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1 %!TEX root = ../blob1.tex |
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2 |
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3 \section{TQFTs via fields} |
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4 \label{sec:fields} |
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5 \label{sec:tqftsviafields} |
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6 |
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7 In this section we review the construction of TQFTs from ``topological fields". |
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8 For more details see \cite{kw:tqft}. |
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9 |
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10 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
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11 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
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12 $\overline{X \setmin Y}$. |
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13 |
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14 |
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15 \subsection{Systems of fields} |
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16 |
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17 Let $\cM_k$ denote the category with objects |
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18 unoriented PL manifolds of dimension |
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19 $k$ and morphisms homeomorphisms. |
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20 (We could equally well work with a different category of manifolds --- |
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21 oriented, topological, smooth, spin, etc. --- but for definiteness we |
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22 will stick with unoriented PL.) |
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23 |
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24 %Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$. |
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25 |
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26 A $n$-dimensional {\it system of fields} in $\cS$ |
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27 is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ |
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28 together with some additional data and satisfying some additional conditions, all specified below. |
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29 |
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30 Before finishing the definition of fields, we give two motivating examples |
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31 (actually, families of examples) of systems of fields. |
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32 |
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33 The first examples: Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps |
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34 from X to $B$. |
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35 |
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36 The second examples: Fix an $n$-category $C$, and let $\cC(X)$ be |
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37 the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by |
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38 $j$-morphisms of $C$. |
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39 One can think of such sub-cell-complexes as dual to pasting diagrams for $C$. |
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40 This is described in more detail below. |
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41 |
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42 Now for the rest of the definition of system of fields. |
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43 \begin{enumerate} |
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44 \item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, |
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45 and these maps are a natural |
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46 transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. |
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47 For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of |
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48 $\cC(X)$ which restricts to $c$. |
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49 In this context, we will call $c$ a boundary condition. |
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50 \item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$. (This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), then this extra structure is considered part of the definition of $\cC_n$. Any maps mentioned below between top level fields must be morphisms in $\cS$. |
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51 \item $\cC_k$ is compatible with the symmetric monoidal |
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52 structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
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53 compatibly with homeomorphisms, restriction to boundary, and orientation reversal. |
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54 We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ |
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55 restriction maps. |
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56 \item Gluing without corners. |
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57 Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. |
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58 Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
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59 Using the boundary restriction, disjoint union, and (in one case) orientation reversal |
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60 maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two |
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61 copies of $Y$ in $\bd X$. |
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62 Let $\Eq_Y(\cC_k(X))$ denote the equalizer of these two maps. |
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63 Then (here's the axiom/definition part) there is an injective ``gluing" map |
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64 \[ |
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65 \Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) , |
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66 \] |
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67 and this gluing map is compatible with all of the above structure (actions |
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68 of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
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69 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
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70 the gluing map is surjective. |
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71 From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
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72 gluing surface, we say that fields in the image of the gluing map |
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73 are transverse to $Y$ or splittable along $Y$. |
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74 \item Gluing with corners. |
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75 Let $\bd X = Y \cup -Y \cup W$, where $\pm Y$ and $W$ might intersect along their boundaries. |
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76 Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
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77 Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself |
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78 (without corners) along two copies of $\bd Y$. |
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79 Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a splittable field on $W\sgl$ and let |
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80 $c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$. |
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81 Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$. |
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82 (This restriction map uses the gluing without corners map above.) |
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83 Using the boundary restriction, gluing without corners, and (in one case) orientation reversal |
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84 maps, we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two |
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85 copies of $Y$ in $\bd X$. |
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86 Let $\Eq^c_Y(\cC_k(X))$ denote the equalizer of these two maps. |
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87 Then (here's the axiom/definition part) there is an injective ``gluing" map |
