Givental’s Lagrangian Cone and S¹-Equivariant Gromov–Witten Theory

In the approach to Gromov-Witten theory developed by Givental, genuszero Gromov-Witten invariants of a manifold X are encoded by a Lagrangian cone in a certain infinite-dimensional symplectic vector space. We give a construction of this cone, in the spirit of S 1 -equivariant Floer theory, in terms of S 1 -equivariant Gromov-Witten theory of X P 1 . This gives a conceptual understanding of the "dilaton shift": a change-of-variables which plays an essential role in Givental's theory.