243 as a ``string diagram". |
243 as a ``string diagram". |
244 It can be thought of as (geometrically) dual to a pasting diagram. |
244 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 |
245 One of the advantages of string diagrams over pasting diagrams is that one has more |
246 flexibility in slicing them up in various ways. |
246 flexibility in slicing them up in various ways. |
247 In addition, string diagrams are traditional in quantum topology. |
247 In addition, string diagrams are traditional in quantum topology. |
248 The diagrams predate by many years the terms ``string diagram" and ``quantum topology". |
248 The diagrams predate by many years the terms ``string diagram" and ``quantum topology", e.g. \cite{ |
249 \nn{?? cite penrose, kauffman, jones(?)} |
249 MR0281657,MR776784 % penrose |
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250 } |
250 |
251 |
251 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 |
252 position with respect to the boundary --- the boundary intersects each cell |
253 position with respect to the boundary --- the boundary intersects each cell |
253 transversely, so cells meeting the boundary are mere half-cells. |
254 transversely, so cells meeting the boundary are mere half-cells. |
254 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 |
313 As in the $n=1$ case, we allow formal linear combinations of fields on 2-manifolds, |
314 As in the $n=1$ case, we allow formal linear combinations of fields on 2-manifolds, |
314 so long as their restrictions to the boundary coincide. |
315 so long as their restrictions to the boundary coincide. |
315 |
316 |
316 In addition, we regard the labelings as being equivariant with respect to the * structure |
317 In addition, we regard the labelings as being equivariant with respect to the * structure |
317 on 1-morphisms and pivotal structure on 2-morphisms. |
318 on 1-morphisms and pivotal structure on 2-morphisms. |
318 That is, we mod out my the relation which flips the transverse orientation of a 1-cell |
319 That is, we mod out by the relation which flips the transverse orientation of a 1-cell |
319 and replaces its label $a$ by $a^*$, as well as the relation which changes the parameterization of the link |
320 and replaces its label $a$ by $a^*$, as well as the relation which changes the parameterization of the link |
320 of a 0-cell and replaces its label by the appropriate pivotal conjugate. |
321 of a 0-cell and replaces its label by the appropriate pivotal conjugate. |
321 |
322 |
322 \medskip |
323 \medskip |
323 |
324 |
376 \label{sec:constructing-a-tqft} |
377 \label{sec:constructing-a-tqft} |
377 |
378 |
378 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. |
379 As usual, see \cite{kw:tqft} for more details. |
380 As usual, see \cite{kw:tqft} for more details. |
380 |
381 |
381 Let $W$ be an $n{+}1$-manifold. |
382 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 the path integral $Z(W)$ as assigning to each |
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383 boundary condition $x\in \cC(\bd W)$ a complex number $Z(W)(x)$. |
383 boundary condition $x\in \cC(\bd W)$ a complex number $Z(W)(x)$. |
384 In other words, $Z(W)$ lies in $\c^{\lf(\bd W)}$, the vector space of linear |
384 In other words, $Z(W)$ lies in $\c^{\lf(\bd W)}$, the vector space of linear |
385 maps $\lf(\bd W)\to \c$. |
385 maps $\lf(\bd W)\to \c$. |
386 (We haven't defined a path integral in this context; this is just for motivation.) |
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387 |
386 |
388 The locality of the TQFT implies that $Z(W)$ in fact lies in a subspace |
387 The locality of the TQFT implies that $Z(W)$ in fact lies in a subspace |
389 $Z(\bd W) \sub \c^{\lf(\bd W)}$ defined by local projections. |
388 $Z(\bd W) \sub \c^{\lf(\bd W)}$ defined by local projections. |
390 The linear dual to this subspace, $A(\bd W) = Z(\bd W)^*$, |
389 The linear dual to this subspace, $A(\bd W) = Z(\bd W)^*$, |
391 can be thought of as finite linear combinations of fields modulo local relations. |
390 can be thought of as finite linear combinations of fields modulo local relations. |
398 %$\bc_0(X) = \lf(X)$. |
397 %$\bc_0(X) = \lf(X)$. |
399 \begin{defn} |
398 \begin{defn} |
400 \label{defn:TQFT-invariant} |
399 \label{defn:TQFT-invariant} |
401 The TQFT invariant of $X$ associated to a system of fields $\cF$ and local relations $\cU$ is |
