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\author[Scott Morrison]{Scott Morrison \\ \texttt{http://tqft.net/} \\ joint work with Kevin Walker}

\institute{UC Berkeley / Miller Institute for Basic Research}

\begin{document}

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\section{Overview}

   \begin{frame}<beamer>

       \begin{quote}

      ... homotopical topology and TQFT have grown so close that I have started thinking that they are turning into the language of new foundations. 

        \end{quote}

        \flushright{--- \href{http://www.ams.org/notices/200910/rtx091001268p.pdf}{Yuri Manin, September 2008}}

      \tableofcontents

\end{frame}

\begin{block}{}

The blob complex takes an $n$-manifold $\cM$ and an `$n$-category with strong duality' $\cC$ and produces a chain complex, $\bc_*(\cM; \cC)$.

\end{block}

\tikzstyle{description}=[gray, font=\tiny, text centered, text width=2cm]

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\node[red] (blobs) at (0,0) {$H(\bc_*(\cM; \cC))$};

\node[blue] (skein) at (4,0) {$\cA(\cM; \cC)$};

\node[below=5pt, description] (skein-label) at (skein) {(the usual TQFT Hilbert space)};

\path[->](blobs) edge node[above] {$*= 0$} (skein);

}

  \node[blue] (hoch) at (0,3) {$HH_*(\cC)$};

  \node[right=20pt, description] (hoch-label) at (hoch) {(the Hochschild homology)};

  \path[->](blobs) edge node[right] {$\cM = S^1$} (hoch);

}

  \node[blue] (comm) at (-2.4, -1.8) {$H_*(\Delta^\infty(\cM), k)$};

  \node[description, below=5pt] (comm-label) at (comm) {(singular homology of the infinite configuration space)};

  \path[->](blobs) edge node[right=5pt] {$\cC = k[t]$} (comm);

}

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\end{frame}

\begin{frame}{$n$-categories}

\begin{block}{Defining $n$-categories is fraught with difficulties}

\end{block}

\begin{block}{}

\end{block}

\begin{block}{}

\begin{itemize}

\item

Defining $n$-categories: a choice of `shape' for morphisms.

\item

We allow all shapes! A vector space for every ball.

\item

`Strong duality' is integral in our definition.

\end{itemize}

\end{block}

\end{frame}

\newcommand{\roundframe}[1]{\begin{tikzpicture}[baseline=-2pt]\node[rectangle,inner sep=1pt,rounded corners,fill=white] {#1};\end{tikzpicture}}

\section{Definition}

\begin{frame}{Fields and pasting diagrams}

\begin{block}{Pasting diagrams}

Fix an $n$-category with strong duality $\cC$. A \emph{field} on $\cM$ is a pasting diagram drawn on $\cM$, with cells labelled by morphisms from $\cC$.

\end{block}

\begin{example}[$\cC = \text{TL}_d$ the Temperley-Lieb category]

\end{example}

\begin{block}{}

Given a pasting diagram on a ball, we can evaluate it to a morphism. We call the kernel the \emph{null fields}.

\vspace{-3mm}

$$\text{ev}\Bigg(\roundframe{\mathfig{0.12}{definition/evaluation1}} - \frac{1}{d}\roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$

\end{block}

\end{frame}

\begin{frame}{Background: TQFT invariants}

\begin{defn}

A decapitated $n+1$-dimensional TQFT associates a vector space $\cA(\cM)$ to each $n$-manifold $\cM$.

\end{defn}

(`decapitated': no numerical invariants of $n+1$-manifolds.)

\begin{block}{}

If the manifold has boundary, we get a category. Objects are boundary data, $\Hom{\cA(\cM)}{x}{y} = \cA(\cM; x,y)$.

\end{block}

\begin{block}{}

We want to extend `all the way down'. The $k$-category associated to the $n-k$-manifold $\cY$ is $\cA(\cY \times B^k)$.

\end{block}

\begin{defn}

Given an $n$-category $\cC$, the associated TQFT is 

\vspace{-3mm}

$$\cA(\cM) = \cF(M) / \ker{ev},$$

\vspace{-3mm}

fields modulo fields which evaluate to zero inside some ball.

\end{defn}

\end{frame}

\begin{frame}{\emph{Definition} of the blob complex, $k=0,1$}

\begin{block}{Motivation}

A \emph{local} construction, such that when $\cM$ is a ball, $\bc_*(\cM; \cC)$ is a resolution of $\cA(\cM,; \cC)$.

\end{block}

\begin{block}{}

\center

$\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary pasting diagrams on $\cM$.

