Circle-actions, quantum cohomology, and the Fukaya category of Fano toric varieties

TL;DR

This paper establishes a mirror symmetry correspondence for a class of non-compact Fano toric manifolds, linking symplectic cohomology with the Jacobian ring of the superpotential and analyzing the Fukaya category's generation properties.

Contribution

It introduces admissible toric manifolds, proves mirror symmetry for them, and demonstrates the toric generation criterion for their Fukaya categories under certain conditions.

Findings

Symplectic cohomology is isomorphic to the Jacobian ring of the superpotential.

Wrapped Fukaya categories are split-generated by toric fibers under Morse conditions.

Extended Hamiltonian classes allow for broader application of maximum principles.

Abstract

We define a class of non-compact Fano toric manifolds, called admissible toric manifolds, for which Floer theory and quantum cohomology are defined. The class includes Fano toric negative line bundles, and it allows blow-ups along fixed point sets. We prove closed-string mirror symmetry for this class of manifolds: the Jacobian ring of the superpotential is the symplectic cohomology (not the quantum cohomology). Moreover, SH(M) is obtained from QH(M) by localizing at the toric divisors. We give explicit presentations of SH(M) and QH(M), using ideas of Batyrev, McDuff and Tolman. Assuming that the superpotential is Morse (or a milder semisimplicity assumption), we prove that the wrapped Fukaya category for this class of manifolds satisfies the toric generation criterion, i.e. is split-generated by the natural Lagrangian torus fibres of the moment map with suitable holonomies. In particular, the wrapped category is compactly generated and cohomologically finite. The proof uses a deformation argument, via a generic generation theorem and an argument about continuity of eigenspaces. We also prove that for any closed Fano toric manifold, if the superpotential is Morse (or a milder semisimplicity assumption) then the Fukaya category satisfies the toric generation criterion. The key ingredients are non-vanishing results for the open-closed string map, using tools from the paper by Ritter-Smith (we also prove a conjecture from that paper that any monotone toric negative line bundle contains a non-displaceable monotone Lagrangian torus). We also need to extend the class of Hamiltonians for which the maximum principle holds for symplectic manifolds conical at infinity, thus extending the class of Hamiltonian circle actions for which invertible elements can be constructed in SH(M).