring}$, the category of rings.
\pause 

For a ring $A$ its inner gauge group is $\opn{IG}(A) := A^{\times}$.
\pause 

For an element $g \in A^{\times}$ we define  
\[ \opn{ig}(g) := \opn{Ad}(g) , \]
namely conjugation by the invertible element $g$. 
\pause 

Thus $\cat{Ring}$ is a category with inner gauge groups.
\end{exa}
\end{frame} 


\section{4. Twisted Objects}
\begin{frame}
\sect{4. Twisted Objects}
\pause

\begin{dfn}
Suppose $\cat{P}$ is a category with inner gauge groups.
\pause 

A \hilite{dblue}{twisted object} in $\cat{P}$ is the data 
$(\cat{G}, \cat{A}, \opn{cp})$, where:
\pause 

\begin{itemize}
\item $\cat{G}$ is a nonempty connected groupoid, called the 
\hilite{rust}{gauge groupoid}.
\pause 
\item $\cat{A} : \cat{G} \to \cat{P}$ is a functor, called the 
\hilite{bgreen}{representation}. 
\pause 
\item For any object $i \in \opn{Ob}(\cat{G})$,  
\[ \opn{cp} : \cat{G}(i, i) \iso \opn{IG} (\cat{A}(i)) \]
is an isomorphism of groups,
called the \hilite{rust}{coupling}.
\end{itemize}
\pause 

The coupling isomorphism must satisfy certain consistency conditions.
(See Appendix C for details.)
\end{dfn}
\end{frame} 

\begin{frame} 
Here is an explanation. 
\pause 

We have a groupoid $\cat{G}$. 
\pause 

To any object $i \in \opn{Ob}(\cat{G})$
we assign an algebra $\cat{A}(i)$ in $\cat{P}$.
\pause 

To any arrow $g : i \to j$ in $\cat{G}$ we assign an algebra
isomorphism
\[ \cat{A}(g) : \cat{A}(i) \iso \cat{A}(j) \]
in $\cat{P}$.
This is the representation $\cat{A}$ of $\cat{G}$ in $\cat{P}$.
\pause 

The coupling isomorphism 
\[ \opn{cp} : \cat{G}(i, i) \iso \opn{IG} ( \cat{A}(i) ) \]
forces the groupoid $\cat{G}$ to be made up of gauge groups.
\end{frame} 

\begin{frame} 
We sometimes write $\cat{A}$ instead of $(\cat{G}, \cat{A}, \opn{cp})$; and
then we say that $\cat{A}$ is a twisted object of $\cat{P}$.
\pause 

Here are two examples of twisted objects. 
\pause 

\begin{exa}
As expected, an algebroid $\cat{A}$ is the same as a twisted
object in $\cat{Ring}$. The gauge groupoid is
$\cat{G} := \cat{A}^{\times}$; the representation 
$\cat{A} : \cat{G} \to \cat{Ring}$ is 
$\cat{A}(i) := \cat{A}(i,i)$; and the coupling
$\opn{cp} : \cat{G}(i,i) \iso \cat{A}(i,i)^{\times}$
is the identity map.
\end{exa}
\pause 

\begin{exa}
Any object $A$ of $\cat{P}$ can be made into a twisted object, by
taking $\cat{G}$ to be the groupoid with a single object $0$,
and with group $\cat{G}(0, 0) := \opn{IG}(A)$.
\end{exa}
\pause 

\begin{rem}
All these twisted objects are equivalent to usual objects.
They can be decomposed, just as an algebroid decomposes
(noncanonically) by a choice a base point and paths.
Things only become interesting when we geometrize -- as in Section 2.
%\ref{sec:stacks}.
\end{rem}
\end{frame} 




\section{5. Deformations}
\begin{frame}
\sect{5. Deformations}
\pause

In this section I will introduce two important categories with inner gauge
groups.
\pause 

Let $\K$ be a field of characteristic $0$, and let $C$
be a commutative $\K$-algebra.
\pause 

\begin{dfn}
An \hilite{dblue}{associative deformation of $C$} is a flat,
$\hbar$-adically complete, unital associative
$\K[[\hbar]]$-algebra $A$, 
together with an isomorphism of $\K$-algebras
\[ A / \hbar A \iso C \]
called an \hilite{rust}{augmentation}.
\pause 

