85 of the two ways of erasing one of the blobs. |
85 of the two ways of erasing one of the blobs. |
86 It's easy to check that $\bd^2 = 0$. |
86 It's easy to check that $\bd^2 = 0$. |
87 |
87 |
88 A nested 2-blob diagram consists of |
88 A nested 2-blob diagram consists of |
89 \begin{itemize} |
89 \begin{itemize} |
90 \item A pair of nested balls (blobs) $B_1 \sub B_2 \sub X$. |
90 \item A pair of nested balls (blobs) $B_1 \subseteq B_2 \subseteq X$. |
91 \item A field $r' \in \cC(B_2 \setminus B_1; c_1, c_2)$ |
91 \item A field $r' \in \cC(B_2 \setminus B_1; c_1, c_2)$ |
92 (for some $c_1 \in \cC(\bdy B_1)$ and $c_2 \in \cC(\bdy B_2)$). |
92 (for some $c_1 \in \cC(\bdy B_1)$ and $c_2 \in \cC(\bdy B_2)$). |
93 \item A field $r \in \cC(X \setminus B_2; c_2)$. |
93 \item A field $r \in \cC(X \setminus B_2; c_2)$. |
94 \item A local relation field $u \in U(B_1; c_1)$. |
94 \item A local relation field $u \in U(B_1; c_1)$. |
95 \end{itemize} |
95 \end{itemize} |
107 |
107 |
108 As with the $1$-blob diagrams, in order to get the linear structure correct the actual definition is |
108 As with the $1$-blob diagrams, in order to get the linear structure correct the actual definition is |
109 \begin{eqnarray*} |
109 \begin{eqnarray*} |
110 \bc_2(X) & \deq & |
110 \bc_2(X) & \deq & |
111 \left( |
111 \left( |
112 \bigoplus_{B_1, B_2 \text{disjoint}} \bigoplus_{c_1, c_2} |
112 \bigoplus_{B_1, B_2\; \text{disjoint}} \bigoplus_{c_1, c_2} |
113 U(B_1; c_1) \otimes U(B_2; c_2) \otimes \lf(X\setmin (B_1\cup B_2); c_1, c_2) |
113 U(B_1; c_1) \otimes U(B_2; c_2) \otimes \lf(X\setmin (B_1\cup B_2); c_1, c_2) |
114 \right) \\ |
114 \right) \bigoplus \\ |
115 && \bigoplus \left( |
115 && \quad\quad \left( |
116 \bigoplus_{B_1 \subset B_2} \bigoplus_{c_1, c_2} |
116 \bigoplus_{B_1 \subset B_2} \bigoplus_{c_1, c_2} |
117 U(B_1; c_1) \otimes \lf(B_2 \setmin B_1; c_1) \tensor \cC(X \setminus B_2; c_2) |
117 U(B_1; c_1) \otimes \lf(B_2 \setmin B_1; c_1) \tensor \cC(X \setminus B_2; c_2) |
118 \right) . |
118 \right) . |
119 \end{eqnarray*} |
119 \end{eqnarray*} |
120 For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign |
120 For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign |
121 (rather than a new, linearly independent 2-blob diagram). |
121 (rather than a new, linearly independent 2-blob diagram). |
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122 \noop{ |
122 \nn{Hmm, I think we should be doing this for nested blobs too -- |
123 \nn{Hmm, I think we should be doing this for nested blobs too -- |
123 we shouldn't force the linear indexing of the blobs to have anything to do with |
124 we shouldn't force the linear indexing of the blobs to have anything to do with |
124 the partial ordering by inclusion -- this is what happens below} |
125 the partial ordering by inclusion -- this is what happens below} |
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126 \nn{KW: I think adding that detail would only add distracting clutter, and the statement is true as written (in the sense that it yields a vector space isomorphic to the general def below} |
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127 } |
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128 |
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129 Before describing the general case we should say more precisely what we mean by |
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130 disjoint and nested blobs. |
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131 Disjoint will mean disjoint interiors. |
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132 Nested blobs are allowed to coincide, or to have overlapping boundaries. |
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133 Blob are allowed to intersect $\bd X$. |
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134 However, we require of any collection of blobs $B_1,\ldots,B_k \subseteq X$ that |
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135 $X$ is decomposable along the union of the boundaries of the blobs. |
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136 \nn{need to say more here. we want to be able to glue blob diagrams, but avoid pathological |
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137 behavior} |
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138 \nn{need to allow the case where $B\to X$ is not an embedding |
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139 on $\bd B$. this is because any blob diagram on $X_{cut}$ should give rise to one on $X_{gl}$ |
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140 and blobs are allowed to meet $\bd X$.} |
125 |
141 |
126 Now for the general case. |
142 Now for the general case. |
127 A $k$-blob diagram consists of |
143 A $k$-blob diagram consists of |
128 \begin{itemize} |
144 \begin{itemize} |
129 \item A collection of blobs $B_i \sub X$, $i = 1, \ldots, k$. |
145 \item A collection of blobs $B_i \sub X$, $i = 1, \ldots, k$. |
130 For each $i$ and $j$, we require that either $B_i$ and $B_j$ have disjoint interiors or |
146 For each $i$ and $j$, we require that either $B_i$ and $B_j$ have disjoint interiors or |
131 $B_i \sub B_j$ or $B_j \sub B_i$. |
147 $B_i \sub B_j$ or $B_j \sub B_i$. |
132 (The case $B_i = B_j$ is allowed. |
148 (The case $B_i = B_j$ is allowed. |
133 If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
149 If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
134 If a blob has no other blobs strictly contained in it, we call it a twig blob. |
150 If a blob has no other blobs strictly contained in it, we call it a twig blob. |
135 \nn{need to allow the case where $B\to X$ is not an embedding |
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136 on $\bd B$. this is because any blob diagram on $X_{cut}$ should give rise to one on $X_{gl}$ |
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137 and blobs are allowed to meet $\bd X$.} |
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138 \item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
151 \item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
139 (These are implied by the data in the next bullets, so we usually |
152 (These are implied by the data in the next bullets, so we usually |
140 suppress them from the notation.) |
153 suppress them from the notation.) |
141 The fields $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
154 The fields $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
142 if the latter space is not empty. |
155 if the latter space is not empty. |