1 %!TEX root = ../blob1.tex |
1 %!TEX root = ../blob1.tex |
2 |
2 |
3 \section{Definitions} |
3 \section{TQFTs via fields} |
4 \label{sec:definitions} |
4 %\label{sec:definitions} |
5 |
5 |
6 \nn{this section is a bit out of date; needs to be updated |
6 In this section we review the construction of TQFTs from ``topological fields". |
7 to fit with $n$-category definition given later} |
7 For more details see xxxx. |
8 |
8 |
9 \subsection{Systems of fields} |
9 \subsection{Systems of fields} |
10 \label{sec:fields} |
10 \label{sec:fields} |
11 |
11 |
12 Let $\cM_k$ denote the category (groupoid, in fact) with objects |
12 Let $\cM_k$ denote the category with objects |
13 oriented PL manifolds of dimension |
13 unoriented PL manifolds of dimension |
14 $k$ and morphisms homeomorphisms. |
14 $k$ and morphisms homeomorphisms. |
15 (We could equally well work with a different category of manifolds --- |
15 (We could equally well work with a different category of manifolds --- |
16 unoriented, topological, smooth, spin, etc. --- but for definiteness we |
16 oriented, topological, smooth, spin, etc. --- but for definiteness we |
17 will stick with oriented PL.) |
17 will stick with unoriented PL.) |
18 |
18 |
19 Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$. |
19 %Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$. |
20 |
20 |
21 A $n$-dimensional {\it system of fields} in $\cS$ |
21 A $n$-dimensional {\it system of fields} in $\cS$ |
22 is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ |
22 is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ |
23 together with some additional data and satisfying some additional conditions, all specified below. |
23 together with some additional data and satisfying some additional conditions, all specified below. |
24 |
24 |
43 transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. |
43 transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. |
44 For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of |
44 For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of |
45 $\cC(X)$ which restricts to $c$. |
45 $\cC(X)$ which restricts to $c$. |
46 In this context, we will call $c$ a boundary condition. |
46 In this context, we will call $c$ a boundary condition. |
47 \item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$. (This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), then this extra structure is considered part of the definition of $\cC_n$. Any maps mentioned below between top level fields must be morphisms in $\cS$. |
47 \item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$. (This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), then this extra structure is considered part of the definition of $\cC_n$. Any maps mentioned below between top level fields must be morphisms in $\cS$. |
48 \item There are orientation reversal maps $\cC_k(X) \to \cC_k(-X)$, and these maps |
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49 again comprise a natural transformation of functors. |
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50 In addition, the orientation reversal maps are compatible with the boundary restriction maps. |
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51 \item $\cC_k$ is compatible with the symmetric monoidal |
48 \item $\cC_k$ is compatible with the symmetric monoidal |
52 structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
49 structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
53 compatibly with homeomorphisms, restriction to boundary, and orientation reversal. |
50 compatibly with homeomorphisms, restriction to boundary, and orientation reversal. |
54 We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ |
51 We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ |
55 restriction maps. |
52 restriction maps. |
68 of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
65 of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
69 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
66 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
70 the gluing map is surjective. |
67 the gluing map is surjective. |
71 From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
68 From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
72 gluing surface, we say that fields in the image of the gluing map |
69 gluing surface, we say that fields in the image of the gluing map |
73 are transverse to $Y$ or cuttable along $Y$. |
70 are transverse to $Y$ or splittable along $Y$. |
74 \item Gluing with corners. |
71 \item Gluing with corners. |
75 Let $\bd X = Y \cup -Y \cup W$, where $\pm Y$ and $W$ might intersect along their boundaries. |
72 Let $\bd X = Y \cup -Y \cup W$, where $\pm Y$ and $W$ might intersect along their boundaries. |
76 Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
73 Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
77 Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself |
74 Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself |
78 (without corners) along two copies of $\bd Y$. |
75 (without corners) along two copies of $\bd Y$. |
79 Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a cuttable field on $W\sgl$ and let |
76 Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a splittable field on $W\sgl$ and let |
80 $c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$. |
77 $c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$. |
81 Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$. |
78 Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$. |
82 (This restriction map uses the gluing without corners map above.) |
79 (This restriction map uses the gluing without corners map above.) |