A \hilite{dblue}{gauge transformation} $A \to A'$ is 
an isomorphism of $\K$-algebras that commutes with the
augmentations to $C$.
\pause 

We denote by $\cat{AssDef}(C)$ the category of 
associative deformations of $C$, where the morphisms are the
gauge transformations.
\end{dfn}
\end{frame} 

\begin{frame} 
The category $\cat{AssDef}(C)$ has an inner gauge group
structure that
is similar to that of $\cat{Ring}$.
\pause 

For an associative deformation $A$ we take the group
\[ \opn{IG}(A) := \{ g \in A \mid g \equiv 1
\opn{mod} \hbar \} \subset A^{\times} . \]
The inner action $\opn{ig}(g)$ is conjugation. 
\pause 

A twisted object of $\cat{AssDef}(C)$ is called a \hilite{rust}{twisted
associative
deformation of $C$}.
\end{frame} 

\begin{frame} 
Let $A$ be a commutative $\K$-algebra. 
\pause 

Recall that a Poisson bracket on $A$ is
a $\K$-bilinear function
\[ \{ -,- \} : A \times A \to A \]
which is a derivation in each argument, anti-symmetric, and satisfies
the Jacobi identity. 
\pause 

A Poisson $\K[[\hbar]]$-algebra is a commutative $\K[[\hbar]]$-algebra $A$,
endowed with a $\K[[\hbar]]$-bilinear Poisson bracket. 
\pause 

We consider $C$ as a Poisson $\K$-algebra with the zero bracket.
\end{frame} 

\begin{frame} 
\begin{dfn}
A \hilite{dblue}{Poisson deformation of $C$} is a flat
$\hbar$-adically complete Poisson $\K[[\hbar]]$-algebra
$A$, together with a Poisson $\K$-algebra isomorphism 
\[ A / \hbar A \iso C \]
called an \hilite{rust}{augmentation}.
\pause 

A \hilite{dblue}{gauge transformation} $A \to A'$ is a Poisson
algebra isomorphism that commutes with the augmentations to $C$.
\pause 

We denote by $\cat{PoisDef}(C)$ the category of Poisson deformations of $C$,
where the morphisms are the gauge transformations.
\end{dfn}
\pause 

The category $\cat{PoisDef}(C)$ also has a natural inner
gauge group
structure, which we now describe. 
\end{frame} 

\begin{frame} 
Take a Poisson deformation $A$. 
\pause 

The ideal $\hbar A$ is a
pronilpotent Lie algebra, with respect to the Poisson bracket $\{ -,- \}$. 
\pause 

According to the theory of nilpotent Lie algebras in characteristic
$0$, there is an associated pronilpotent group
$\opn{exp}(\hbar A)$, with abstract exponential map
\[ \opn{exp} : \hbar A \to \opn{exp}(\hbar A) . \]
\pause 

We define the inner gauge group to be
\[ \opn{IG}(A) := \opn{exp}(\hbar A) . \]
\end{frame} 

\begin{frame} 
An element $a \in \hbar A$ determines the \hilite{rust}{hamiltonian
derivation}
\[ \opn{ad}(a) := \{ a,- \} \]
of $A$. 
\pause 

Writing
$g := \opn{exp}(a)$, we let
\[ \opn{ig}(g) := \opn{exp}(\opn{ad}(a)) . \]
This is a gauge transformation of $A$. 
\pause 

In the Poisson case it is a bit harder to verify, but indeed this too
is an inner gauge group structure. 
\pause 

A twisted object of $\cat{PoisDef}(C)$ is called a \hilite{dblue}{twisted
Poisson
deformation of $C$}.
\end{frame} 

\begin{frame}
Two remarks to end this section: 
\pause 

$\bullet$ Twisted Poisson deformations do not have
any other description -- unlike twisted associative deformations, that
can be described as a special kind of algebroid, or by Morita theory.
\pause 

