Floer-rescaled residue pairing for Kodaira–Spencer map as a Frobenius algebra isomorphism

Establish that the Kodaira–Spencer map from the quantum cohomology ring QH^*(X) of a symplectic manifold X to the Jacobian ring Jac(W) of the mirror Landau–Ginzburg superpotential W becomes an isomorphism of Frobenius algebras when the residue pairing on Jac(W) is rescaled by the constant equal to the ratio of the Floer volume form to the usual volume form on the relevant Lagrangian submanifold.

Background

Closed string mirror symmetry predicts an isomorphism between the quantum cohomology of a symplectic manifold and the Jacobian ring of a mirror Landau–Ginzburg potential. Both sides naturally carry Frobenius algebra structures via the Poincaré pairing and the residue pairing, respectively.

The paper discusses the relationship between these pairings in the context of the Kodaira–Spencer map, referencing prior work which conjectured that a specific rescaling of the residue pairing—by a constant given by the ratio of the Floer volume form to the usual volume form on a Lagrangian—would make the Kodaira–Spencer map an isomorphism of Frobenius algebras. The authors verify this for elliptic orbispheres under this modification, but the conjecture is presented in general terms.