author | Kevin Walker <kevin@canyon23.net> |
Sun, 14 Nov 2010 18:33:03 -0800 | |
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%!TEX root = ../blob1.tex |
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\section{TQFTs via fields} |
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\label{sec:fields} |
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\label{sec:tqftsviafields} |
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In this section we review the construction of TQFTs from fields and local relations. |
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For more details see \cite{kw:tqft}. |
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For our purposes, a TQFT is {\it defined} to be something which arises |
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from this construction. |
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This is an alternative to the more common definition of a TQFT |
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as a functor on cobordism categories satisfying various conditions. |
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A fully local (``down to points") version of the cobordism-functor TQFT definition |
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should be equivalent to the fields-and-local-relations definition. |
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A system of fields is very closely related to an $n$-category. |
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In one direction, Example \ref{ex:traditional-n-categories(fields)} |
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shows how to construct a system of fields from a (traditional) $n$-category. |
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We do this in detail for $n=1,2$ (\S\ref{sec:example:traditional-n-categories(fields)}) |
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and more informally for general $n$. |
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In the other direction, |
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our preferred definition of an $n$-category in \S\ref{sec:ncats} is essentially |
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just a system of fields restricted to balls of dimensions 0 through $n$; |
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one could call this the ``local" part of a system of fields. |
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Since this section is intended primarily to motivate |
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the blob complex construction of \S\ref{sec:blob-definition}, |
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we suppress some technical details. |
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In \S\ref{sec:ncats} the analogous details are treated more carefully. |
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\medskip |
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We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
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submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
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$\overline{X \setmin Y}$. |
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\subsection{Systems of fields} |
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\label{ss:syst-o-fields} |
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Let $\cM_k$ denote the category with objects |
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unoriented PL manifolds of dimension |
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$k$ and morphisms homeomorphisms. |
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(We could equally well work with a different category of manifolds --- |
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oriented, topological, smooth, spin, etc. --- but for simplicity we |
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will stick with unoriented PL.) |
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Fix a symmetric monoidal category $\cS$. |
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Fields on $n$-manifolds will be enriched over $\cS$. |
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Good examples to keep in mind are $\cS = \Set$ or $\cS = \Vect$. |
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The presentation here requires that the objects of $\cS$ have an underlying set, |
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but this could probably be avoided if desired. |
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A $n$-dimensional {\it system of fields} in $\cS$ |
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is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ |
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together with some additional data and satisfying some additional conditions, all specified below. |
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Before finishing the definition of fields, we give two motivating examples of systems of fields. |
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\begin{example} |
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\label{ex:maps-to-a-space(fields)} |
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Fix a target space $T$, and let $\cC(X)$ be the set of continuous maps |
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from $X$ to $T$. |
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\end{example} |
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\begin{example} |
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\label{ex:traditional-n-categories(fields)} |
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Fix an $n$-category $C$, and let $\cC(X)$ be |
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the set of embedded cell complexes in $X$ with codimension-$j$ cells labeled by |
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