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99 \section{Definition} |
99 \section{Definition} |
100 \begin{frame}{Fields and pasting diagrams} |
100 \begin{frame}{Fields and pasting diagrams} |
101 \begin{block}{Pasting diagrams} |
101 \begin{block}{Pasting diagrams} |
102 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 $\cC$. |
102 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 $\cC$. |
103 \end{block} |
103 \end{block} |
104 \begin{example}[$\cC = \text{TL}_d$ the Temperley-Lieb category] |
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105 $$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF^{\text{TL}_d}\left(T^2\right)$$ |
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106 \end{example} |
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107 \begin{block}{} |
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108 Given a pasting diagram on a ball, we can evaluate it to a morphism. We call the kernel the \emph{null fields}. |
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109 \vspace{-3mm} |
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110 $$\text{ev}\Bigg(\roundframe{\mathfig{0.12}{definition/evaluation1}} - \frac{1}{d}\roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$ |
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111 \end{block} |
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112 \end{frame} |
104 \end{frame} |
113 |
105 |
114 \begin{frame}{Background: TQFT invariants} |
106 \begin{frame}{Background: TQFT invariants} |
115 \begin{defn} |
107 \begin{defn} |
116 A decapitated $n+1$-dimensional TQFT associates a vector space $\cA(\cM)$ to each $n$-manifold $\cM$. |
108 A decapitated $n+1$-dimensional TQFT associates a vector space $\cA(\cM)$ to each $n$-manifold $\cM$. |