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moduli of representations (Betti moduli), moduli of Higgs bundles (Dolbeault moduli), moduli of connections (de Rham moduli space), moduli of principal bundles (Čech moduli space), moduli of polygons, moduli of geometric structures, spin networks, skein theory, A-polynomial, higher Teichmüller theory
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What is the state in the WRT TQFT associated to a handlebody?
(Making this an answer, since it will be a little long, even though it doesn't actually answer the question.)
One thing to be aware of is that there are multiple ways to talk about this same TQFT. In …