\end{block}

\begin{block}{}

\vspace{-1mm}

$$\bc_1(\cM; \cC) = \Complex\setcr{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$

\end{block}

\vspace{-3.5mm}

$$\mathfig{.5}{definition/single-blob}$$

\vspace{-3mm}

\begin{block}{}

\vspace{-6mm}

\begin{align*}

d_1 : (B, u, r) & \mapsto u \circ r & \bc_0 / \im(d_1) \iso A(\cM; \cC)

\end{align*}

\end{block}

\end{frame}

\begin{frame}{Definition, $k=2$}

\begin{block}{}

\vspace{-1mm}

$$\bc_2 = \bc_2^{\text{disjoint}} \oplus \bc_2^{\text{nested}}$$

\end{block}

\begin{block}{}

\vspace{-5mm}

\begin{align*}

\bc_2^{\text{disjoint}} & =  \Complex\setcl{\roundframe{\mathfig{0.5}{definition/disjoint-blobs}}}{\text{ev}_{B_i}(u_i) = 0}

\end{align*}

\vspace{-4mm}

$$d_2 : (B_1, B_2, u_1, u_2, r) \mapsto (B_2, u_2, r \circ u_1) - (B_1, u_1, r \circ u_2)$$

\end{block}

\begin{block}{}

\vspace{-5mm}

\begin{align*}

\bc_2^{\text{nested}} & = \Complex\setcl{\roundframe{\mathfig{0.5}{definition/nested-blobs}}}{\text{ev}_{B_1}(u)=0}

\end{align*}

\vspace{-4mm}

$$d_2 : (B_1, B_2, u, r', r) \mapsto (B_2, u \circ r', r) - (B_1, u, r \circ r')$$

\end{block}

\end{frame}

\begin{frame}{Definition, general case}

\begin{block}{}

$$\bc_k = \Complex\set{\roundframe{\mathfig{0.7}{definition/k-blobs}}}$$

$k$ blobs, properly nested or disjoint, with ``innermost'' blobs labelled by pasting diagrams that evaluate to zero.

\end{block}

\begin{block}{}

\vspace{-2mm}

$$d_k : \bc_k \to \bc_{k-1} = {\textstyle \sum_i} (-1)^i (\text{erase blob $i$})$$

\end{block}

\end{frame}

\section{Properties}

\begin{frame}{Hochschild homology}

\begin{block}{TQFT on $S^1$ is `coinvariants'}

\vspace{-3mm}

$$\cA(S^1, A) = \Complex\set{\roundframe{\mathfig{0.1}{hochschild/m-a-b}}}\scalebox{2}{$/$}\set{\roundframe{\mathfig{0.065}{hochschild/ma}} - \roundframe{\mathfig{0.12}{hochschild/m-a}}} = A/(ab-ba)$$

\end{block}

\begin{block}{}

The Hochschild complex is `coinvariants of the bar resolution'

\vspace{-2mm}

$$ \cdots \to A \tensor A \tensor A \to A \tensor A \xrightarrow{m \tensor a \mapsto ma-am} A$$

\end{block}

\begin{thm}[$ \HC_*(A) \iso \bc_*(S^1; A)$]

$$m \tensor a \mapsto

\roundframe{\mathfig{0.35}{hochschild/1-chains}}

$$

\vspace{-5mm}

\begin{align*}

u_1 & = \mathfig{0.05}{hochschild/u_1-1} - \mathfig{0.05}{hochschild/u_1-2} & u_2  &= \mathfig{0.05}{hochschild/u_2-1} - \mathfig{0.05}{hochschild/u_2-2} 

\end{align*}

\end{thm}

\end{frame}

\begin{frame}{An action of $\CH{\cM}$}

\begin{thm}

There's a chain map

$$\CH{\cM} \tensor \bc_*(\cM) \to \bc_*(\cM).$$

which is associative up to homotopy, and compatible with gluing.

\end{thm}

\begin{block}{}

Taking $H_0$, this is the mapping class group acting on a TQFT skein module.

\end{block}

\end{frame}

\begin{frame}{Gluing}

\begin{block}{$\bc_*(Y \times [0,1])$ is naturally an $A_\infty$ category}

\begin{itemize}

\item[$m_2$:] gluing $[0,1] \simeq [0,1] \cup [0,1]$

\item[$m_k$:] reparametrising $[0,1]$

\end{itemize}

\end{block}

\begin{block}{}

If $Y \subset \bdy X$ then $\bc_*(X)$ is an $A_\infty$ module over $\bc_*(Y)$.

\end{block}

\begin{thm}[Gluing formula]

When $Y \sqcup Y^{\text{op}} \subset \bdy X$,

\vspace{-5mm}

\[

	\bc_*(X \bigcup_Y \selfarrow) \iso \bc_*(X) \bigotimes_{\bc_*(Y)}^{A_\infty} \selfarrow.

\]

\end{thm}

In principle, we can compute blob homology from a handle decomposition, by iterated Hochschild homology.

\end{frame}

\end{document}

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