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] |
|
105 $$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF^{\text{TL}_d}\left(T^2\right)$$ |
|
106 \end{example} |
|
107 \begin{block}{} |
|
108 Given a pasting diagram on a ball, we can evaluate it to a morphism. We call the kernel the \emph{null fields}. |
|
109 \vspace{-3mm} |
|
110 $$\text{ev}\Bigg(\roundframe{\mathfig{0.12}{definition/evaluation1}} - \frac{1}{d}\roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$ |
|
111 \end{block} |
104 \end{frame} |
112 \end{frame} |
105 |
113 |
106 \begin{frame}{Background: TQFT invariants} |
114 \begin{frame}{Background: TQFT invariants} |
107 \begin{defn} |
115 \begin{defn} |
108 A decapitated $n+1$-dimensional TQFT associates a vector space $\cA(\cM)$ to each $n$-manifold $\cM$. |
116 A decapitated $n+1$-dimensional TQFT associates a vector space $\cA(\cM)$ to each $n$-manifold $\cM$. |
137 $\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary pasting diagrams on $\cM$. |
145 $\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary pasting diagrams on $\cM$. |
138 \end{block} |
146 \end{block} |
139 |
147 |
140 \begin{block}{} |
148 \begin{block}{} |
141 \vspace{-1mm} |
149 \vspace{-1mm} |
142 $$\bc_1(\cM; \cC) = \setcr{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$ |
150 $$\bc_1(\cM; \cC) = \Complex\setcr{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$ |
143 \end{block} |
151 \end{block} |
144 \vspace{-3.5mm} |
152 \vspace{-3.5mm} |
145 $$\mathfig{.5}{definition/single-blob}$$ |
153 $$\mathfig{.5}{definition/single-blob}$$ |
146 \vspace{-3mm} |
154 \vspace{-3mm} |
147 \begin{block}{} |
155 \begin{block}{} |
158 $$\bc_2 = \bc_2^{\text{disjoint}} \oplus \bc_2^{\text{nested}}$$ |
166 $$\bc_2 = \bc_2^{\text{disjoint}} \oplus \bc_2^{\text{nested}}$$ |
159 \end{block} |
167 \end{block} |
160 \begin{block}{} |
168 \begin{block}{} |
161 \vspace{-5mm} |
169 \vspace{-5mm} |
162 \begin{align*} |
170 \begin{align*} |
163 \bc_2^{\text{disjoint}} & = \setcl{\roundframe{\mathfig{0.5}{definition/disjoint-blobs}}}{\text{ev}_{B_i}(u_i) = 0} |
171 \bc_2^{\text{disjoint}} & = \Complex\setcl{\roundframe{\mathfig{0.5}{definition/disjoint-blobs}}}{\text{ev}_{B_i}(u_i) = 0} |
164 \end{align*} |
172 \end{align*} |
165 \vspace{-4mm} |
173 \vspace{-4mm} |
166 $$d_2 : (B_1, B_2, u_1, u_2, r) \mapsto (B_2, u_2, r \circ u_1) - (B_1, u_1, r \circ u_2)$$ |
174 $$d_2 : (B_1, B_2, u_1, u_2, r) \mapsto (B_2, u_2, r \circ u_1) - (B_1, u_1, r \circ u_2)$$ |
167 \end{block} |
175 \end{block} |
168 \begin{block}{} |
176 \begin{block}{} |
169 \vspace{-5mm} |
177 \vspace{-5mm} |
170 \begin{align*} |
178 \begin{align*} |
171 \bc_2^{\text{nested}} & = \setcl{\roundframe{\mathfig{0.5}{definition/nested-blobs}}}{\text{ev}_{B_1}(u)=0} |
179 \bc_2^{\text{nested}} & = \Complex\setcl{\roundframe{\mathfig{0.5}{definition/nested-blobs}}}{\text{ev}_{B_1}(u)=0} |
172 \end{align*} |
180 \end{align*} |
173 \vspace{-4mm} |
181 \vspace{-4mm} |
174 $$d_2 : (B_1, B_2, u, r', r) \mapsto (B_2, u \circ r', r) - (B_1, u, r \circ r')$$ |
182 $$d_2 : (B_1, B_2, u, r', r) \mapsto (B_2, u \circ r', r) - (B_1, u, r \circ r')$$ |
175 \end{block} |
183 \end{block} |
176 \end{frame} |
184 \end{frame} |
177 |
185 |
178 \begin{frame}{Definition, general case} |
186 \begin{frame}{Definition, general case} |
179 \begin{block}{} |
187 \begin{block}{} |
180 $$\bc_k = \set{\roundframe{\mathfig{0.7}{definition/k-blobs}}}$$ |
188 $$\bc_k = \Complex\set{\roundframe{\mathfig{0.7}{definition/k-blobs}}}$$ |
181 $k$ blobs, properly nested or disjoint, with ``innermost'' blobs labelled by pasting diagrams that evaluate to zero. |
189 $k$ blobs, properly nested or disjoint, with ``innermost'' blobs labelled by pasting diagrams that evaluate to zero. |
182 \end{block} |
190 \end{block} |
183 \begin{block}{} |
191 \begin{block}{} |
184 \vspace{-2mm} |
192 \vspace{-2mm} |
185 $$d_k : \bc_k \to \bc_{k-1} = {\textstyle \sum_i} (-1)^i (\text{erase blob $i$})$$ |
193 $$d_k : \bc_k \to \bc_{k-1} = {\textstyle \sum_i} (-1)^i (\text{erase blob $i$})$$ |
217 \begin{block}{} |
225 \begin{block}{} |
218 Taking $H_0$, this is the mapping class group acting on a TQFT skein module. |
226 Taking $H_0$, this is the mapping class group acting on a TQFT skein module. |
219 \end{block} |
227 \end{block} |
220 \end{frame} |
228 \end{frame} |
221 |
229 |
|
230 \begin{frame}{Higher Deligne conjecture} |
|
231 \begin{block}{Deligne conjecture} |
|
232 Chains on the little discs operad acts on Hochschild cohomology. |
|
233 \end{block} |
|
234 |
|
235 \begin{block}{} |
|
236 Call $\Hom{A_\infty}{\bc_*(\cM)}{\bc_*(\cM)}$ `blob cochains on $\cM$'. |
|
237 \end{block} |
|
238 |
|
239 \begin{block}{Theorem* (Higher Deligne conjecture)} |
|
240 \scalebox{0.96}{Chains on the $n$-dimensional fat graph operad acts on blob cochains.} |
|
241 \vspace{-3mm} |
|
242 $$\mathfig{.85}{tempkw/delfig2}$$ |
|
243 \end{block} |
|
244 \end{frame} |
|
245 |
222 \begin{frame}{Gluing} |
246 \begin{frame}{Gluing} |
223 \begin{block}{$\bc_*(Y \times [0,1])$ is naturally an $A_\infty$ category} |
247 \begin{block}{$\bc_*(Y \times [0,1])$ is naturally an $A_\infty$ category} |
224 \begin{itemize} |
248 \begin{itemize} |
225 \item[$m_2$:] gluing $[0,1] \simeq [0,1] \cup [0,1]$ |
249 \item[$m_2$:] gluing $[0,1] \simeq [0,1] \cup [0,1]$ |
226 \item[$m_k$:] reparametrising $[0,1]$ |
250 \item[$m_k$:] reparametrising $[0,1]$ |