On a Conjecture of Givental

These brief notes record our puzzles and findings surrounding Givental’s recent conjecture which expresses higher genus Gromov-Witten invariants in terms of the genus-0 data. We limit our considerations to the case of a complex projective line, whose Gromov-Witten invariants are well-known and easy to compute. We make some simple checks supporting his conjecture. E-mail: [email protected]. E-mail: [email protected]. 1 1 Brief Summary These notes are brief sketches of our troubles and findings surrounding a work of Givental [5]. Let Fg be the generating function in the small phase space for genus-g Gromov-Witten (GW) invariants of a manifold X with a semi-simple Frobenius structure on H∗(X,Q). Then, Givental’s conjecture, whose equivariant counter-part he has proved [5], is