$\bullet$ In the associative case the inner
gauge group $\opn{IG}(A)$ can also be described in terms of nilpotent
Lie theory. 
\end{frame} 




\section{6. Twisted Objects in Stacks}
\begin{frame}
\sect{6. Twisted Objects in Stacks}
\pause

In this final section we paint the theory of twisted deformations of sheaves in
broad brush strokes.
\pause 

Again $\K$ is a field of characteristic $0$. Let $X$ be a smooth
algebraic variety over $\K$.
\pause 

In this context we talk about a \hilite{rust}{stack with inner gauge groups}
$\twocat{P}$ on $X$. 
\pause 

This means that for every open set
$U \subset X$ there is a category with inner gauge groups
$\twocat{P}(U)$; and these categories behave correctly when we restrict to
smaller open sets.
\end{frame} 

\begin{frame} 
Recall that the geometric version of a nonempty connected groupoid is a
gerbe. 
\pause 

A \hilite{dblue}{twisted object in $\twocat{P}$} is 
a triple 
$(\gerbe{G}, \gerbe{A}, \opn{cp})$,
consisting of a \hilite{rust}{gauge gerbe} $\gerbe{G}$; 
a functor of stacks
$\gerbe{A} : \gerbe{G} \to \twocat{P}$
called the \hilite{bgreen}{representation};
and a \hilite{rust}{coupling isomorphism} $\opn{cp}$. 
\pause 

On any open set $U$, these yield a pre-twisted object
$\bigl( \gerbe{G}(U), \gerbe{A}(U), \opn{cp} \bigr)$
in the category with inner gauge groups $\twocat{P}(U)$.
\pause 

(By writing ``pre-twisted'' I am allowing for the possibility that the groupoid 
$\gerbe{G}(U)$ is empty of not connected.)
\end{frame} 

\begin{frame} 
\begin{exa}
Consider the stack
$\twocat{P} := \twocat{Ring} X$. 
This is the stack on $X$
which assigns to an open set $U$ the  category
$\twocat{P}(U) := \cat{Ring} U$ of sheaves of
rings on $U$. 
\pause 

$\twocat{P}$ is a stack with inner gauge groups.
\pause 

A twisted object in $\twocat{Ring} X$ is precisely a stack of
algebroids.
\end{exa}
\pause 

In a similar (but technically much harder) way we can define the stacks with
inner gauge groups
$\twocat{AssDef}(\mcal{O}_X)$ and $\twocat{PoisDef}(\mcal{O}_X)$ on $X$.
\pause 

A \hilite{dblue}{twisted associative  deformation of $\mcal{O}_X$} is a twisted
object of
$\twocat{AssDef}(\mcal{O}_X)$.
\pause 

A \hilite{rust}{twisted Poisson  deformation of $\mcal{O}_X$} is a twisted
object of
$\twocat{PoisDef}(\mcal{O}_X)$. 
\end{frame} 

\begin{frame} 
A twisted object $(\gerbe{G}, \gerbe{A}, \opn{cp})$ is said to be
\hilite{dblue}{really
twisted} if there are no global objects; i.e.\ if the groupoid
$\gerbe{G}(X)$ is empty.
\pause 

\begin{exa}
The stack of algebroids $\til{\gerbe{A}\ }$ constructed in Appendix B,
%\ref{sec:cech}, 
from a nonzero cohomology class in $\mrm{H}^2(X, \mcal{O}_X)$, 
is a twisted associative deformation of $\mcal{O}_X$.
\pause 

It is also a twisted Poisson  deformation of $\mcal{O}_X$.
\pause 

This deformation is really twisted.
\end{exa}
\end{frame} 

\begin{frame} 
In the paper \cite{Ye2} I proved the following theorem:
\pause 

\begin{thm}
\hilite{bgreen}{There is a canonical bijection
\[ \begin{aligned}
& \opn{tw{.}quant} : 
\frac{ \{ \text{twisted Poisson deformations of $\mcal{O}_X$} \} }
{\text{equivalence}} \\
& \hspace{15ex} \xto{\, \simeq \, } \
\frac{ \{ \text{twisted associative deformations of $\mcal{O}_X$} \} }
{\text{equivalence}} 
\end{aligned} \]
called the} \hilite{rust}{twisted quantization map}.
\end{thm}
\end{frame} 

\begin{frame} 
Let me end with a question from \cite{Ye2}, that in my opinion is very
interesting:
\pause 

\begin{que}
Suppose  $X$ is a Calabi-Yau surface, and
$\beta  = \{ -,- \}$ is a nonzero Poisson bracket on $\mcal{O}_X$,
\pause 

