In (for example) Semisimple Frobenius structures at higher genus (section 1.2) and Gromov-Witten invariants and quantization of quadratic Hamiltonians (section 6.8), Givental gives a conjectural formula for the higher genus Gromov-Witten potentials in terms of data coming from the Frobenius manifold of quantum cohomology, assuming that the quantum cohomology is (generically) semisimple. (The formula was proven to be correct by Teleman.) Givental does not give much explanation as to how or where he obtained these mysterious formulas. Can anybody here give some explanation or background?
Another question, more general, that I have is: Where does the quantized quadratic Hamiltonians formalism come from? How does it naturally arise? Presently (and still now, several months later after first asking this question...) everything just seems to me like a bunch of magical formulas that are pulled out a hat. I'd like to have this magic explained...
