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\section*{Oberwolfach Workshop, June 2009 -- Monday, June 8}

Here are notes from [[Urs Schreiber]] for Monday, June 8, from [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology|Oberwolfach]].

\hypertarget{schweigert_cft_and_algebra_in_braided_tensor_categories_i}{}\subsection*{{Schweigert: CFT and algebra in braided tensor categories, I}}\label{schweigert_cft_and_algebra_in_braided_tensor_categories_i}

\hypertarget{1_modular_tensor_categories_and_rational_cft}{}\subsubsection*{{1. Modular tensor categories and rational CFT}}\label{1_modular_tensor_categories_and_rational_cft}

\hypertarget{11}{}\paragraph*{{1.1}}\label{11}

\begin{itemize}%
\item consider a \emph{rational semisimple conformal vertex algebra} $V$


\item $\to$ from this we get a representation category $C$, which is a [[braided monoidal category]]

\begin{itemize}%
\item that $V$ is semisimple and rarional makes $C$ in fact a [[modular tensor category]]

\end{itemize}

\item $\to$ this gives rise to conformal blocks



\end{itemize}
\textbf{Definition: [[modular tensor category]]}

\begin{itemize}%
\item [[abelian category]], $\mathbb{C}$-linear (i.e. $Vect_{\mathbb{C}}$ [[enriched category]]), [[semisimple category]] [[monoidal category|tensor category]]


\item the tensor unit is a simple object, $I$ a finite set of representatives of isomorphism classes of simple objects


\item [[fusion category]]


\item [[braided monoidal category]]


\item [[ribbon category]], in particular objects have duals


\item \textbf{modularity} a non-degeneracy condition on the braiding given by an isomorphism of algebras

\begin{displaymath}
K(C) \otimes_{\mathbb{Z}} \stackrel{\simeq}{\to} End(Id_C)
\end{displaymath}
where

\begin{displaymath}
[U] \mapsto \alpha_U
\end{displaymath}
where the transformation $\alpha_U$ is given on the simple object $V$ by

\begin{displaymath}
\alpha_U(V) = straight V-line encircled by U-loop
\end{displaymath}
(on the right we use string diagram notation)



\end{itemize}
\hypertarget{12}{}\paragraph*{{1.2}}\label{12}

\textbf{Fact ([[Reshitikhin-Turaev model]])} for any [[modular tensor category]] $C$ there is a [[monoidal functor]]

\begin{displaymath}
tft_C : cobord_{3,2}^C \to vect_f(\mathcal{C})
\end{displaymath}
\begin{itemize}%
\item 1) factorization homomorphism: given a surface $\Sigma_{U,U^*}$ with two marked points labeled by $U$ and $U^*$ consider the 3d cobordism from $\Sigma$ to the result of gluing a handle to $\Sigma$ that connects the two marked points with a $U$-line running inside the handle. the corresponding linear map given by the tft

\begin{displaymath}
fact_U := tft_C(\Sigma_{U,U^*} \to \Sigma)
\end{displaymath}
we have


\item \begin{displaymath}
\oplus_{i \in I} fact_{U_i} 
 : 
 \oplus_{i \in I} tft_C(\Sigma_{U,U^*})
 \stackrel{\simeq}{\to}
 tft_C(X)
\end{displaymath}

\item a representation of the mapping class group $Map(\Sigma)$ on $tft_C(X)$



\end{itemize}
\hypertarget{2_cft_correlator}{}\subsubsection*{{2. CFT correlator}}\label{2_cft_correlator}

\hypertarget{21}{}\paragraph*{{2.1}}\label{21}

\begin{itemize}%
\item $X$ a 2-dimensional conformal manifold, either oriented or unoriented without boundary


\item for the purpose of this talk restrict to the oriented case (but unoriented case has been dealt with, too)


\item also, from now on $X$ regarded just as a topological surface, no longer a conformal one



