Mathematical Research Letters

Volume 15, Issue 1, January 2008 pp. 15-31.

Givental's Lagrangian Cone and $S^1$-Equivariant Gromov--Witten Theory

Authors: Tom Coates
Author institution: Harvard University

Summary: 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.

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