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In his landmark paper Quantum field theory and the Jones polynomial, Witten proposed that the Jones polynomial can be obtained by the expectation value of the Wilson loop operators over links in the Chern Simons action.

He wrote that one can separate the manifold $M$ into two pieces $M_1$ and $M_2$, so that the functional integral is obtained by the inner product

$$Z(M) = \langle Z(M_1), Z(M_2) \rangle$$

I have two questions:

  1. Why is this so?

  2. Does the common boundary of $M_1$ and $M_2$ have to have trivial topology? Can it be other than $S^2$, say, can it be a torus $S^1 \times S^1$?

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