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In his book on topological quantum field theories Steven Simon writes that 2+1D TQFTs are objects that assign topologically invariant numbers to labeled links embedded in arbitrary 3-manifolds. They assign vector spaces $V(\Sigma)$ to 2-manifolds ($\Sigma$) and assign to 3-manifolds vectors in their boundary vector spaces $Z(\mathcal{M}) = \psi \in V(\partial\mathcal{M})$. Here the maps (Z, V) define the TQFT.

He then goes on to introduce fusion diagrams like the following:

Anyon fusion and splitting

And writes

for each fusion $N_{ab}^{c}$ we define a space $V_{ab}^{c}$ known as a fusion space and a space $V_{c}^{ab}$ known as a splitting space.

However to me the relation to the original definition of a TQFT is not clear. Do these spaces $V_{ab}^{c}$ arise as the spaces to the boundaries of certain 3-manifolds. I.e. is there a 3-manifold $M$ or a 2-manifold $\Sigma$ potentially with particles such that $V_{a}^{bc} = V(\partial M)$ or $V_{a}^{bc} = V(\Sigma)$?

Phrased differently, what is the exact relationship of the fusion and splitting diagrams and spaces to the TQFT. Given two functions $Z, V$ fully describing a TQFT on what manifolds must I evaluate these to obtain the fusion/splitting spaces. If these arise differently then what is the relationship between TQFTs and fusion diagrams and spaces.

I have searched for some literature where these things are explained however most things seem to be quite inaccessible to the average physicist without diving deeply into such things as category theory.

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$V^{ab}_c$ should be viewed as the space of states on a 3-punctured sphere, where the punctures are labeled by $a,b$ and $c$.

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  • $\begingroup$ Is the space $V_{ab}^{c}$ then the dual vector space of $V_{c}^{ab}$ which is assigned to the manifold with opposite orientation? $\endgroup$ Commented Oct 29, 2022 at 11:14
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    $\begingroup$ Strictly speaking, not always, but if the category is unitary, yes! In this case, the category is unitary as it represents a TQFT. So, yes! $\endgroup$ Commented Nov 25, 2022 at 23:16

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