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1 \newcommand{\manifolds}[1]{\cM_{#1}} |
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2 \newcommand{\closedManifolds}[1]{\cM_{#1}^{\text{closed}}} |
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3 \newcommand{\boundaryConditions}[1]{\cM_{#1}^{\bdy}} |
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4 Let $\manifolds{k}$ be the groupoid of manifolds (possibly with boundary) of dimension $k$ and diffeomorphisms between them. Write |
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5 $\closedManifolds{k}$ for the subgroupoid of closed manifolds. Taking the boundary gives a functor $\bdy : \manifolds{k} \to \closedManifolds{k-1}$. |
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6 Both $\manifolds{k}$ and $\closedManifolds{k}$ are symmetric tensor categories under the operation of disjoint union. |
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7 \begin{defn} |
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8 A \emph{system of fields} is a collection of functors associating a `set of fields' to each manifold of dimension at most $n$. |
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9 |
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10 First, there are functors $\cC_k : \closedManifolds{k} \to \Set$ for each $0 \leq k < n$. We ask that these are tensor functors, so they |
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11 take disjoint unions of manifolds to cartesian products of sets. In particular, this means that $\cC_k(\eset)$ is a point; there's only one field |
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12 on the empty manifold of any dimension. |
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13 |
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14 Define the groupoid $\boundaryConditions{k}$ of `manifolds with boundary conditions' as |
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15 \begin{equation*} |
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16 \setc{(Y; c)}{\begin{array}{c} \text{$Y$ a $k$-manifold} \\ c \in \cC_{k-1}(\bdy Y) \end{array}} |
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17 \xymatrix{ \ar@(ru,rd)@<-1ex>[]} |
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18 \set{Y \diffeoto Y'} |
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19 \end{equation*} |
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20 where we think of $f: Y \diffeoto Y'$ as a morphism $(Y; c) \isoto (Y'; \cC_{k-1}(\restrict{f}{\bdy Y})(c))$. |
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21 % |
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22 %The objects are pairs $(Y; c)$ with $Y$ a manifold (possibly with boundary) of dimension $k$ and $c \in \cC_{k-1}(\bdy Y)$ |
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23 %a field on the boundary of $Y$. A morphism $(Y; c) \to (Y'; c')$ is any diffeomorphism $f: Y \to Y'$ such that $\cC_{k-1}(\restrict{f}{\bdy Y})(c) = c'$. |
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24 Notice that $\closedManifolds{k}$ is naturally a subgroupoid of $\boundaryConditions{k}$, since a closed manifold has a unique field on its (empty) boundary. |
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25 |
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26 We now ask that the functors $\cC_k$ above extend to functors $\cC_k : \boundaryConditions{k} \to \Set$ for each $0 \leq k < n$, |
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27 and that there is an extra functor at the top level, $\cC_n : \boundaryConditions{n} \to \Vect$. (Notice that for $n$-manifolds we ask for a vector space, not just a set. This isn't essential for the definition, but we will only be interested in this case hereafter.) |
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28 We still require that these are tensor functors, and so take disjoint unions of manifolds to cartesian products of sets, or tensor products of vector spaces, as appropriate. |
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29 |
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30 \scott{Not sure how to say product fields in this setup.} |
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31 Finally, notice there are functors $- \times I : \manifolds{k} \to \manifolds{k+1}$ |
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32 Finally (?) we ask for natural transformations $- \times I : \cC_k \to \cC_{k+1} \compose (- \times I)$. Thus for each pair $(Y^k; c)$ we have a map $\cC_k($ |
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33 \end{defn} |
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34 \begin{rem} |
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35 Where the dimension of the manifold is clear, we'll often leave off the subscript on $\cC_k$. |
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36 \end{rem} |