Poisson structure on moduli space of CFTs

Question

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?

Answer 1

Due to the existence of symmetries, one would want to look for a BV-BRST structure (from which symplectic structures would be constructed).

There is the famous old article

• Edward Witten, On Background Independent Open-String Field Theory (arXiv:hep-th/9208027)

where it is pointed out that the space of open 2d CFTs should carry a natural BV-bracket.