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Readers should be warned that the author is not an expert of CFT and, in fact, not even a novice in physics. What follows should be taken with a hefty pinch of salt.

Given any Riemann surface X (as a target manifold) one is able to associate to it an SCFT(X) or a super conformal field theory. The word super can just be construed as a \mathbb{Z}_2-grading of the theory. This is a simplistic version as normally one should also take into account several other parameters like a B-field and so on. Within an SCFT there is a topological sector called a TQFT (topological quantum field theory) which is insensitive to the metric on the target space. There are axiomatic descriptions of this theory due to Atiyah and Segal and in its latest version possibly due to Costello. Roughly an n-dimensional TQFT is a functor satisfying a lot of axioms from n-manifolds with labelled boundaries (incoming and outgoing) to symmetric monoidal DG (differential graded) categories with some twisting.

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