264 of points in the interior of $S$); |
264 of points in the interior of $S$); |
265 \item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$) |
265 \item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$) |
266 by an object (0-morphism) of $C$; |
266 by an object (0-morphism) of $C$; |
267 \item a transverse orientation of each 0-cell, thought of as a choice of |
267 \item a transverse orientation of each 0-cell, thought of as a choice of |
268 ``domain" and ``range" for the two adjacent 1-cells; and |
268 ``domain" and ``range" for the two adjacent 1-cells; and |
269 \item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with |
269 \item a labeling of each 0-cell by a 1-morphism of $C$, with |
270 domain and range determined by the transverse orientation and the labelings of the 1-cells. |
270 domain and range determined by the transverse orientation and the labelings of the 1-cells. |
271 \end{itemize} |
271 \end{itemize} |
272 |
272 |
273 If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels |
273 We want fields on 1-manifolds to be enriched over Vect, so we also allow formal linear combinations |
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274 of the above fields on a 1-manifold $X$ so long as these fields restrict to the same field on $\bd X$. |
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275 |
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276 In addition, we mod out by the relation which replaces |
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277 a 1-morphism label $a$ of a 0-cell $p$ with $a^*$ and reverse the transverse orientation of $p$. |
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278 |
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279 If $C$ is a *-algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels |
274 of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the |
280 of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the |
275 interior of $S$, each transversely oriented and each labeled by an element (1-morphism) |
281 interior of $S$, each transversely oriented and each labeled by an element (1-morphism) |
276 of the algebra. |
282 of the algebra. |
277 |
283 |
278 \medskip |
284 \medskip |
295 \item a labeling of each 1-cell by a 1-morphism of $C$, with |
301 \item a labeling of each 1-cell by a 1-morphism of $C$, with |
296 domain and range determined by the transverse orientation of the 1-cell |
302 domain and range determined by the transverse orientation of the 1-cell |
297 and the labelings of the 2-cells; |
303 and the labelings of the 2-cells; |
298 \item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood |
304 \item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood |
299 of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped |
305 of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped |
300 to $\pm 1 \in S^1$; and |
306 to $\pm 1 \in S^1$ |
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307 (this amounts to splitting of the link of the 0-cell into domain and range); and |
301 \item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range |
308 \item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range |
302 determined by the labelings of the 1-cells and the parameterizations of the previous |
309 determined by the labelings of the 1-cells and the parameterizations of the previous |
303 bullet. |
310 bullet. |
304 \end{itemize} |
311 \end{itemize} |
305 \nn{need to say this better; don't try to fit everything into the bulleted list} |
312 |
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313 As in the $n=1$ case, we allow formal linear combinations of fields on 2-manifolds, |
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314 so long as their restrictions to the boundary coincide. |
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315 |
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316 In addition, we regard the labelings as being equivariant with respect to the * structure |
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317 on 1-morphisms and pivotal structure on 2-morphisms. |
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318 That is, we mod out my the relation which flips the transverse orientation of a 1-cell |
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319 and replaces its label $a$ by $a^*$, as well as the relation which changes the parameterization of the link |
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320 of a 0-cell and replaces its label by the appropriate pivotal conjugate. |
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321 |
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322 \medskip |
306 |
323 |
307 For general $n$, a field on a $k$-manifold $X^k$ consists of |
324 For general $n$, a field on a $k$-manifold $X^k$ consists of |
308 \begin{itemize} |
325 \begin{itemize} |
309 \item A cell decomposition of $X$; |
326 \item A cell decomposition of $X$; |
310 \item an explicit general position homeomorphism from the link of each $j$-cell |
