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Kevin Walker
Kevin Walker
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The TQFT in question should probably be called the Crane-Yetter or Crane-Yetter-Kauffman TQFT.
Crane-Yetter-Kauffman didn't work it out as a fully extended theory, and didn't notice (so far as I can tell) the relation to Witten-Reshetikhin-Turaev theories, but they definitely were the first to write down the 4d part of the theory.

Contrary to what David Ben-Zvi wrote, I would say that theThe CYK TQFT contains all of the information of the WRT TQFT. (This disagrees with David Ben Zvi's answer, but I think the difference is due to our using different axiomatic frameworks for TQFTs, not disagreement about mathematical facts.) More specifically, $$ Z_{WRT}(X, \Gamma) = Z_{CYK}(\partial^{-1}(X); \Gamma). $$ Here $X$ is a manifold of dimension 1, 2 or 3 (not necessarily closed). $X$ is equipped with extra structure ($p_1$ structure, signature structure, null-bordism structure, ...) which makes $\partial^{-1}(X)$ sufficiently unambiguous. (For example, if $X$ is a closed 3-manifold, then the choice of $\partial^{-1}(X)$ only matters up to bordism.) The $\Gamma$ on the left hand side is a collection of "Wilson loops" or more generally a Wilson (labeled) graph. The $\Gamma$ on the right hand side is a boundary condition. (Same graph, but different interpretation.)

For more details, see Chapter 9 of these notes.

One way of looking at this is as follows. We expect, roughly, a correspondence $$ \mbox{$n$-category} \;\; \leftrightarrow \;\; \mbox{$(n{+}1)$-dimensional TQFT}. $$ The input data for for a WRT TQFT is a modular tensor category, which is a particular type of 3-category. But the WRT TQFT is a (2+1)-dimensional theory, not a (3+1)-dimensional theory, so something weird is going on here. The natural thing to do with a modular tensor category is to build the (3+1)-dimensional CYK TQFT, which is fully extended (a "0-1-2-3-4" theory) and anomaly-free. One then notices that the CYK theory is almost trivial for closed manifolds (more specifically, the dimensional reduction by $S^1$ is 2-Morita trivial), so we can derive from the CYK TQFT the (2+1)-dimensional WRT TQFT via the slogan $$ Z_{WRT}(X, \Gamma) = Z_{CYK}(\partial^{-1}(X); \Gamma). $$ But note that since CYK is merely almost trivial and not completely trivial, the WRT TQFT acquires an anomaly (i.e. manifolds need to be equipped with extra structure). Also, since it's hard to make sense of $\partial^{-1}(X)$ when $X$ is a point, the WRT theory is not fully extended; it's a 1-2-3 theory rather than a 0-1-2-3 theory.

I should also note that the input data for the CYK can be a premodular category ($S$-matrix perhaps degenerate). When the input is premodular but not modular, then the CYK TQFT is not almost trivial and we cannot construct a (2+1)-dimensional TQFT as above.

The TQFT in question should probably be called the Crane-Yetter or Crane-Yetter-Kauffman TQFT.
Crane-Yetter-Kauffman didn't work it out as a fully extended theory, and didn't notice (so far as I can tell) the relation to Witten-Reshetikhin-Turaev theories, but they definitely were the first to write down the 4d part of the theory.

Contrary to what David Ben-Zvi wrote, I would say that the CYK TQFT contains all of the information of the WRT TQFT. More specifically, $$ Z_{WRT}(X, \Gamma) = Z_{CYK}(\partial^{-1}(X); \Gamma). $$ Here $X$ is a manifold of dimension 1, 2 or 3 (not necessarily closed). $X$ is equipped with extra structure ($p_1$ structure, signature structure, null-bordism structure, ...) which makes $\partial^{-1}(X)$ sufficiently unambiguous. (For example, if $X$ is a closed 3-manifold, then the choice of $\partial^{-1}(X)$ only matters up to bordism.) The $\Gamma$ on the left hand side is a collection of "Wilson loops" or more generally a Wilson (labeled) graph. The $\Gamma$ on the right hand side is a boundary condition. (Same graph, but different interpretation.)

For more details, see Chapter 9 of these notes.

One way of looking at this is as follows. We expect, roughly, a correspondence $$ \mbox{$n$-category} \;\; \leftrightarrow \;\; \mbox{$(n{+}1)$-dimensional TQFT}. $$ The input data for for a WRT TQFT is a modular tensor category, which is a particular type of 3-category. But the WRT TQFT is a (2+1)-dimensional theory, not a (3+1)-dimensional theory, so something weird is going on here. The natural thing to do with a modular tensor category is to build the (3+1)-dimensional CYK TQFT, which is fully extended (a "0-1-2-3-4" theory) and anomaly-free. One then notices that the CYK theory is almost trivial for closed manifolds (more specifically, the dimensional reduction by $S^1$ is 2-Morita trivial), so we can derive from the CYK TQFT the (2+1)-dimensional WRT TQFT via the slogan $$ Z_{WRT}(X, \Gamma) = Z_{CYK}(\partial^{-1}(X); \Gamma). $$ But note that since CYK is merely almost trivial and not completely trivial, the WRT TQFT acquires an anomaly (i.e. manifolds need to be equipped with extra structure). Also, since it's hard to make sense of $\partial^{-1}(X)$ when $X$ is a point, the WRT theory is not fully extended; it's a 1-2-3 theory rather than a 0-1-2-3 theory.

I should also note that the input data for the CYK can be a premodular category ($S$-matrix perhaps degenerate). When the input is premodular but not modular, then the CYK TQFT is not almost trivial and we cannot construct a (2+1)-dimensional TQFT as above.