83 Using the boundary restriction, gluing without corners, and (in one case) orientation reversal |
80 Using the boundary restriction, gluing without corners, and (in one case) orientation reversal |
84 maps, we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two |
81 maps, we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two |
92 of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
89 of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
93 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
90 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
94 the gluing map is surjective. |
91 the gluing map is surjective. |
95 From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
92 From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
96 gluing surface, we say that fields in the image of the gluing map |
93 gluing surface, we say that fields in the image of the gluing map |
97 are transverse to $Y$ or cuttable along $Y$. |
94 are transverse to $Y$ or splittable along $Y$. |
98 \item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted |
95 \item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted |
99 $c \mapsto c\times I$. |
96 $c \mapsto c\times I$. |
100 These maps comprise a natural transformation of functors, and commute appropriately |
97 These maps comprise a natural transformation of functors, and commute appropriately |
101 with all the structure maps above (disjoint union, boundary restriction, etc.). |
98 with all the structure maps above (disjoint union, boundary restriction, etc.). |
102 Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism |
99 Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism |
109 Using the functoriality and $\bullet\times I$ properties above, together |
106 Using the functoriality and $\bullet\times I$ properties above, together |
110 with boundary collar homeomorphisms of manifolds, we can define the notion of |
107 with boundary collar homeomorphisms of manifolds, we can define the notion of |
111 {\it extended isotopy}. |
108 {\it extended isotopy}. |
112 Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
109 Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
113 of $\bd M$. |
110 of $\bd M$. |
114 Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is cuttable along $\bd Y$. |
111 Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is splittable along $\bd Y$. |
115 Let $c$ be $x$ restricted to $Y$. |
112 Let $c$ be $x$ restricted to $Y$. |
116 Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$. |
113 Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$. |
117 Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$. |
114 Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$. |
118 Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism. |
115 Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism. |
119 Then we say that $x$ is {\it extended isotopic} to $f(x \bullet (c\times I))$. |
116 Then we say that $x$ is {\it extended isotopic} to $f(x \bullet (c\times I))$. |
121 on $M$ generated by isotopy plus all instance of the above construction |
118 on $M$ generated by isotopy plus all instance of the above construction |
122 (for all appropriate $Y$ and $x$). |
119 (for all appropriate $Y$ and $x$). |
123 |
120 |
124 \nn{should also say something about pseudo-isotopy} |
121 \nn{should also say something about pseudo-isotopy} |
125 |
122 |
126 %\bigskip |
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127 %\hrule |
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128 %\bigskip |
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129 % |
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130 %\input{text/fields.tex} |
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131 % |
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132 % |
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133 %\bigskip |
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134 %\hrule |
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135 %\bigskip |
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136 |
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137 \nn{note: probably will suppress from notation the distinction |
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138 between fields and their (orientation-reversal) duals} |
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139 |
123 |
140 \nn{remark that if top dimensional fields are not already linear |
124 \nn{remark that if top dimensional fields are not already linear |
141 then we will soon linearize them(?)} |
125 then we will soon linearize them(?)} |
142 |
126 |
143 We now describe in more detail systems of fields coming from sub-cell-complexes labeled |
127 We now describe in more detail systems of fields coming from sub-cell-complexes labeled |
289 $\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into |
273 $\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into |
290 domain and range. |
274 domain and range. |
291 |
275 |
292 \nn{maybe examples of local relations before general def?} |
276 \nn{maybe examples of local relations before general def?} |
293 |
277 |
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278 \subsection{Constructing a TQFT} |
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279 |
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280 \nn{need to expand this; use $\bc_0/\bc_1$ notation (maybe); also introduce |
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281 cylinder categories and gluing formula} |
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282 |
294 Given a system of fields and local relations, we define the skein space |