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88 \[ |
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89 \Eq^c_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl, c\sgl) , |
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90 \] |
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91 and this gluing map is compatible with all of the above structure (actions |
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92 of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
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93 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
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94 the gluing map is surjective. |
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95 From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
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96 gluing surface, we say that fields in the image of the gluing map |
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97 are transverse to $Y$ or splittable along $Y$. |
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98 \item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted |
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99 $c \mapsto c\times I$. |
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100 These maps comprise a natural transformation of functors, and commute appropriately |
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101 with all the structure maps above (disjoint union, boundary restriction, etc.). |
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102 Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism |
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103 covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. |
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104 \end{enumerate} |
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105 |
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106 There are two notations we commonly use for gluing. |
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107 One is |
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108 \[ |
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109 x\sgl \deq \gl(x) \in \cC(X\sgl) , |
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110 \] |
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111 for $x\in\cC(X)$. |
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112 The other is |
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113 \[ |
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114 x_1\bullet x_2 \deq \gl(x_1\otimes x_2) \in \cC(X\sgl) , |
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115 \] |
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116 in the case that $X = X_1 \du X_2$, with $x_i \in \cC(X_i)$. |
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117 |
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118 \medskip |
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119 |
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120 Using the functoriality and $\cdot\times I$ properties above, together |
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121 with boundary collar homeomorphisms of manifolds, we can define the notion of |
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122 {\it extended isotopy}. |
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123 Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
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124 of $\bd M$. |
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125 Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is splittable along $\bd Y$. |
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126 Let $c$ be $x$ restricted to $Y$. |
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127 Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$. |
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128 Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$. |
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129 Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism. |
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130 Then we say that $x$ is {\it extended isotopic} to $f(x \bullet (c\times I))$. |
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131 More generally, we define extended isotopy to be the equivalence relation on fields |
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132 on $M$ generated by isotopy plus all instance of the above construction |
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133 (for all appropriate $Y$ and $x$). |
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134 |
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135 \nn{should also say something about pseudo-isotopy} |
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136 |
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137 |
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138 \nn{remark that if top dimensional fields are not already linear |
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139 then we will soon linearize them(?)} |
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140 |
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141 We now describe in more detail systems of fields coming from sub-cell-complexes labeled |
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142 by $n$-category morphisms. |
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143 |
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144 Given an $n$-category $C$ with the right sort of duality |
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145 (e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), |
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146 we can construct a system of fields as follows. |
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147 Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
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148 with codimension $i$ cells labeled by $i$-morphisms of $C$. |
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149 We'll spell this out for $n=1,2$ and then describe the general case. |
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150 |
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151 If $X$ has boundary, we require that the cell decompositions are in general |
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152 position with respect to the boundary --- the boundary intersects each cell |
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153 transversely, so cells meeting the boundary are mere half-cells. |
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154 |
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155 Put another way, the cell decompositions we consider are dual to standard cell |
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156 decompositions of $X$. |
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157 |
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158 We will always assume that our $n$-categories have linear $n$-morphisms. |
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159 |
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160 For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
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161 an object (0-morphism) of the 1-category $C$. |
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162 A field on a 1-manifold $S$ consists of |
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163 \begin{itemize} |
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164 \item A cell decomposition of $S$ (equivalently, a finite collection |
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165 of points in the interior of $S$); |
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166 \item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$) |
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167 by an object (0-morphism) of $C$; |
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168 \item a transverse orientation of each 0-cell, thought of as a choice of |
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169 ``domain" and ``range" for the two adjacent 1-cells; and |
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170 \item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with |
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171 domain and range determined by the transverse orientation and the labelings of the 1-cells. |
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172 \end{itemize} |
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173 |
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174 If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels |
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175 of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the |
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176 interior of $S$, each transversely oriented and each labeled by an element (1-morphism) |
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177 of the algebra. |