\end{itemize}
\hypertarget{22}{}\paragraph*{{2.2}}\label{22}

strategy

\begin{itemize}%
\item decorate $X$


\item appropriate spaces of ``functions'' for correlators



\end{itemize}
\textbf{holomorphic factorization}

\begin{itemize}%
\item pass from $X$ to its $\mathbb{Z}_2$ \emph{orientation bundle} $\hat X$, the \emph{orientation double cover} (identifying points on the boundary, though) (an orientation twisted version of what is called the \textbf{double} )


\item double accounts for what physicists call left- and rightmoving degrees of freedom



\end{itemize}
\textbf{goal}

\begin{itemize}%
\item 1) find a decoration for $X$ such that $\hat X \in cobord_{3,2}^C$


\item 2) specify the correlator $Cor(X) \in tft_C(\hatt X)$

\begin{itemize}%
\item a) $Cor(X)$ invariant under $Map(\hat X)^{\mathbb{Z}_2}$ \textbf{modular invariance}


\item b) compatibility with factorization homomorphism (technical to state, dropped here)



\end{itemize}


\end{itemize}
\hypertarget{23}{}\paragraph*{{2.3}}\label{23}

\textbf{Insight.} decoration bicategory of special symmetric Frobenius algebras in $C$

\begin{itemize}%
\item Frobenius algebra in $C$: an object

\begin{displaymath}
(A \in Obj(C), \eta : 1 \to A,\epsilon : A \to 1,  
  m : A \otimes A \to A, \Delta : A \to A \otimes A)
\end{displaymath}
which is a unital associative algebra and counital coassociative coalgebra in $C$


\item being Frobenius means that the coproduct $\Delta : A \to A \otimes A$ is a homomorphism of $A$-bimodules


\item being \emph{symmetric} means that the two obvious nontriviall morphisms $A \to A^*$ that one can build using unit and counit are equal


\item being special means that $m \circ \Delta = Id_A$ and $\epsilon \circ \eta = dim(A) Id_1$



\end{itemize}
\hypertarget{24}{}\paragraph*{{2.4}}\label{24}

A typical worldsheet $X$: higher genus surface with defect lines and marked points drawn on it

\textbf{decoration}

\begin{itemize}%
\item to 2-dimensional cells assign special symmetric Frobenius algebras $A$;


\item to 1-dimensional cells

\begin{itemize}%
\item to boundary lines a left or right module over the respective Frobenius algebra (boundary is oriented);


\item to a defect line: a bimodule over the respective Frobenius algebras



\end{itemize}

\item to 0-dimensional cells

\begin{itemize}%
\item to a junction of three defect lines labeled by three bimodules $D_i$, associate an element in $Hom_{A_1| A_3}(D_1 \otimes_{A_2} D_2, D_3)$


\item boundary field insertions: for marked point on the boundary $x \in \partial X$ attach a simple object $U \in Obj(C)$ to a marked point on a given defect line (or rather, on a junction of two defect lines) and an element in $Hom_A(D_1 \otimes U, D_2)$


\item bulk field insertion: for $x$ not in the boundary but in the inside of $X$, consider the two preimages in $\hat X$, assign simple objects $U, V$ to these, respectively and assign an element in

\begin{displaymath}
Hom_{A_1|A_2}(U \otimes A_1 \otimes V, A_2)
\end{displaymath}

\item here the bimodule structure on these tensor products are given by over- and underbraiding, respectively (depending on orientation)



\end{itemize}


\end{itemize}
\hypertarget{25}{}\paragraph*{{2.5}}\label{25}

correlators from cobordisms

given $X$ consider the cobordism from the empty surface

\begin{displaymath}
\emptyset \stackrel{M_X}{\to} \hat X
\end{displaymath}
given by

\begin{displaymath}
M_X = (\hat X \times [-1,1])/_{\sigma t \sim -t}
\end{displaymath}
and set

\begin{displaymath}
Cor(X) = tft_C(M_X) 1  \in tft_C(\hat H)
\end{displaymath}
\textbf{Example}