Consider the Poisson deformation 
$\mcal{A} := \mcal{O}_X[[\hbar]]$ 
of $\mcal{O}_X$ with bracket $\hbar \beta$.
\pause 

Is the resulting twisted associative deformation
\[ \gerbe{B} := \opn{tw{.}quant} (\mcal{A}) \]
really twisted?
\end{que}
\pause 

\begin{center}
\hilite{dblue}{\bfseries \red - END -}
\end{center}
\end{frame} 

% 
% 
% \appendix
% 
% \section{Morita Theory} 
% \label{sec:morita}
% 
% In this appendix I will explain the relation between algebroids and
% Morita theory. 
% 
% Let $\cat{C}$ be an abelian category with infinite direct sums.
% 
% Given an object $P \in \opn{Ob}(\cat{C})$, let $A$ be the ring
% $A := \opn{Hom}_{\cat{C}}(P, P)$.
% 
% For any $M \in \opn{Ob}(\cat{C})$ the set
% $\opn{Hom}_{\cat{C}}(P, M)$ is a right $A$-module (by
% pre-composition). So 
% there is a functor
% \[ \opn{Hom}_{\cat{C}}(P, -) : \cat{C} \to \cat{Mod} A^{\mrm{op}} . \]
% 
% An object $P$ is called a \hilite{dblue}{progenerator} of $\cat{C}$
% if it has these properties:
% \begin{itemize}
% \item Projective; namely the functor
% $\opn{Hom}_{\cat{C}}(P, -)$ is exact.
% \item Generator; namely $M \neq 0$ implies 
% $\opn{Hom}_{\cat{C}}(P, M) \neq 0$.
% \item Compact; namely the functor $\opn{Hom}_{\cat{C}}(P, -)$ commutes
% with direct sums.
% \end{itemize}
% 
% Morita Theory says that $P$ is a progenerator if and only if the
% functor $\opn{Hom}_{\cat{C}}(P, -)$ is an equivalence.
% 
% Now suppose we are given a collection $\{ P_i \}_{i \in I}$ of
% progenerators of $\cat{C}$, indexed by a nonempty set $I$, all
% isomorphic to each other. 
% 
% Consider the algebroid
% $\cat{A}$ with $\opn{Ob}(\cat{A}) := I$, and 
% \[ \cat{A}(i, j) := \opn{Hom}_{\cat{C}}(P_i, P_j) . \]
% 
% Define
% $\cat{Mod} \cat{A}^{\mrm{op}}$ to be the category of contravariant
% additive functors 
% $\cat{A} \to \cat{Ab}$. This turns out to be an abelian category.
% 
% The ``modern Morita Theory'' says that $\cat{C}$ is
%  \hilite{dblue}{canonically}
% equivalent to $\cat{Mod} \cat{A}^{\mrm{op}}$.
%  
% For more about these ideas see \cite{KS, LV, Lo}.
% 
% If we choose a decomposition of the algebroid $\cat{A}$, then it becomes
% isomorphic to a path algebroid (see next example).
% 
% \begin{exa}
% Take a ring $A$ and a nonempty set $I$. Define an algebroid $\cat{A}$
% as follows. 
% 
% As object set we take
% $\opn{Ob}(\cat{A}) := I$.
% 
% The morphism sets are
% $\cat{A}(i, j) := A$, with addition and composition coming from
% addition and multiplication in $A$.
% 
% We call this algebroid the \hilite{dblue}{path algebroid} over the set $I$,
% with 
% coefficients in $A$. 
% 
% In case $I = \{ 1, \ldots, n \}$, then by ``totalizing'' we get the
% ring
% \[ \prod_{i, j} \cat{A}(i, j)  , \]
% which is canonically isomorphic to the matrix ring $\opn{M}_n(A)$. 
% 
% Symbolically the isomorphism is
% \[ \bmat{
% \cat{A}(1, 1)
% &
% \cat{A}(2, 1)
% &
% \cdots
% \\
% \cat{A}(1, 2)
% &
% \cat{A}(2, 2)
% \\
% \vdots
% } \]
% \end{exa}
% 
% 
% 
% 
% \section{A Really Twisted Stack of Algebroids} \label{sec:cech}
% 
% Let $X$ be an algebraic variety over a field $\K$, with structure
% sheaf $\mcal{O}_X$.
% 
% Suppose $c \in \mrm{H}^2(X, \mcal{O}_X)$ is a nonzero cohomology
% class.  (This happens often.)
% 
% Choose some affine open covering $\{ U_i \}_{i \in I}$ of $X$.
% 
