No abstract

Goodwillie [16] introduced a periodic cyclic homology group associated to a mixed complex. In this paper, we apply this construction to the symplectic cochain complex of a Liouville domain M and obtain two periodic symplectic cohomology theories, denoted as HP * S 1 (M ) and HP * S 1 ,loc (M ). Our main result is that both cohomology theories are invariant under Liouville isomorphisms and there is a natural isomorphism HP * S 1 ,loc (M, Q) ∼ = H * (M, Q)((u)), which can be seen as a localization theorem for HP * S 1 ,loc (M, Q).

We provide an inductive algorithm to compute the bulk-deformed potentials for toric Fano surfaces via wall-crossing techniques and a tropicalholomorphic correspondence theorem for holomorphic discs. As an application of the correspondence theorem, we also prove a big quantum period theorem for toric Fano surfaces which relates the log descendant Gromov-Witten invariants with the oscillatory integrals of the bulk-deformed potentials.

We provide an inductive algorithm to compute the bulk-deformed potentials for toric Fano surfaces via wall-crossing techniques and a tropical–holomorphic correspondence theorem for holomorphic discs. As an application of the correspondence theorem, we also prove a big quantum period theorem for toric Fano surfaces, which relates the log descendant Gromov–Witten invariants with the oscillatory integrals of the bulk-deformed potentials.