$X$ the disk with a defect line running across it from boundary to boundary.

then

\begin{itemize}%
\item $\hat X$ is the sphere


\item $M_X$ is the 3-ball


\item $X$ sits inside the equatorial plane of $M_X$: one copy of $[-1,1]$ over each of its interior points connectint its two preimages in $\hat X$, in addtion one copy of the interval connecting each boundary point to its unique preimage in $\hat X$



\end{itemize}
\begin{quote}%
long discussion with audience ensues: audience wants to better understand why all these constructions are being undertaken. The answer is in the theorem to come: every assignment of correlators as above (compatible with factorization morphism and invariant in the above sense under mapping class group (preserving defect decoration)) is obtained by the recipe presented here


\end{quote}
\hypertarget{runkel_cft_and_modular_tensor_categories_ii}{}\subsection*{{Runkel: CFT and modular tensor categories, II}}\label{runkel_cft_and_modular_tensor_categories_ii}

\begin{itemize}%
\item Ingo Runkel, Monday morning 11.15. CFT and algebra in braided tensor categories II; notes by [[Bruce Bartlett]]: [[talk2.pdf:file]]

\end{itemize}
\hypertarget{freed}{}\subsection*{{Freed}}\label{freed}

joint work with [[Jacques Distler]] and Greg Moore.

(on [[differential cohomology]] of background fields in type II [[string theory]], in particular on [[orientifold]] backgrounds)

\begin{itemize}%
\item $\Sigma$ in this talk a compact 2-dimensional manifold: the \emph{worldsheet}

\begin{itemize}%
\item (called $X$ in the morning talks)

\end{itemize}

\item $X$ smooth 10-dimensional manifold: \emph{spacetime}


\item fields $\Sigma \stackrel{\phi}{\to} X$ (see [[sigma-model]] )


\item 2-dimensional theory $\to$ 10-dimensional theory


\item more generally: $X$ is an [[orbifold]]


\item today: a variation of this called \emph{orientiold} : $X_W \to X$ is a double cover

\begin{itemize}%
\item physicists think of $X_W$ as the spacetime equipped with an involution, but really spacetime is the $\mathbb{Z}_2$-quotient $X$

\end{itemize}

\item I) $\sigma \in \mathbb{Z}_2$ acts trivially : \emph{type I string}


\item II) section $X_W \stackrel{\leftarrow}{\to} X$ \emph{type II string theory}



\end{itemize}
this project started when Jacques Distler showed Dan Freed a certain formula, namely

let

\begin{displaymath}
i : F \hookrightarrow  X_w
\end{displaymath}
be the fixed point

\begin{itemize}%
\item set\begin{displaymath}
RR-charge = \pm 2^{something} i_*
  ( \sqrt{\frac{L'(F)}{L'(\nu)}} )
\end{displaymath}
where

\begin{displaymath}
L' = \prod \frac{x/4 u}{tanh x/4u}
\end{displaymath}
where * $u$ is the Bott generator of K-theory * $\nu$ is normal bundle of $F \hookrightarrow X_W$ * with another factor this would be the Hrizebruch L-function, this way it is not



\end{itemize}
what is it?

this led to thinking about the following

\begin{itemize}%
\item 1) definition of fields/theory


\item 2) derive RR-charge formula over $\mathbb{Z}[1/2]$ from $10 d$


\item 3) anomaly cancellation in 2d


\item there is paper on arXiv with a summary, but this is still work in progress



\end{itemize}
diff geoemtric structures that one needs to make sense of this:

\begin{itemize}%
\item differential cohomology


\item twistings



\end{itemize}
\hypertarget{differential_cohomology}{}\subsubsection*{{differential cohomology}}\label{differential_cohomology}