% Let $\{ c_{i_0, i_1, i_2} \}$ be a \v{C}ech $2$-cocycle 
% representing $c$ on this open covering. Thus
% \[ c_{i_0, i_1, i_2} \in \Gamma(U_{i_0, i_1, i_2}, \mcal{O}_X) , \]
% where we write
% \[ U_{i_0, i_1, i_2} := U_{i_0} \cap U_{i_1} \cap U_{i_2} . \]
% 
% We may assume that 
% \[ c_{i_0, i_1, i_1} = c_{i_0, i_0, i_1} = 0 . \]
% 
% Let $\hbar$ be a variable, and consider the sheaf of commutative
% $\K[[\hbar]]$-algebras $\mcal{O}_X[[\hbar]]$.
% 
% For any open set $V$ let us define a category 
% $\gerbe{A}(V)$ as follows. 
% 
% The set of objects is
% \[ \opn{Ob} \bigl( \gerbe{A}(V) \bigr) :=
% \{ i \in I \mid V \subset U_i \} . \]
% See figures.
% 
% \begin{figure}
% \includegraphics[scale=0.8, viewport = 100 535 500 750, clip = true]
% {covering_0_plus.ps}
% %\label{fig1}
% \caption{An open covering $\{ U_0, U_1, \ldots \}$ of the space $X$.}
% \end{figure}
% 
% \begin{figure}
% \includegraphics[scale=0.8, viewport = 100 535 500 750, clip = true]
% {covering_1_plus.ps}
% %\label{fig1}
% \caption{For this open set $V$, the set of objects of the category
% $\gerbe{A}(V)$ is $\{ 1 \}$.}
% \end{figure}
% 
% \begin{figure}
% \includegraphics[scale=0.8, viewport = 100 535 500 750, clip = true]
% {covering_2_plus.ps}
% %\label{fig1}
% \caption{But now the set of objects of $\gerbe{A}(V)$ is $\{ 0, 1 \}$.}
% \end{figure}
% 
% For $i_0, i_1 \in \opn{Ob} \bigl( \gerbe{A}(V) \bigr)$,
% the set of morphisms is the $\K[[\hbar]]$-module
% \[ \gerbe{A}(V)(i_0, i_1) := \Gamma(V, \mcal{O}_X[[\hbar]]) . \]
% 
% The composition rule 
% \[ \gerbe{A}(V)(i_0, i_1) \times \gerbe{A}(V)(i_1, i_2) \to 
% \gerbe{A}(V)(i_0, i_2) \]
% is
% \[ a_2 \circ a_1 := a_2 \cdot a_1 \cdot 
% \opn{exp}(\hbar c_{i_0, i_1, i_2}) , \]
% for $a_1, a_2 \in \Gamma(V, \mcal{O}_X[[\hbar]])$.
% Here $\cdot$ is the multiplication in the ring
% $\Gamma(V, \mcal{O}_X[[\hbar]])$, and $\opn{exp}$ is the usual
% exponential power series. 
% 
% The composition is associative because 
% $\{ c_{i_0, i_1, i_2} \}$ is a $2$-cocycle.
% 
% Thus $\gerbe{A}(V)$ is a linear category.
% 
% If $W \subset V$ is a smaller open set, then there is an additive functor
% \[ \gerbe{A}(V) \to \gerbe{A}(W) . \]
% On objects it is the inclusion
% \[ \opn{Ob} \bigl( \gerbe{A}(V) \bigr) \subset
% \opn{Ob} \bigl( \gerbe{A}(W) \bigr) , \]
% and on morphisms it is the canonical homomorphism 
% \[ \Gamma(V, \mcal{O}_X[[\hbar]]) \to 
% \Gamma(W, \mcal{O}_X[[\hbar]]) . \]
% 
% What this means is the the collection of categories
% $\gerbe{A} := \{ \gerbe{A}(V) \}$ is a \hilite{dblue}{prestack of algebroids}
% on $X$, as defined by Kontsevich in \cite{Ko}.
% 
% For the precise definition of a prestack of algebroids see 
% \cite{Ko, KS, DP, Lo, Ye2}.
% 
% {}From the prestack $\gerbe{A}$ we can pass to the associated
% \hilite{dblue}{stack
% of algebroids} $\til{\gerbe{A}\ }$. 
% (This is analogous to the sheaf associated to a presheaf.)
% 
% One can show that the stack $\til{\gerbe{A}\ }$ depends only on the cohomology
% class $c$, not on the cocycle that we chose.
% 
% The stack $\til{\gerbe{A}\ }$ will have more local objects than the prestack
% $\gerbe{A}$. Indeed, for an open set $V$, the set of objects 
% $\opn{Ob} \bigl( \til{\gerbe{A}\ }(V) \bigr)$
% is nonempty if and only if the cohomology class
% $c|_V \in \mrm{H}^2(V, \mcal{O}_X)$ vanishes. This includes all affine open