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178 |
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179 \medskip |
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180 |
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181 For $n=2$, fields are just the sort of pictures based on 2-categories (e.g.\ tensor categories) |
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182 that are common in the literature. |
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183 We describe these carefully here. |
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184 |
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185 A field on a 0-manifold $P$ is a labeling of each point of $P$ with |
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186 an object of the 2-category $C$. |
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187 A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$. |
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188 A field on a 2-manifold $Y$ consists of |
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189 \begin{itemize} |
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190 \item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such |
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191 that each component of the complement is homeomorphic to a disk); |
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192 \item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$) |
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193 by a 0-morphism of $C$; |
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194 \item a transverse orientation of each 1-cell, thought of as a choice of |
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195 ``domain" and ``range" for the two adjacent 2-cells; |
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196 \item a labeling of each 1-cell by a 1-morphism of $C$, with |
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197 domain and range determined by the transverse orientation of the 1-cell |
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198 and the labelings of the 2-cells; |
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199 \item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood |
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200 of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped |
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201 to $\pm 1 \in S^1$; and |
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202 \item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range |
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203 determined by the labelings of the 1-cells and the parameterizations of the previous |
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204 bullet. |
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205 \end{itemize} |
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206 \nn{need to say this better; don't try to fit everything into the bulleted list} |
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207 |
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208 For general $n$, a field on a $k$-manifold $X^k$ consists of |
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209 \begin{itemize} |
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210 \item A cell decomposition of $X$; |
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211 \item an explicit general position homeomorphism from the link of each $j$-cell |
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212 to the boundary of the standard $(k-j)$-dimensional bihedron; and |
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213 \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with |
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214 domain and range determined by the labelings of the link of $j$-cell. |
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215 \end{itemize} |
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216 |
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217 %\nn{next definition might need some work; I think linearity relations should |
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218 %be treated differently (segregated) from other local relations, but I'm not sure |
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219 %the next definition is the best way to do it} |
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220 |
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221 \medskip |
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222 |
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223 For top dimensional ($n$-dimensional) manifolds, we're actually interested |
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224 in the linearized space of fields. |
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225 By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is |
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226 the vector space of finite |
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227 linear combinations of fields on $X$. |
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228 If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$. |
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229 Thus the restriction (to boundary) maps are well defined because we never |
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230 take linear combinations of fields with differing boundary conditions. |
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231 |
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232 In some cases we don't linearize the default way; instead we take the |
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233 spaces $\lf(X; a)$ to be part of the data for the system of fields. |
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234 In particular, for fields based on linear $n$-category pictures we linearize as follows. |
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235 Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by |
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236 obvious relations on 0-cell labels. |
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237 More specifically, let $L$ be a cell decomposition of $X$ |
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238 and let $p$ be a 0-cell of $L$. |
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239 Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that |
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240 $\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. |
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241 Then the subspace $K$ is generated by things of the form |
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242 $\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader |
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243 to infer the meaning of $\alpha_{\lambda c + d}$. |
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244 Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms. |
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245 |
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246 \nn{Maybe comment further: if there's a natural basis of morphisms, then no need; |
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247 will do something similar below; in general, whenever a label lives in a linear |
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248 space we do something like this; ? say something about tensor |
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249 product of all the linear label spaces? Yes:} |
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250 |
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251 For top dimensional ($n$-dimensional) manifolds, we linearize as follows. |
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252 Define an ``almost-field" to be a field without labels on the 0-cells. |
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253 (Recall that 0-cells are labeled by $n$-morphisms.) |
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254 To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism |
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255 space determined by the labeling of the link of the 0-cell. |
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256 (If the 0-cell were labeled, the label would live in this space.) |
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257 We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). |
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258 We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the |
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259 above tensor products. |
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260 |
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261 |
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262 |
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263 \subsection{Local relations} |
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264 \label{sec:local-relations} |