suppose $h$ is any cohomology theory

then $h$ with rational coefficients $h(-; \mathbb{Q}) = H(-;h(pt, \mathbb{Q}))^\bullet$

here the coefficient object

\begin{displaymath}
h_Q := h(pt;\mathbb{Q})
\end{displaymath}
on the right is a graded ring and the bulleted-degree is the corresponding total degree of that and the cohomology degree

consider the [[homotopy limit]] which gives [[differential cohomology]]

\begin{displaymath}
\itexarray{
    h^{v \bullet} &\to& \Omega(M, h_{\mathbb{R}})_{closed}
    \\
    \downarrow && \downarrow
    \\
    h^\bullet(M) &\to& H(M; h_{\mathbb{R}})
  }
\end{displaymath}
\begin{quote}%
question from Mike Hopkins: are there choices in the bottom horizontal map; doesn't one have to choose a basis of cocycles?


\end{quote}
we get from this two exact sequences

\begin{displaymath}
0 \to h(M;h_{\mathbb{R}} \otimes \mathbb{R}/\mathbb{Z})^{q-1}
  \stackrel{curv}{\to}\Omega(M;h_{\mathbb{R}})^q_{\mathbb{Z}} \to 0
\end{displaymath}
\begin{displaymath}
0 \to forms \to h^{v \bullet}(M) \to h^q(M) \to 0
\end{displaymath}
\begin{itemize}%
\item this was introduced for ordinary cohomology by Cheeger and Simons and generalized by Hopkins and Singer


\item we can think of the cohomology group as components of some space

\begin{displaymath}
h^q(M) = \pi_0(Map(M, h_q))
\end{displaymath}
and

\begin{displaymath}
h^{v q}(M) = \pi_0(--)
\end{displaymath}

\item various people constructed various models for this, such as Bunke and Schick; by Gomi for equivariant cohomology, also Szabo and Alessandro



\end{itemize}
\textbf{examples}

\begin{displaymath}
H^{v 1}(M) = Map(M, \mathbb{T})
\end{displaymath}
\begin{displaymath}
H^{1}(M) = \pi_0 Map(M, \mathbb{T})
\end{displaymath}
\begin{displaymath}
H^{v 2}(M) = \{line bundles with connection\}
\end{displaymath}
\begin{displaymath}
H^{v 3}(M) = \{line bundle gerbes with connection\}
\end{displaymath}
\textbf{remarks}

\begin{itemize}%
\item the application to string theory here will have completely topological flavor


\item defining certain terms in an action like Wess-Zumino-Witten term and chern-Simons term, these are nicely understood in terms of differential cohomology: observation goes back to Gawedzki


\item in physics the forms are ``currents'' which say where charges are located, the class in real cohomology is the total charge


\item in quantum physics this total charge has to be quantized (sit on a lattice inside the real cohomology)


\item so the above pullback diagram says that classical charges are to be combined with quantization condition in order to give physical fields



\end{itemize}
\hypertarget{twistings_of_}{}\subsubsection*{{twistings of $KR(X_w)$}}\label{twistings_of_}

so consider again $\pi : X_w \to X$ a double cover

\begin{itemize}%
\item an object in $KR^0(X_W)$ is represented by

\begin{itemize}%
\item a $\mathbb{Z}_2$-graded complex

\end{itemize}


\end{itemize}
vector bundle $E \to X_W$ (in terms of pseudobundles: even part minus odd part)

\begin{itemize}%
\item $\tilde \sigma : \sigma^* \bar E \to E$

\end{itemize}
recall that $\sigma$ may have fixedpoints

special cases

\begin{itemize}%
\item $\sigma$ acts trivially: we get just $KO^0(X_w)$-theory


\item $X_w \to X$ has a section: $K^0(X)$



\end{itemize}
\textbf{twisting}

pass to a locally equivalent groupoid

\begin{displaymath}
Y_W \to Y
\end{displaymath}
\begin{displaymath}
Y : (Y_0  \stackrel{\leftarrow}{\leftarrow} Y_1 \stackrel{\stackrel{\leftarrow}{\leftarrow}}{\leftarrow}
 \cdots)
\end{displaymath}
notation: $V^\phi =$ $V$ if $\phi = 0$ and $\bar V$ otherwise

