Mathematical Research Letters
Volume 15 (2008)
Givental’s Lagrangian Cone and $S^1$-Equivariant Gromov–Witten Theory
Pages: 15 – 31
In the approach to Gromov–Witten theory developed by Givental, genus-zero 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 $X \times \PP^1$. This gives a conceptual understanding of the “dilaton shift”: a change-of-variables which plays an essential role in Givental’s theory.