A perturbative construction of primitive forms from log Landau-Ginzburg mirrors of toric manifolds

Abstract

We introduce the notion of a logarithmic Landau-Ginzburg (log LG) model, which is essentially given by equipping the central degenerate fiber of the family of Landau-Ginzburg (LG) models mirror to a projective toric manifold with a natural log structure. We show that the state space of the mirror log LG model is naturally isomorphic to that of the original toric manifold. Following Li-Li-Saito, we give a perturbative construction of primitive forms by studying the deformation theory of such a log LG model, which involves both smoothing of the central degenerate fiber and unfolding of the superpotential. This yields a logarithmic Frobenius manifold structure on the base space of the universal unfolding. The primitive forms and flat coordinates we obtained are computable and closely related to the bulk-deformed Lagrangian Floer superpotential of a projective toric manifold, at least in the semi-Fano case.

arXiv comment40 pages; v3: final version to appear in Adv. Math

References in corpus (11)

• Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes
• Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields
• Lagrangian Floer theory on compact toric manifolds I
• Quantum D-modules and generalized mirror transformations
• Open Gromov-Witten invariants, mirror maps, and Seidel representations for toric manifolds
• A construction of Frobenius manifolds with logarithmic poles and applications
• Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties
• Hodge-Theoretic Mirror Symmetry for Toric Stacks
• A mirror construction for the big equivariant quantum cohomology of toric manifolds
• Open Gromov-Witten invariants and superpotentials for semi-Fano toric surfaces
• Seidel elements and potential functions of holomorphic disc counting