\textbf{definition}

a twsiting of $KR(X_W)$ is an equivalent thing $Y_w \to Y$ as above

where

\begin{displaymath}
d : Y_0 \to \mathbb{Z}
\end{displaymath}
continuous

\begin{displaymath}
L \to Y_1
\end{displaymath}
hermitian line bundle, $\mathbb{Z}_2$-graded

\begin{displaymath}
\theta: L_g^{\phi(f)} \otimes L_f \to L_{g f}
\end{displaymath}
cocycle condition for $\theta$

recognize these twistings as classified by some cohomology theory

\textbf{cohomology group}

For $K(X)$: $\pi_{\{0,1,2,3\}h \simeq \{\mathbb{Z}, \mathbb{Z}_2, 0 , \mathbb{Z}\}$

for some $h$ that we are not being told about

For $KO(X_W)$: $\pi_{\{0,1,2\}} k_{0 \lt 0..2\gt} \simeq \{\mathbb{Z},  \mathbb{Z}_2, \mathbb{Z}_2\}$

for $KR(X_W)$ the iso classes are

\begin{displaymath}
H^0(X, \mathbb{Z}) \times H^1(X; \mathbb{Z}) \times
   H^{w+3}(X, \mathbb{Z})
\end{displaymath}
as a set

\begin{displaymath}
u \in K^2(pt) \in KR^{\tau_1 +2}(pt)
\end{displaymath}
\begin{quote}%
speaker is running out of time, coherence is being lost a bit\ldots{} notetaker misses to take notes on some central statement on these twisted cohomology classes, but see the arXiv article


\end{quote}
\hypertarget{string_backgrounds}{}\subsubsection*{{string backgrounds}}\label{string_backgrounds}

the differential cocycles are background data for the 2-d theory and field data for the 10 d theory (see [[sigma-model]])

\textbf{def} an NS-NS superstring background is

\begin{itemize}%
\item i) a smooth 10d orbifold with metruc and real function (dilaton field)


\item ii) $\pi : X_W \to X$ orientifold double cover


\item iii) $\beta^v$ a differential twisting of $KR(X_w)$: the $B$-field


\item iv) K : $R(\beta) \to \tau^{KO}(T X -2)$ iso of twistings of $KO(X)$: twisted Spin-structure



\end{itemize}
Bott shift, leading to equivalent theories $\beta^v \to \beta^v + (\tau^v + 2)$

and something else

Stiefel-Whitney classes

\begin{displaymath}
w_1(X) = t w
\end{displaymath}
\begin{displaymath}
w_2(X) =  t w^2 + a w
\end{displaymath}
aim: mix iii) and iv)

2d theory: A worldsheet $\Sigma$ with metric, a spin structure on $\hat \Sigma$: the orientation double cover

\begin{displaymath}
\itexarray{
     \hat \Sigma &\to& X_w
     \\
     \downarrow && \downarrow
    \\
    \Sigma &\to& X
  }
\end{displaymath}
\begin{displaymath}
\phi^* w \simeq \hat w
\end{displaymath}
and spme spinor fields on $\hat \Sigma$

in the [[path integral]]: integrate over all of these pieces of data

$\to$ effective action Pfaffian of Dirac operator

\begin{displaymath}
Pfaff  D_{\hat \Sigma}(\phi^*T X - 2)
  \cdot
    exp 2 \pi i \int_{\Sigma/S} \phi^* \beta^v
\end{displaymath}
Pfaffian bit is section of a Pfaffian line bundle over $S$

where?