283 Given a system of fields and local relations, we define the skein space |
295 $A(Y^n; c)$ to be the space of all finite linear combinations of fields on |
284 $A(Y^n; c)$ to be the space of all finite linear combinations of fields on |
296 the $n$-manifold $Y$ modulo local relations. |
285 the $n$-manifold $Y$ modulo local relations. |
297 The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations |
286 The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations |
298 is defined to be the dual of $A(Y; c)$. |
287 is defined to be the dual of $A(Y; c)$. |
302 |
291 |
303 The blob complex is in some sense the derived version of $A(Y; c)$. |
292 The blob complex is in some sense the derived version of $A(Y; c)$. |
304 |
293 |
305 |
294 |
306 |
295 |
307 \subsection{The blob complex} |
296 \section{The blob complex} |
308 \label{sec:blob-definition} |
297 \label{sec:blob-definition} |
309 |
298 |
310 Let $X$ be an $n$-manifold. |
299 Let $X$ be an $n$-manifold. |
311 Assume a fixed system of fields and local relations. |
300 Assume a fixed system of fields and local relations. |
312 In this section we will usually suppress boundary conditions on $X$ from the notation |
301 In this section we will usually suppress boundary conditions on $X$ from the notation |
377 |
366 |
378 A nested 2-blob diagram consists of |
367 A nested 2-blob diagram consists of |
379 \begin{itemize} |
368 \begin{itemize} |
380 \item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. |
369 \item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. |
381 \item A field $r \in \cC(X \setmin B_0; c_0)$ |
370 \item A field $r \in \cC(X \setmin B_0; c_0)$ |
382 (for some $c_0 \in \cC(\bd B_0)$), which is cuttable along $\bd B_1$. |
371 (for some $c_0 \in \cC(\bd B_0)$), which is splittable along $\bd B_1$. |
383 \item A local relation field $u_0 \in U(B_0; c_0)$. |
372 \item A local relation field $u_0 \in U(B_0; c_0)$. |
384 \end{itemize} |
373 \end{itemize} |
385 Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
374 Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
386 (for some $c_1 \in \cC(B_1)$) and |
375 (for some $c_1 \in \cC(B_1)$) and |
387 $r' \in \cC(X \setmin B_1; c_1)$. |
376 $r' \in \cC(X \setmin B_1; c_1)$. |
406 && \bigoplus \left( |
395 && \bigoplus \left( |
407 \bigoplus_{B_0 \subset B_1} \bigoplus_{c_0} |
396 \bigoplus_{B_0 \subset B_1} \bigoplus_{c_0} |
408 U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0) |
397 U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0) |
409 \right) . |
398 \right) . |
410 \end{eqnarray*} |
399 \end{eqnarray*} |
411 The final $\lf(X\setmin B_0; c_0)$ above really means fields cuttable along $\bd B_1$, |
400 The final $\lf(X\setmin B_0; c_0)$ above really means fields splittable along $\bd B_1$, |
412 but we didn't feel like introducing a notation for that. |
401 but we didn't feel like introducing a notation for that. |
413 For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign |
402 For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign |
414 (rather than a new, linearly independent 2-blob diagram). |
403 (rather than a new, linearly independent 2-blob diagram). |
415 |
404 |
416 Now for the general case. |
405 Now for the general case. |
428 $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
417 $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
429 if the latter space is not empty. |
418 if the latter space is not empty. |
430 \item A field $r \in \cC(X \setmin B^t; c^t)$, |
419 \item A field $r \in \cC(X \setmin B^t; c^t)$, |
431 where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ |
420 where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ |
432 is determined by the $c_i$'s. |
421 is determined by the $c_i$'s. |
433 $r$ is required to be cuttable along the boundaries of all blobs, twigs or not. |
422 $r$ is required to be splittable along the boundaries of all blobs, twigs or not. |
434 \item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
423 \item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
435 where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
424 where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
436 If $B_i = B_j$ then $u_i = u_j$. |
425 If $B_i = B_j$ then $u_i = u_j$. |
437 \end{itemize} |
426 \end{itemize} |
438 |
427 |
447 \bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} |
436 \bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} |
448 \left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . |
437 \left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . |
449 \] |
438 \] |
450 Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
439 Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
451 $\overline{c}$ runs over all boundary conditions, again as described above. |
440 $\overline{c}$ runs over all boundary conditions, again as described above. |
452 $j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are cuttable along all of the blobs in $\overline{B}$. |
441 $j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are splittable along all of the blobs in $\overline{B}$. |
453 |
442 |
454 The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. |
443 The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. |
455 Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
444 Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
456 Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
445 Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
457 If $B_j$ is not a twig blob, this involves only decrementing |
446 If $B_j$ is not a twig blob, this involves only decrementing |