45 will stick with unoriented PL.) |
45 will stick with unoriented PL.) |
46 |
46 |
47 Fix a symmetric monoidal category $\cS$. |
47 Fix a symmetric monoidal category $\cS$. |
48 Fields on $n$-manifolds will be enriched over $\cS$. |
48 Fields on $n$-manifolds will be enriched over $\cS$. |
49 Good examples to keep in mind are $\cS = \Set$ or $\cS = \Vect$. |
49 Good examples to keep in mind are $\cS = \Set$ or $\cS = \Vect$. |
50 The presentation here requires that the objects of $\cS$ have an underlying set, but this could probably be avoided if desired. |
50 The presentation here requires that the objects of $\cS$ have an underlying set, |
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51 but this could probably be avoided if desired. |
51 |
52 |
52 A $n$-dimensional {\it system of fields} in $\cS$ |
53 A $n$-dimensional {\it system of fields} in $\cS$ |
53 is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ |
54 is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ |
54 together with some additional data and satisfying some additional conditions, all specified below. |
55 together with some additional data and satisfying some additional conditions, all specified below. |
55 |
56 |
224 above tensor products. |
225 above tensor products. |
225 |
226 |
226 } % end \noop |
227 } % end \noop |
227 |
228 |
228 |
229 |
229 \subsection{Systems of fields from $n$-categories} |
230 \subsection{Systems of fields from \texorpdfstring{$n$}{n}-categories} |
230 \label{sec:example:traditional-n-categories(fields)} |
231 \label{sec:example:traditional-n-categories(fields)} |
231 We now describe in more detail Example \ref{ex:traditional-n-categories(fields)}, |
232 We now describe in more detail Example \ref{ex:traditional-n-categories(fields)}, |
232 systems of fields coming from embedded cell complexes labeled |
233 systems of fields coming from embedded cell complexes labeled |
233 by $n$-category morphisms. |
234 by $n$-category morphisms. |
234 |
235 |
243 as a ``string diagram". |
244 as a ``string diagram". |
244 It can be thought of as (geometrically) dual to a pasting diagram. |
245 It can be thought of as (geometrically) dual to a pasting diagram. |
245 One of the advantages of string diagrams over pasting diagrams is that one has more |
246 One of the advantages of string diagrams over pasting diagrams is that one has more |
246 flexibility in slicing them up in various ways. |
247 flexibility in slicing them up in various ways. |
247 In addition, string diagrams are traditional in quantum topology. |
248 In addition, string diagrams are traditional in quantum topology. |
248 The diagrams predate by many years the terms ``string diagram" and ``quantum topology", e.g. \cite{MR0281657,MR776784} % both penrose |
249 The diagrams predate by many years the terms ``string diagram" and |
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250 ``quantum topology", e.g. \cite{MR0281657,MR776784} % both penrose |
249 |
251 |
250 If $X$ has boundary, we require that the cell decompositions are in general |
252 If $X$ has boundary, we require that the cell decompositions are in general |
251 position with respect to the boundary --- the boundary intersects each cell |
253 position with respect to the boundary --- the boundary intersects each cell |
252 transversely, so cells meeting the boundary are mere half-cells. |
254 transversely, so cells meeting the boundary are mere half-cells. |
253 Put another way, the cell decompositions we consider are dual to standard cell |
255 Put another way, the cell decompositions we consider are dual to standard cell |
375 \label{sec:constructing-a-tqft} |
377 \label{sec:constructing-a-tqft} |
376 |
378 |
377 In this subsection we briefly review the construction of a TQFT from a system of fields and local relations. |
379 In this subsection we briefly review the construction of a TQFT from a system of fields and local relations. |
378 As usual, see \cite{kw:tqft} for more details. |
380 As usual, see \cite{kw:tqft} for more details. |
379 |
381 |
380 We can think of a path integral $Z(W)$ of an $n+1$-manifold (which we're not defining in this context; this is just motivation) as assigning to each |
382 We can think of a path integral $Z(W)$ of an $n+1$-manifold |
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383 (which we're not defining in this context; this is just motivation) as assigning to each |
381 boundary condition $x\in \cC(\bd W)$ a complex number $Z(W)(x)$. |
384 boundary condition $x\in \cC(\bd W)$ a complex number $Z(W)(x)$. |
382 In other words, $Z(W)$ lies in $\c^{\lf(\bd W)}$, the vector space of linear |
385 In other words, $Z(W)$ lies in $\c^{\lf(\bd W)}$, the vector space of linear |
383 maps $\lf(\bd W)\to \c$. |
386 maps $\lf(\bd W)\to \c$. |
384 |
387 |
385 The locality of the TQFT implies that $Z(W)$ in fact lies in a subspace |
388 The locality of the TQFT implies that $Z(W)$ in fact lies in a subspace |
412 We describe only the case $k=1$ below. |
415 We describe only the case $k=1$ below. |
413 The construction of the $n{+}1$-dimensional part of the theory (the path integral) |
416 The construction of the $n{+}1$-dimensional part of the theory (the path integral) |
414 requires that the starting data (fields and local relations) satisfy additional |
417 requires that the starting data (fields and local relations) satisfy additional |
415 conditions. |
418 conditions. |
416 We do not assume these conditions here, so when we say ``TQFT" we mean a decapitated TQFT |
419 We do not assume these conditions here, so when we say ``TQFT" we mean a decapitated TQFT |
417 that lacks its $n{+}1$-dimensional part. Such a ``decapitated'' TQFT is sometimes also called an $n+\epsilon$ or $n+\frac{1}{2}$ dimensional TQFT, referring to the fact that it assigns maps to mapping cylinders between $n$-manifolds, but nothing to arbitrary $n{+}1$-manifolds. |
420 that lacks its $n{+}1$-dimensional part. |
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421 Such a ``decapitated'' TQFT is sometimes also called an $n+\epsilon$ or |
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422 $n+\frac{1}{2}$ dimensional TQFT, referring to the fact that it assigns maps to |
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423 mapping cylinders between $n$-manifolds, but nothing to arbitrary $n{+}1$-manifolds. |
418 |
424 |
419 Let $Y$ be an $n{-}1$-manifold. |
425 Let $Y$ be an $n{-}1$-manifold. |
420 Define a linear 1-category $A(Y)$ as follows. |
426 Define a linear 1-category $A(Y)$ as follows. |
421 The set of objects of $A(Y)$ is $\cC(Y)$. |
427 The set of objects of $A(Y)$ is $\cC(Y)$. |
422 The morphisms from $a$ to $b$ are $A(Y\times I; a, b)$, |
428 The morphisms from $a$ to $b$ are $A(Y\times I; a, b)$, |
432 we have left and right actions of $A(Y)$ on $A(X_1; -)$ and $A(X_2; -)$. |
438 we have left and right actions of $A(Y)$ on $A(X_1; -)$ and $A(X_2; -)$. |
433 The gluing theorem for $n$-manifolds states that there is a natural isomorphism |
439 The gluing theorem for $n$-manifolds states that there is a natural isomorphism |
434 \[ |
440 \[ |
435 A(X) \cong A(X_1; -) \otimes_{A(Y)} A(X_2; -) . |
441 A(X) \cong A(X_1; -) \otimes_{A(Y)} A(X_2; -) . |
436 \] |
442 \] |
437 A proof of this gluing formula appears in \cite{kw:tqft}, but it also becomes a special case of Theorem \ref{thm:gluing} by taking $0$-th homology. |
443 A proof of this gluing formula appears in \cite{kw:tqft}, but it also becomes a |
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444 special case of Theorem \ref{thm:gluing} by taking $0$-th homology. |