work over some parameter space $S$

both factors above are sections of a line bundle over $S$

\textbf{theorem} (in preparation) there is a hopefully canonical trivialization of $L_\Psi \otimes L_B \to S$

action being section of bundle instead of function: annomaly:

sources

\begin{itemize}%
\item 1) integrals over fermions $L_\Psi$: spin structure need not be equivariant under $\mathbb{Z}_2$-action


\item 2) simultaneous electric and magnetic current or alternatively self-dual current



\end{itemize}
interply between 1 and 2 leads to anomaly cancellation

\begin{itemize}%
\item 3) boundary of topological terms, like WZW, Chern-Simons

\end{itemize}
what about $L_B$: exotic orientation

\hypertarget{kevin_costello_part_i}{}\subsection*{{Kevin Costello; part I}}\label{kevin_costello_part_i}

\hypertarget{deformation_quantization}{}\subsubsection*{{deformation quantization}}\label{deformation_quantization}

\begin{itemize}%
\item classical mechanics: $A^{cl}$ commutative algebra with Poisson brackets $\{-,-\}$


\item this is the classical observable algebra


\item to quantize this we need to find some associative algebra $A^q$ over the ring $\mathbb{R}[\![\hbar]\!]$


\item such that

\begin{itemize}%
\item 1) $A^q/\hbar A^q = A^cl$


\item 2) if $a,b \in A^c{l}$, $\tilde a, \tilde b$ are lifts in $A^q$ then

\begin{displaymath}
\{a,b\} = \frac{1}{\hbar} [\tilde a, \tilde b] mod \hbar
\end{displaymath}


\end{itemize}


\end{itemize}
\textbf{goal} of these lectures: want to give an analog of ths picture for QFT

\begin{itemize}%
\item 1) need to explain wha plays the role of commutative, Poisson and associative algebras


\item 2) explain how classical field theory is encoded in commutative and Poisson


\item 3) explain how to quantize



\end{itemize}
structure that play the role of associative algebras is a [[factorization algebra]]

this is a $C^\infty$-analog (i.e. differential geometric analog) of a \textbf{chiral algebra} in the sense of Beilinson and Drinfeld

\begin{itemize}%
\item let $M$ be a manifold (on which we do QFT)


\item Let $B(M) = \{smooth balls in M\}$; this is an $\infty$-dim manifold



\end{itemize}
\begin{quote}%
audience: would open balls here form a manifold? does it matter? {\tt \symbol{62}}answer: well, really we don't think of manifolds but of [[diffeological space]]s, of course (sheaves on manifold)


\end{quote}
\begin{itemize}%
\item Let $B_n(M) = \{n disjoint smooth balls embedded in larger ball in M\}$

\end{itemize}
there are obvious projection maps

\begin{displaymath}
B(M) \stackrel{q}{\leftarrow} B_n(M) \stackrel{p}{\to} B(M)^n
\end{displaymath}
everything now joint work with O. Gwilliam

\textbf{def} A \textbf{factorization algebra} is

\begin{itemize}%
\item a [[vector bundle]] $F$ on $B(M)$

\begin{itemize}%
\item equipped with maps $p^*(F^{\times n}) \to q^*(F)$

\begin{itemize}%
\item satisfying some evident compatibility condition


\item such that everything is invariant under the obvious $S_n$-action on $B_n(M)$ that exchanges the order of the balls



\end{itemize}


\end{itemize}

\item concretely:

\begin{itemize}%
\item $F$ assigns a vector space to every ball $B \subset M$


\item if we have some configuration of balls, like 2 balls $B_1, B_2$ inside a big one $B_3$ we get a map $F(B_1)\otimes F(B_2) \to F(B_3)$


