Correspondences, quilts and the wrapped Fukaya category

Details Correspondences, quilts and the wrapped Fukaya category Correspondences, quilts and the wrapped Fukaya category

2012-01-27

arxiv.org/abs/1201.5880v2

Mathematics

Symplectic Geometry

49 pages, 21 figures. Version 2 incorporates eigenvalue splittings, whilst the material on line bundles and mirror symmetry (w

Scientific paper

Let G be a monotone Lagrangian correpondence in the product BxE, where B and E are monotone symplectic manifolds with B closed and E convex at infinity. We obtain a sufficient condition for (an idempotent summand of) the wrapped Fukaya category of E to be cohomologically finite, split-generated by lifts via G of Lagrangians from B. The main theorem constructs a commutative diagram entwining quilt maps on Hochschild homology and on quantum cohomology with the open-closed string maps of B and E. En route, we establish the symplectic cohomology module structure of the open-closed string map, and the extension of Abouzaid's generation criterion from the exact to the monotone case. A sequel to this paper applies these results to negative line bundles over toric Fano varieties, and discusses the relation to homological mirror symmetry.

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