275 |
275 |
276 \nn{maybe examples of local relations before general def?} |
276 \nn{maybe examples of local relations before general def?} |
277 |
277 |
278 \subsection{Constructing a TQFT} |
278 \subsection{Constructing a TQFT} |
279 |
279 |
280 \nn{need to expand this; use $\bc_0/\bc_1$ notation (maybe); also introduce |
280 In this subsection we briefly review the construction of a TQFT from a system of fields and local relations. |
281 cylinder categories and gluing formula} |
281 (For more details, see \cite{kw:tqft}.) |
282 |
282 |
283 Given a system of fields and local relations, we define the skein space |
283 Let $W$ be an $n{+}1$-manifold. |
284 $A(Y^n; c)$ to be the space of all finite linear combinations of fields on |
284 We can think of the path integral $Z(W)$ as assigning to each |
285 the $n$-manifold $Y$ modulo local relations. |
285 boundary condition $x\in \cC(\bd W)$ a complex number $Z(W)(x)$. |
286 The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations |
286 In other words, $Z(W)$ lies in $\c^{\lf(\bd W)}$, the vector space of linear |
287 is defined to be the dual of $A(Y; c)$. |
287 maps $\lf(\bd W)\to \c$. |
288 (See \cite{kw:tqft} or xxxx for details.) |
288 |
289 |
289 The locality of the TQFT implies that $Z(W)$ in fact lies in a subspace |
290 \nn{should expand above paragraph} |
290 $Z(\bd W) \sub \c^{\lf(\bd W)}$ defined by local projections. |
291 |
291 The linear dual to this subspace, $A(\bd W) = Z(\bd W)^*$, |
292 The blob complex is in some sense the derived version of $A(Y; c)$. |
292 can be thought of as finite linear combinations of fields modulo local relations. |
293 |
293 (In other words, $A(\bd W)$ is a sort of generalized skein module.) |
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294 This is the motivation behind the definition of fields and local relations above. |
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295 |
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296 In more detail, let $X$ be an $n$-manifold. |
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297 %To harmonize notation with the next section, |
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298 %let $\bc_0(X)$ be the vector space of finite linear combinations of fields on $X$, so |
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299 %$\bc_0(X) = \lf(X)$. |
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300 Define $U(X) \sub \lf(X)$ to be the space of local relations in $\lf(X)$; |
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301 $U(X)$ is generated by things of the form $u\bullet r$, where |
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302 $u\in U(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$. |
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303 Define |
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304 \[ |
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305 A(X) \deq \lf(X) / U(X) . |
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306 \] |
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307 (The blob complex, defined in the next section, |
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308 is in some sense the derived version of $A(X)$.) |
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309 If $X$ has boundary we can similarly define $A(X; c)$ for each |
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310 boundary condition $c\in\cC(\bd X)$. |
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311 |
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312 The above construction can be extended to higher codimensions, assigning |
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313 a $k$-category $A(Y)$ to an $n{-}k$-manifold $Y$, for $0 \le k \le n$. |
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314 These invariants fit together via actions and gluing formulas. |
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315 We describe only the case $k=1$ below. |
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316 (The construction of the $n{+}1$-dimensional part of the theory (the path integral) |
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317 requires that the starting data (fields and local relations) satisfy additional |
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318 conditions. |
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319 We do not assume these conditions here, so when we say ``TQFT" we mean a decapitated TQFT |
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320 that lacks its $n{+}1$-dimensional part.) |
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321 |
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322 Let $Y$ be an $n{-}1$-manifold. |
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323 Define a (linear) 1-category $A(Y)$ as follows. |
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324 The objects of $A(Y)$ are $\cC(Y)$. |
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325 The morphisms from $a$ to $b$ are $A(Y\times I; a, b)$, where $a$ and $b$ label the two boundary components of the cylinder $Y\times I$. |
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326 Composition is given by gluing of cylinders. |
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327 |
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328 Let $X$ be an $n$-manifold with boundary and consider the collection of vector spaces |
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329 $A(X; \cdot) \deq \{A(X; c)\}$ where $c$ ranges through $\cC(\bd X)$. |
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330 This collection of vector spaces affords a representation of the category $A(\bd X)$, where |
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331 the action is given by gluing a collar $\bd X\times I$ to $X$. |
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332 |
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333 Given a splitting $X = X_1 \cup_Y X_2$ of a closed $n$-manifold $X$ along an $n{-}1$-manifold $Y$, |
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334 we have left and right actions of $A(Y)$ on $A(X_1; \cdot)$ and $A(X_2; \cdot)$. |
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335 The gluing theorem for $n$-manifolds states that there is a natural isomorphism |
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336 \[ |
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337 A(X) \cong A(X_1; \cdot) \otimes_{A(Y)} A(X_2; \cdot) . |
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338 \] |
294 |
339 |
295 |
340 |
296 \section{The blob complex} |
341 \section{The blob complex} |
297 \label{sec:blob-definition} |
342 \label{sec:blob-definition} |
298 |
343 |