400 The TQFT invariant of $X$ associated to a system of fields $\cF$ and local relations $\cU$ is |
402 $$A(X) \deq \lf(X) / U(X),$$ |
401 $$A(X) \deq \lf(X) / U(X),$$ |
403 where $\cU(X) \sub \lf(X)$ to be the space of local relations in $\lf(X)$; |
402 where $\cU(X) \sub \lf(X)$ is the space of local relations in $\lf(X)$: |
404 $\cU(X)$ is generated by things of the form $u\bullet r$, where |
403 $\cU(X)$ is generated by fields of the form $u\bullet r$, where |
405 $u\in U(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$. |
404 $u\in U(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$. |
406 \end{defn} |
405 \end{defn} |
407 (The blob complex, defined in the next section, |
406 The blob complex, defined in the next section, |
408 is in some sense the derived version of $A(X)$.) |
407 is in some sense the derived version of $A(X)$. |
409 If $X$ has boundary we can similarly define $A(X; c)$ for each |
408 If $X$ has boundary we can similarly define $A(X; c)$ for each |
410 boundary condition $c\in\cC(\bd X)$. |
409 boundary condition $c\in\cC(\bd X)$. |
411 |
410 |
412 The above construction can be extended to higher codimensions, assigning |
411 The above construction can be extended to higher codimensions, assigning |
413 a $k$-category $A(Y)$ to an $n{-}k$-manifold $Y$, for $0 \le k \le n$. |
412 a $k$-category $A(Y)$ to an $n{-}k$-manifold $Y$, for $0 \le k \le n$. |
414 These invariants fit together via actions and gluing formulas. |
413 These invariants fit together via actions and gluing formulas. |
415 We describe only the case $k=1$ below. |
414 We describe only the case $k=1$ below. |
416 (The construction of the $n{+}1$-dimensional part of the theory (the path integral) |
415 The construction of the $n{+}1$-dimensional part of the theory (the path integral) |
417 requires that the starting data (fields and local relations) satisfy additional |
416 requires that the starting data (fields and local relations) satisfy additional |
418 conditions. |
417 conditions. |
419 We do not assume these conditions here, so when we say ``TQFT" we mean a decapitated TQFT |
418 We do not assume these conditions here, so when we say ``TQFT" we mean a decapitated TQFT |
420 that lacks its $n{+}1$-dimensional part.) |
419 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. |
421 |
420 |
422 Let $Y$ be an $n{-}1$-manifold. |
421 Let $Y$ be an $n{-}1$-manifold. |
423 Define a (linear) 1-category $A(Y)$ as follows. |
422 Define a linear 1-category $A(Y)$ as follows. |
424 The objects of $A(Y)$ are $\cC(Y)$. |
423 The set of objects of $A(Y)$ is $\cC(Y)$. |
425 The morphisms from $a$ to $b$ are $A(Y\times I; a, b)$, |
424 The morphisms from $a$ to $b$ are $A(Y\times I; a, b)$, |
426 where $a$ and $b$ label the two boundary components of the cylinder $Y\times I$. |
425 where $a$ and $b$ label the two boundary components of the cylinder $Y\times I$. |
427 Composition is given by gluing of cylinders. |
426 Composition is given by gluing of cylinders. |
428 |
427 |
429 Let $X$ be an $n$-manifold with boundary and consider the collection of vector spaces |
428 Let $X$ be an $n$-manifold with boundary and consider the collection of vector spaces |
430 $A(X; \cdot) \deq \{A(X; c)\}$ where $c$ ranges through $\cC(\bd X)$. |
429 $A(X; -) \deq \{A(X; c)\}$ where $c$ ranges through $\cC(\bd X)$. |
431 This collection of vector spaces affords a representation of the category $A(\bd X)$, where |
430 This collection of vector spaces affords a representation of the category $A(\bd X)$, where |
432 the action is given by gluing a collar $\bd X\times I$ to $X$. |
431 the action is given by gluing a collar $\bd X\times I$ to $X$. |
433 |
432 |
434 Given a splitting $X = X_1 \cup_Y X_2$ of a closed $n$-manifold $X$ along an $n{-}1$-manifold $Y$, |
433 Given a splitting $X = X_1 \cup_Y X_2$ of a closed $n$-manifold $X$ along an $n{-}1$-manifold $Y$, |
435 we have left and right actions of $A(Y)$ on $A(X_1; \cdot)$ and $A(X_2; \cdot)$. |
434 we have left and right actions of $A(Y)$ on $A(X_1; -)$ and $A(X_2; -)$. |
436 The gluing theorem for $n$-manifolds states that there is a natural isomorphism |
435 The gluing theorem for $n$-manifolds states that there is a natural isomorphism |
437 \[ |
436 \[ |
438 A(X) \cong A(X_1; \cdot) \otimes_{A(Y)} A(X_2; \cdot) . |
437 A(X) \cong A(X_1; -) \otimes_{A(Y)} A(X_2; -) . |
439 \] |
438 \] |
440 |
439 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. |