The TQFT in question should probably be called the Crane-Yetter or Crane-Yetter-Kauffman TQFT.
Crane-Yetter-Kauffman didn't work it out as a fully extended theory, and didn't notice (so far as I can tell) the relation to Witten-Reshetikhin-Turaev theories, but they definitely were the first to write down the 4d part of the theory.

The CYK TQFT contains all of the information of the WRT TQFT. (This disagrees with David Ben Zvi's answer, but I think the difference is due to our using different axiomatic frameworks for TQFTs, not disagreement about mathematical facts.) More specifically, $$ Z_{WRT}(X, \Gamma) = Z_{CYK}(\partial^{-1}(X); \Gamma). $$ Here $X$ is a manifold of dimension 1, 2 or 3 (not necessarily closed). $X$ is equipped with extra structure ($p_1$ structure, signature structure, null-bordism structure, ...) which makes $\partial^{-1}(X)$ sufficiently unambiguous. (For example, if $X$ is a closed 3-manifold, then the choice of $\partial^{-1}(X)$ only matters up to bordism.) The $\Gamma$ on the left hand side is a collection of "Wilson loops" or more generally a Wilson (labeled) graph. The $\Gamma$ on the right hand side is a boundary condition. (Same graph, but different interpretation.)

For more details, see Chapter 9 of these notes.

One way of looking at this is as follows. We expect, roughly, a correspondence $$ \mbox{$n$-category} \;\; \leftrightarrow \;\; \mbox{$(n{+}1)$-dimensional TQFT}. $$ The input data for for a WRT TQFT is a modular tensor category, which is a particular type of 3-category. But the WRT TQFT is a (2+1)-dimensional theory, not a (3+1)-dimensional theory, so something weird is going on here. The natural thing to do with a modular tensor category is to build the (3+1)-dimensional CYK TQFT, which is fully extended (a "0-1-2-3-4" theory) and anomaly-free. One then notices that the CYK theory is almost trivial for closed manifolds (more specifically, the dimensional reduction by $S^1$ is 2-Morita trivial), so we can derive from the CYK TQFT the (2+1)-dimensional WRT TQFT via the slogan $$ Z_{WRT}(X, \Gamma) = Z_{CYK}(\partial^{-1}(X); \Gamma). $$ But note that since CYK is merely almost trivial and not completely trivial, the WRT TQFT acquires an anomaly (i.e. manifolds need to be equipped with extra structure). Also, since it's hard to make sense of $\partial^{-1}(X)$ when $X$ is a point, the WRT theory is not fully extended; it's a 1-2-3 theory rather than a 0-1-2-3 theory.

I should also note that the input data for the CYK can be a premodular category ($S$-matrix perhaps degenerate). When the input is premodular but not modular, then the CYK TQFT is not almost trivial and we cannot construct a (2+1)-dimensional TQFT as above.

Source Link
Kevin Walker
Kevin Walker
  • 12.8k
  • 2
  • 42
  • 91

The TQFT in question should probably be called the Crane-Yetter or Crane-Yetter-Kauffman TQFT.
Crane-Yetter-Kauffman didn't work it out as a fully extended theory, and didn't notice (so far as I can tell) the relation to Witten-Reshetikhin-Turaev theories, but they definitely were the first to write down the 4d part of the theory.

Contrary to what David Ben-Zvi wrote, I would say that the CYK TQFT contains all of the information of the WRT TQFT. More specifically, $$ Z_{WRT}(X, \Gamma) = Z_{CYK}(\partial^{-1}(X); \Gamma). $$ Here $X$ is a manifold of dimension 1, 2 or 3 (not necessarily closed). $X$ is equipped with extra structure ($p_1$ structure, signature structure, null-bordism structure, ...) which makes $\partial^{-1}(X)$ sufficiently unambiguous. (For example, if $X$ is a closed 3-manifold, then the choice of $\partial^{-1}(X)$ only matters up to bordism.) The $\Gamma$ on the left hand side is a collection of "Wilson loops" or more generally a Wilson (labeled) graph. The $\Gamma$ on the right hand side is a boundary condition. (Same graph, but different interpretation.)

For more details, see Chapter 9 of these notes.

One way of looking at this is as follows. We expect, roughly, a correspondence $$ \mbox{$n$-category} \;\; \leftrightarrow \;\; \mbox{$(n{+}1)$-dimensional TQFT}. $$ The input data for for a WRT TQFT is a modular tensor category, which is a particular type of 3-category. But the WRT TQFT is a (2+1)-dimensional theory, not a (3+1)-dimensional theory, so something weird is going on here. The natural thing to do with a modular tensor category is to build the (3+1)-dimensional CYK TQFT, which is fully extended (a "0-1-2-3-4" theory) and anomaly-free. One then notices that the CYK theory is almost trivial for closed manifolds (more specifically, the dimensional reduction by $S^1$ is 2-Morita trivial), so we can derive from the CYK TQFT the (2+1)-dimensional WRT TQFT via the slogan $$ Z_{WRT}(X, \Gamma) = Z_{CYK}(\partial^{-1}(X); \Gamma). $$ But note that since CYK is merely almost trivial and not completely trivial, the WRT TQFT acquires an anomaly (i.e. manifolds need to be equipped with extra structure). Also, since it's hard to make sense of $\partial^{-1}(X)$ when $X$ is a point, the WRT theory is not fully extended; it's a 1-2-3 theory rather than a 0-1-2-3 theory.

I should also note that the input data for the CYK can be a premodular category ($S$-matrix perhaps degenerate). When the input is premodular but not modular, then the CYK TQFT is not almost trivial and we cannot construct a (2+1)-dimensional TQFT as above.