% sets.
% 
% On the other hand 
% \[ \opn{Ob} \bigl( \til{\gerbe{A}\ }(X) \bigr) = \emptyset . \]
% Hence we say that the stack of algebroids $\til{\gerbe{A}\ }$ is
% \hilite{dblue}{really
% twisted}.
% 
% 
% 
% 
% 
% \section{Inner Gauge Groups -- the Details}
% \label{sec:whole}
% 
% Let us denote by $\cat{Grp}$ the category of groups. 
% 
% Given a category $\cat{P}$, there is a functor
% \[ \opn{Aut}_{\cat{P}} : \cat{P}^{\times} \to \cat{Grp} \]
% defined in the following silly way. 
% For an object $A \in \cat{P}$ we let
% \[ \opn{Aut}_{\cat{P}}(A) := \opn{Isom}_{\cat{P}}(A, A) = 
% \opn{Hom}_{\cat{P}^{\times}}(A) . \]
% 
% And for an isomorphism $g : A_0 \to A_1$ in $\cat{P}$ we let
% \[ \opn{Aut}_{\cat{P}}(g) := \opn{Ad}(g) : 
% \opn{Aut}_{\cat{P}}(A_0) \iso \opn{Aut}_{\cat{P}}(A_1) .  \]
% Recall that this means that 
% \[ \opn{Aut}_{\cat{P}}(g)(h) = g \circ h \circ g^{-1} \]
% for $h \in \opn{Aut}_{\cat{P}}(A_0)$.
% 
% Continuing with this nonsense, suppose 
% $F : \cat{P} \to \cat{Q}$ is a functor. There there is a natural
% transformation
% \[ \opn{Aut}_{F} : \opn{Aut}_{\cat{P}} \to \opn{Aut}_{\cat{Q}}
% \circ \, F \]
% of functors $\cat{P}^{\times} \to \cat{Grp}$. 
% For an object $A \in \cat{P}$ the formula for the group homomorphism
% \[ \opn{Aut}_{F} : \opn{Aut}_{\cat{P}}(A) \to 
% \opn{Aut}_{\cat{Q}}(F(A)) \]
% is
% \[ \opn{Aut}_{F}(h) := F(h)  . \]
% 
% \begin{dfn}
% Let $\cat{P}$ be category. An \hilite{dblue}{inner gauge group} structure on
% $\cat{P}$ is a functor
% \[ \opn{IG} : \cat{P} \to \cat{Grp} , \]
% together with a natural transformation
% \[ \opn{ig} : \opn{IG} \to \opn{Aut}_{\cat{P}} \]
% of functors $\cat{P}^{\times} \to \cat{Grp}$. 
% \end{dfn}
% 
% We refer to the data $(\cat{P}, \opn{IG}, \opn{ig})$ as 
% \hilite{dblue}{category with inner gauge groups}.
% 
% \begin{rem}
% Let $A$ be an object of the category with inner gauge groups $\cat{P}$. The
% pair of groups
% \[ \bigl( \opn{IG}(A), \opn{Aut}_{\cat{P}}(A) \bigr) , \]
% together with the homomorphism
% $\opn{ig} : \opn{IG}(A) \to \opn{Aut}_{\cat{P}}(A)$,
% and the action of $\opn{Aut}_{\cat{P}}(A)$ on $\opn{IG}(A)$ by functoriality,
% is called a \hilite{dblue}{crossed module}. 
% 
% This notion appears in recent papers on stacks (e.g.\ \cite{BM}) and in
% mathematical physics (e.g.\ \cite{BS}).
% \end{rem}
% 
% Here is the full definition of twisted objects.
% 
% \begin{dfn}
% Suppose $(\cat{P}, \opn{IG}, \opn{ig})$ is a category with inner gauge groups.
% A \hilite{dblue}{twisted object} in $(\cat{P}, \opn{IG}, \opn{ig})$ is the
% data 
% $(\cat{G}, \cat{A}, \opn{cp})$, where:
% \begin{itemize}
% \item $\cat{G}$ is a nonempty connected groupoid, called the {\em
% gauge groupoid}.
% \item $\cat{A} : \cat{G} \to \cat{P}$ is a functor, called the {\em
% representation}. 
% \item For any object $i \in \opn{Ob}(\cat{G})$,  
% \[ \opn{cp} : \cat{G}(i, i) \iso \opn{IG} (\cat{A}(i)) \]
% is an isomorphism of groups,
% called the \hilite{dblue}{coupling}.
% \end{itemize}
% 
% The conditions are:
% \begin{enumerate}
% \rmitem{i} The coupling is functorial in $i$.
% \rmitem{ii} For any $g \in \cat{G}(i, i)$, its functorial action
% $\cat{A}(g)$ on $\cat{A}(i)$ coincides with its inner action
% $\opn{ig}(\opn{cp}(g))$.
% \end{enumerate}
% \end{dfn}
% 
% 
% 
% 



