Given a symplectic cohomology class of degree 1, we define the notion of an "equivariant" Lagrangian submanifold (this roughly corresponds to equivariant coherent sheaves under mirror symmetry). The Floer cohomology of equivariant Lagrangian submanifolds has a natural endomorphism, which induces an ℝs-grading by generalized eigenspaces. Taking Euler characteristics with respect to the induced grading yields a deformation of the intersection number. Dehn twists act naturally on equivariant Lagrangians. Cotangent bundles and Lefschetz fibrations give fully computable examples. A key step in computations is to impose the "dilation" condition stipulating that the BV operator applied to the symplectic cohomology class gives the identity.
Bibliographical noteFunding Information:
Acknowledgments. Roman Bezrukavnikov is the unofficial third author of this paper, and we are deeply indebted to his ideas. The first author would like to thank MSRI for its hospitality, and the NSF for partial financial support through grant DMS-0652620. The second author would like to thank the NSF for partial financial support through grant DMS-0703722 and the ISF for partial financial support through grant 1321/09. The second author was also partially supported by Marie Curie grant No. 239381.
- Fukaya category
- mirror symmetry