Primitive forms via polyvector fields

Abstract

We develop a complex differential geometric approach to the theory of higher residues and primitive forms from the viewpoint of Kodaira-Spencer gauge theory, unifying the semi-infinite period maps for Calabi-Yau models and Landau-Ginzburg models. We give an explicit perturbative construction of primitive forms with respect to opposite filtrations and primitive elements. This leads to a concrete algorithm to compute the Taylor expansions of primitive forms as well as the description of their moduli space for all weighted homogenous cases. As an example, we present unknown perturbative expressions for the primitive form of E_12 singularity and illustrate its application to Landau-Ginzburg mirror symmetry with FJRW-theory.

61 pages, remarks added. Mistake on the analyticity in the previous version is corrected. Details on the correspondence for formal primitive forms added

References in corpus (4)

• Gromov-Witten theory of elliptic orbifold P^1 and quasi-modular forms
• Primitive Forms for Affine Cusp Polynomials
• Schrödinger equations, deformation theory and $t{t}^{\ast }$-geometry
• Mirror symmetry for exceptional unimodular singularities

Cited by in corpus (18)

• On the B-twisted topological sigma model and Calabi-Yau geometry
• Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for P^1_{a,b,c}
• On the structure of Brieskorn lattices, II
• Categorical primitive forms of Calabi-Yau $A_{\infty }$-categories with semi-simple cohomology
• Calabi-Yau structures on categories of matrix factorizations
• A Mirror Theorem between Landau-Ginzburg models
• Flat structures on Frobenius Manifolds in the case of irrelevant deformations
• A Landau-Ginzburg mirror theorem via matrix factorizations
• On the mixed Hodge structure associated to hypersurface singularities
• A twisted ${\bar{\partial }}_f$-Neumann problem and Toeplitz $n$-tuples from singularity theory
• Virtual Residue and an integral formalism
• On the L2-Hodge theory of Landau-Ginzburg models
• Cohomologies of Landau-Ginzburg models
• From Calabi-Yau dg Categories to Frobenius manifolds via Primitive Forms
• Virasoro constraints in quantum singularity theories
• Bulk-deformed potentials for toric Fano surfaces, wall-crossing and period
• From primitive form to mirror symmetry
• Primitive forms for Gepner singularities