\item these must vary smoothly as the configuration of the balls varies



\end{itemize}


\end{itemize}
This is an algebra over the embedded little disk [[operad]], which is a ``colored operad'' (i.e. a [[multicategory]] with more than one object)

\begin{itemize}%
\item where the colors are $B(M)$


\item $n$-ary operations are $B_n(M)$


\item with extra conditions such that the vector space we assign to each color forms a smooth vector bundle

\begin{itemize}%
\item and all operad maps are compatible with this

\end{itemize}


\end{itemize}
notice that it happenss that for given in and out colors, there is at least one morphism in the operad.

\begin{itemize}%
\item there are several different reductions of this structure that are more familiar


\item notion of [[vector bundle]] comes in three natural flavors

\begin{itemize}%
\item 1) $C^\infty$


\item 2) holomorphic


\item 3) locally constant sheaves



\end{itemize}

\item definition of factorization algebra can be modified to the case of 2) and 3)



\end{itemize}
\textbf{def} a locally constant factorization algebra is like a factorization algebra, except that instead of being a vector bundle $F$ is a locally constant sheaf on $B(M)$ of co[[chain complex]]es and the structure maps of locally constant sheaves

\begin{quote}%
question: cohomologically locally constant or really locally constant


\end{quote}
{\tt \symbol{62}} answer: I think cohomologically locally constant

let $F$ be a locally constant factorization algebra in $\mathbb{R}^n$, then since $B(\mathbb{R}^n)$ is contractible $F$ is quasi-isomorphic to a trivial sheaf with fiber $V$ a cochain complex

so then for instance the map $V__{B_1} \otimes V_{B_2}  \to V_{B_3}$ depends only on homotopies of the configuration,

so, a locally constant factorization algebra on $\mathbb{R}^n$ is an $E_n$-[[algebra over an operad|algebra]]

next specialization: \textbf{holomorphic factorization algebras}

Let $\Sigma$ be a Riemannian surface.

We know what it means for a map from a complex manifold to $B(M)$ to be holomorphic; so we can talk about holomorphic algebras on $B(\Sigma)$

a homomorphic map $U \to B(M)$ is a bundle $M \to U$ all of whose fibers are balls and a map

\begin{displaymath}
\itexarray{
    M &\to& \Sigma
    \\
    \downarrow
    \\
    U
  }
\end{displaymath}
actually $B(M)$, too, is also not a complex manifold but a sheaf on complex manifold

\begin{quote}%
Andre Henriquez: but with this definition, won't every map $U \to B(M)$ be holomorphic: {\tt \symbol{62}} answer: oops, right


\end{quote}
??

let's consider a holomorphic factorization algebra on $\mathcal{C}$, which is translation invariant and dilation invariant

let $V = F$ (any round disk)

if we have a configuratoin of disks with $B_1$ and $B_2$ in $B_3$ with radii $\epsilon_i$ with one disk in the center of the big one and the other at complex parameter $z$, the map

\begin{displaymath}
m_z : V \otimes V \to V
\end{displaymath}
must vary holomorphically with $z$

so $z \mapsto m_z$ is a holomorphic map $Annulus \to Hom(V \otimes V, V)$, so it has a Laurent expansion

\begin{displaymath}
m_z \sim \sum_{k \in \mathbb{Z}} Z^k a_k
\end{displaymath}
with $a_k$ in some completion of $Hom(V \otimes V , V)$

\begin{quote}%
question by Ulrich Bunke: algebraic tensor product or not? {\tt \symbol{62}} answer: no, in examples tensor product will be projective tensor product


\end{quote}
reminiscent of vertex operator algebra

notice that Beilinson-Drinfeld make the same def in the algebraic setting, in their case the axioms are equivalent to that of a vertex operator algebra;

they show axioms for chiral algebra on $\mathcal{C}$ are essentially equivalent to those of a vertex operator algebra

\textbf{claim}: structure of factorization algebra: good to encode quantum field theory

notice that factorizations algebras on real line tend to be associativ algebras, so that fits in with the expectation from quantum mechanics.


\vspace{.5em} \hrule \vspace{.5em}
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