327 \item an explicit general position homeomorphism from the link of each $j$-cell |
311 to the boundary of the standard $(k-j)$-dimensional bihedron; and |
328 to the boundary of the standard $(k-j)$-dimensional bihedron; and |
312 \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with |
329 \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with |
313 domain and range determined by the labelings of the link of $j$-cell. |
330 domain and range determined by the labelings of the link of $j$-cell. |
314 \end{itemize} |
331 \end{itemize} |
315 |
332 |
316 %\nn{next definition might need some work; I think linearity relations should |
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317 %be treated differently (segregated) from other local relations, but I'm not sure |
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318 %the next definition is the best way to do it} |
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319 |
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320 \medskip |
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321 |
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322 |
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323 |
333 |
324 |
334 |
325 \subsection{Local relations} |
335 \subsection{Local relations} |
326 \label{sec:local-relations} |
336 \label{sec:local-relations} |
327 Local relations are certain subspaces of the fields on balls, which form an ideal under gluing. |
337 |
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338 For convenience we assume that fields are enriched over Vect. |
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339 |
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340 Local relations are subspaces $U(B; c)\sub \cC(B; c)$ of the fields on balls which form an ideal under gluing. |
328 Again, we give the examples first. |
341 Again, we give the examples first. |
329 |
342 |
330 \addtocounter{prop}{-2} |
343 \addtocounter{prop}{-2} |
331 \begin{example}[contd.] |
344 \begin{example}[contd.] |
332 For maps into spaces, $U(B; c)$ is generated by fields of the form $a-b \in \lf(B; c)$, |
345 For maps into spaces, $U(B; c)$ is generated by fields of the form $a-b \in \lf(B; c)$, |
361 |
374 |
362 \subsection{Constructing a TQFT} |
375 \subsection{Constructing a TQFT} |
363 \label{sec:constructing-a-tqft} |
376 \label{sec:constructing-a-tqft} |
364 |
377 |
365 In this subsection we briefly review the construction of a TQFT from a system of fields and local relations. |
378 In this subsection we briefly review the construction of a TQFT from a system of fields and local relations. |
366 (For more details, see \cite{kw:tqft}.) |
379 As usual, see \cite{kw:tqft} for more details. |
367 |
380 |
368 Let $W$ be an $n{+}1$-manifold. |
381 Let $W$ be an $n{+}1$-manifold. |
369 We can think of the path integral $Z(W)$ as assigning to each |
382 We can think of the path integral $Z(W)$ as assigning to each |
370 boundary condition $x\in \cC(\bd W)$ a complex number $Z(W)(x)$. |
383 boundary condition $x\in \cC(\bd W)$ a complex number $Z(W)(x)$. |
371 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 |
372 maps $\lf(\bd W)\to \c$. |
385 maps $\lf(\bd W)\to \c$. |
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386 (We haven't defined a path integral in this context; this is just for motivation.) |
373 |
387 |
374 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 |
375 $Z(\bd W) \sub \c^{\lf(\bd W)}$ defined by local projections. |
389 $Z(\bd W) \sub \c^{\lf(\bd W)}$ defined by local projections. |
376 The linear dual to this subspace, $A(\bd W) = Z(\bd W)^*$, |
390 The linear dual to this subspace, $A(\bd W) = Z(\bd W)^*$, |
377 can be thought of as finite linear combinations of fields modulo local relations. |
391 can be thought of as finite linear combinations of fields modulo local relations. |
386 \label{defn:TQFT-invariant} |
400 \label{defn:TQFT-invariant} |
387 The TQFT invariant of $X$ associated to a system of fields $\cF$ and local relations $\cU$ is |
401 The TQFT invariant of $X$ associated to a system of fields $\cF$ and local relations $\cU$ is |
388 $$A(X) \deq \lf(X) / U(X),$$ |
402 $$A(X) \deq \lf(X) / U(X),$$ |
389 where $\cU(X) \sub \lf(X)$ to be the space of local relations in $\lf(X)$; |
403 where $\cU(X) \sub \lf(X)$ to be the space of local relations in $\lf(X)$; |
390 $\cU(X)$ is generated by things of the form $u\bullet r$, where |
404 $\cU(X)$ is generated by things of the form $u\bullet r$, where |
391 $u\in \cU(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$. |
405 $u\in U(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$. |
392 \end{defn} |
406 \end{defn} |
393 (The blob complex, defined in the next section, |
407 (The blob complex, defined in the next section, |
394 is in some sense the derived version of $A(X)$.) |
408 is in some sense the derived version of $A(X)$.) |
395 If $X$ has boundary we can similarly define $A(X; c)$ for each |
409 If $X$ has boundary we can similarly define $A(X; c)$ for each |
396 boundary condition $c\in\cC(\bd X)$. |
410 boundary condition $c\in\cC(\bd X)$. |