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...


1 Answer 1


OK, I guess, your first question is addressed to me. The answer is: fixed point localization. In my paper "Elliptic Gromov-Witten invariants and the mirror conjecture", a formula is found for the genus-1 (no descendants) potential of a semisimple target. It is a theorem, discovered and proved by fixed point localization when a torus acts on the target with isolated fixed points, and the GW-invariants are understood as equivariant ones. Since the answer is expressed in genus-0 data making sense for any semisimple Frobenius manifold, the conjectural extension to all such manifold is immediate. (The conjecture was proved by Dubrovin-Zhang in the sense that they showed my formula being the only candidate that would satisfy Getzler's relation.) The paper of mine you are asking about, "Semisimple Frobenius structures at higher genus", does exactly the same that the elliptic paper, but for higher genus GW-invariants, first without, and then with gravitational descendants.

After the fact, there is a more satisfying description of how that formula could have been invented. Dubrovin's connection of a semisimple Frobenius manifod allows for an asymptotical fundamental solution (which looks like the complete stationary phase asymptotics of oscillating integrals on the mirror theory). It's construction ("the $R$-matrix") is contained in the key lemma in that elliptic paper I've mentioned. Another way to interpret this solution is to say - in terms of overruled Lagrangian cones in symplectic loop spaces as the objects that describe genus-0 theory in lieu of Frobenius structures - that the overruled Lagrangian cone of a semisimple Frobenius manifold is isomorphic to the Cartesian product of several such cones corresponding to the one-point target space, and moreover, the isomorphism is accomplished by transformations from the twisted loop group: $$ L = M (L_{pt}\times \cdots \times L_{pt}).$$ The "mysterious" conjectural higher genus formula simply says that the same relation persists for the total descendant potentials of higher genus theory:$$D \sim \hat{M} (D_{pt}\otimes\cdots\otimes D_{pt}),$$ where the elements of the loop group are quantized, and the equality is replaced by proportionality up to a non-zero "central constant".


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