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    Higher Gauge Theory, 
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\bibitem[BM]{BM} L. Breen and W. Messing, 
    Differential geometry of gerbes,
    Adv.\ Math.\ {\bf 198} (2005), no.\ 2, 732-846.
\bibitem[DP]{DP}  A. D'Agnolo and P. Polesello,
    Stacks of twisted modules and integral transforms,
    eprint math.AG/0307387 at http://arxiv.org.
\bibitem[Gi]{Gi} J. Giraud, 
    ``Cohomologie non abelienne,'' 
    Grundlehren der Math.\ Wiss.\ {\bf 179}, Springer (1971).
\bibitem[Ko]{Ko} M. Kontsevich,
    Deformation quantization of algebraic varieties,
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\bibitem[KS]{KS} M. Kashiwara and P. Schapira,
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\bibitem[Lo]{Lo} W. Lowen, 
    Algebroid prestacks and deformations of ringed spaces,
    eprint math.AG/0511197 at http://arxiv.org.  
\bibitem[LV]{LV} W. Lowen and M. Van den Bergh,
    Deformation theory of abelian categories,
    Trans.\ AMS {\bf 358} (2006) no.\ 12, 5441 - 5483.
\bibitem[Ye1]{Ye1} A.\ Yekutieli, 
	Central Extensions of Gerbes,
	eprint arXiv:0801.0083 at http://arxiv.org.
\bibitem[Ye2]{Ye2} A. Yekutieli,
    Twisted Deformation Quantization of Algebraic Varieties,
    in preparation.
\end{thebibliography}

\end{document}