The moduli space of CFTs with central charge 26 forms the classical phase space of bosonic string theory, in some sense. Similarily the moduli space of SCFTs with central charge 10 forms the classical phase space of type II superstring theory

We might expect this should make these moduli spaces symplectic, or at least Poisson (super)manifolds. Is there such structure on them?

More generally, the only geometric structure on CFT moduli spaces I encountered is the Zamolodchikov metric. What other structures do they have?

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