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265 |
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266 |
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267 A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$, |
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268 for all $n$-manifolds $B$ which are |
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269 homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, |
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270 satisfying the following properties. |
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271 \begin{enumerate} |
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272 \item functoriality: |
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273 $f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$ |
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274 \item local relations imply extended isotopy: |
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275 if $x, y \in \cC(B; c)$ and $x$ is extended isotopic |
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276 to $y$, then $x-y \in U(B; c)$. |
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277 \item ideal with respect to gluing: |
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278 if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$ |
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279 \end{enumerate} |
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280 See \cite{kw:tqft} for details. |
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281 |
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282 |
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283 For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \lf(B; c)$, |
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284 where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
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285 |
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286 For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map |
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287 $\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into |
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288 domain and range. |
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289 |
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290 \nn{maybe examples of local relations before general def?} |
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291 |
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292 \subsection{Constructing a TQFT} |
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293 |
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294 In this subsection we briefly review the construction of a TQFT from a system of fields and local relations. |
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295 (For more details, see \cite{kw:tqft}.) |
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296 |
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297 Let $W$ be an $n{+}1$-manifold. |
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298 We can think of the path integral $Z(W)$ as assigning to each |
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299 boundary condition $x\in \cC(\bd W)$ a complex number $Z(W)(x)$. |
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300 In other words, $Z(W)$ lies in $\c^{\lf(\bd W)}$, the vector space of linear |
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301 maps $\lf(\bd W)\to \c$. |
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302 |
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303 The locality of the TQFT implies that $Z(W)$ in fact lies in a subspace |
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304 $Z(\bd W) \sub \c^{\lf(\bd W)}$ defined by local projections. |
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305 The linear dual to this subspace, $A(\bd W) = Z(\bd W)^*$, |
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306 can be thought of as finite linear combinations of fields modulo local relations. |
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307 (In other words, $A(\bd W)$ is a sort of generalized skein module.) |
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308 This is the motivation behind the definition of fields and local relations above. |
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309 |
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310 In more detail, let $X$ be an $n$-manifold. |
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311 %To harmonize notation with the next section, |
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312 %let $\bc_0(X)$ be the vector space of finite linear combinations of fields on $X$, so |
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313 %$\bc_0(X) = \lf(X)$. |
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314 Define $U(X) \sub \lf(X)$ to be the space of local relations in $\lf(X)$; |
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315 $U(X)$ is generated by things of the form $u\bullet r$, where |
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316 $u\in U(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$. |
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317 Define |
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318 \[ |
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319 A(X) \deq \lf(X) / U(X) . |
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320 \] |
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321 (The blob complex, defined in the next section, |
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322 is in some sense the derived version of $A(X)$.) |
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323 If $X$ has boundary we can similarly define $A(X; c)$ for each |
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324 boundary condition $c\in\cC(\bd X)$. |
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325 |
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326 The above construction can be extended to higher codimensions, assigning |
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327 a $k$-category $A(Y)$ to an $n{-}k$-manifold $Y$, for $0 \le k \le n$. |
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328 These invariants fit together via actions and gluing formulas. |
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329 We describe only the case $k=1$ below. |
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330 (The construction of the $n{+}1$-dimensional part of the theory (the path integral) |
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331 requires that the starting data (fields and local relations) satisfy additional |
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332 conditions. |
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333 We do not assume these conditions here, so when we say ``TQFT" we mean a decapitated TQFT |
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334 that lacks its $n{+}1$-dimensional part.) |
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335 |
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336 Let $Y$ be an $n{-}1$-manifold. |
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337 Define a (linear) 1-category $A(Y)$ as follows. |
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338 The objects of $A(Y)$ are $\cC(Y)$. |
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339 The morphisms from $a$ to $b$ are $A(Y\times I; a, b)$, where $a$ and $b$ label the two boundary components of the cylinder $Y\times I$. |
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340 Composition is given by gluing of cylinders. |
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341 |
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342 Let $X$ be an $n$-manifold with boundary and consider the collection of vector spaces |
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343 $A(X; \cdot) \deq \{A(X; c)\}$ where $c$ ranges through $\cC(\bd X)$. |
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344 This collection of vector spaces affords a representation of the category $A(\bd X)$, where |
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345 the action is given by gluing a collar $\bd X\times I$ to $X$. |
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346 |
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347 Given a splitting $X = X_1 \cup_Y X_2$ of a closed $n$-manifold $X$ along an $n{-}1$-manifold $Y$, |
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348 we have left and right actions of $A(Y)$ on $A(X_1; \cdot)$ and $A(X_2; \cdot)$. |
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349 The gluing theorem for $n$-manifolds states that there is a natural isomorphism |
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350 \[ |
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351 A(X) \cong A(X_1; \cdot) \otimes_{A(Y)} A(X_2; \cdot) . |
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352 \] |
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353 |