3 \section{TQFTs via fields} |
3 \section{TQFTs via fields} |
4 %\label{sec:definitions} |
4 %\label{sec:definitions} |
5 |
5 |
6 In this section we review the construction of TQFTs from ``topological fields". |
6 In this section we review the construction of TQFTs from ``topological fields". |
7 For more details see xxxx. |
7 For more details see xxxx. |
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8 |
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9 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
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10 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
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11 $\overline{X \setmin Y}$. |
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12 |
8 |
13 |
9 \subsection{Systems of fields} |
14 \subsection{Systems of fields} |
10 \label{sec:fields} |
15 \label{sec:fields} |
11 |
16 |
12 Let $\cM_k$ denote the category with objects |
17 Let $\cM_k$ denote the category with objects |
344 Let $X$ be an $n$-manifold. |
349 Let $X$ be an $n$-manifold. |
345 Assume a fixed system of fields and local relations. |
350 Assume a fixed system of fields and local relations. |
346 In this section we will usually suppress boundary conditions on $X$ from the notation |
351 In this section we will usually suppress boundary conditions on $X$ from the notation |
347 (e.g. write $\lf(X)$ instead of $\lf(X; c)$). |
352 (e.g. write $\lf(X)$ instead of $\lf(X; c)$). |
348 |
353 |
349 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
354 We want to replace the quotient |
350 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
355 \[ |
351 $\overline{X \setmin Y}$. |
356 A(X) \deq \lf(X) / U(X) |
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357 \] |
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358 of the previous section with a resolution |
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359 \[ |
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360 \cdots \to \bc_2(X) \to \bc_1(X) \to \bc_0(X) . |
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361 \] |
352 |
362 |
353 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. |
363 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. |
354 |
364 |
355 Define $\bc_0(X) = \lf(X)$. |
365 We of course define $\bc_0(X) = \lf(X)$. |
356 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. |
366 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. |
357 We'll omit this sort of detail in the rest of this section.) |
367 We'll omit this sort of detail in the rest of this section.) |
358 In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$. |
368 In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$. |
359 |
369 |
360 $\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$. |
370 $\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$. |
365 \item A field $r \in \cC(X \setmin B; c)$ |
375 \item A field $r \in \cC(X \setmin B; c)$ |
366 (for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). |
376 (for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). |
367 \item A local relation field $u \in U(B; c)$ |
377 \item A local relation field $u \in U(B; c)$ |
368 (same $c$ as previous bullet). |
378 (same $c$ as previous bullet). |
369 \end{itemize} |
379 \end{itemize} |
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380 (See Figure \ref{blob1diagram}.) |
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381 \begin{figure}[!ht]\begin{equation*} |
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382 \mathfig{.9}{tempkw/blob1diagram} |
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383 \end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure} |
370 In order to get the linear structure correct, we (officially) define |
384 In order to get the linear structure correct, we (officially) define |
371 \[ |
385 \[ |
372 \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . |
386 \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . |
373 \] |
387 \] |
374 The first direct sum is indexed by all blobs $B\subset X$, and the second |
388 The first direct sum is indexed by all blobs $B\subset X$, and the second |
398 \item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors. |
412 \item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors. |
399 \item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$ |
413 \item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$ |
400 (where $c_i \in \cC(\bd B_i)$). |
414 (where $c_i \in \cC(\bd B_i)$). |
401 \item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. |
415 \item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. |
402 \end{itemize} |
416 \end{itemize} |
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417 (See Figure \ref{blob2ddiagram}.) |
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418 \begin{figure}[!ht]\begin{equation*} |
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419 \mathfig{.9}{tempkw/blob2ddiagram} |
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420 \end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure} |
403 We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$; |
421 We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$; |
404 reversing the order of the blobs changes the sign. |
422 reversing the order of the blobs changes the sign. |
405 Define $\bd(B_0, B_1, u_0, u_1, r) = |
423 Define $\bd(B_0, B_1, u_0, u_1, r) = |
406 (B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$. |
424 (B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$. |
407 In other words, the boundary of a disjoint 2-blob diagram |
425 In other words, the boundary of a disjoint 2-blob diagram |
414 \item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. |
432 \item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. |
415 \item A field $r \in \cC(X \setmin B_0; c_0)$ |
433 \item A field $r \in \cC(X \setmin B_0; c_0)$ |
416 (for some $c_0 \in \cC(\bd B_0)$), which is splittable along $\bd B_1$. |
434 (for some $c_0 \in \cC(\bd B_0)$), which is splittable along $\bd B_1$. |
417 \item A local relation field $u_0 \in U(B_0; c_0)$. |
435 \item A local relation field $u_0 \in U(B_0; c_0)$. |
418 \end{itemize} |
436 \end{itemize} |
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437 (See Figure \ref{blob2ndiagram}.) |
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438 \begin{figure}[!ht]\begin{equation*} |
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439 \mathfig{.9}{tempkw/blob2ndiagram} |
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440 \end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure} |
419 Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
441 Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
420 (for some $c_1 \in \cC(B_1)$) and |
442 (for some $c_1 \in \cC(B_1)$) and |
421 $r' \in \cC(X \setmin B_1; c_1)$. |
443 $r' \in \cC(X \setmin B_1; c_1)$. |
422 Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$. |
444 Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$. |
423 Note that the requirement that |
445 Note that the requirement that |
424 local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$. |
446 local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$. |
425 As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
447 As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
426 sum of the two ways of erasing one of the blobs. |
448 sum of the two ways of erasing one of the blobs. |
427 If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$. |
449 If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$. |
428 It is again easy to check that $\bd^2 = 0$. |
450 It is again easy to check that $\bd^2 = 0$. |
429 |
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430 \nn{should draw figures for 1, 2 and $k$-blob diagrams} |
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431 |
451 |
432 As with the 1-blob diagrams, in order to get the linear structure correct it is better to define |
452 As with the 1-blob diagrams, in order to get the linear structure correct it is better to define |
433 (officially) |
453 (officially) |
434 \begin{eqnarray*} |
454 \begin{eqnarray*} |
435 \bc_2(X) & \deq & |
455 \bc_2(X) & \deq & |
467 $r$ is required to be splittable along the boundaries of all blobs, twigs or not. |
487 $r$ is required to be splittable along the boundaries of all blobs, twigs or not. |
468 \item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
488 \item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
469 where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
489 where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
470 If $B_i = B_j$ then $u_i = u_j$. |
490 If $B_i = B_j$ then $u_i = u_j$. |
471 \end{itemize} |
491 \end{itemize} |
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492 (See Figure \ref{blobkdiagram}.) |
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493 \begin{figure}[!ht]\begin{equation*} |
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494 \mathfig{.9}{tempkw/blobkdiagram} |
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495 \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
472 |
496 |
473 If two blob diagrams $D_1$ and $D_2$ |
497 If two blob diagrams $D_1$ and $D_2$ |
474 differ only by a reordering of the blobs, then we identify |
498 differ only by a reordering of the blobs, then we identify |
475 $D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. |
499 $D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. |
476 |
500 |