"Gromov Witten potential", when does "potential" mean here? What does the whole thing mean in physics? Thanks!
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Gromov-Witten potential is a generating function for Gromov-Witten numbers, which are integrals of Gromov-Witten classes over the moduli space. According to Kontsevich and Manin this numbers define all the classes under some rather general assumptions. This potential defines a structure of formal Frobenius manifold on the cohomology. This is an example of cohomological field theory. Standard refernce is the book Frobenius manifolds, quantum cohomology and moduli spaces, written by Yuri Manin.