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A perturbative algorithm for flat F-manifolds associated with Landau-Ginzburg models

Jeehoon Park, Jaewon Yoo

Abstract

We develop a perturbative algorithm for constructing formal flat $F$-manifold structures on the cohomologies of dGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebras associated with Landau-Ginzburg models. As an application, this approach provides a perturbative construction of formal flat $F$-manifold structures on two important objects: the Jacobian algebra of a homogeneous polynomial with an isolated singularity at the origin, and the primitive cohomology of smooth projective Calabi-Yau complete intersections.

A perturbative algorithm for flat F-manifolds associated with Landau-Ginzburg models

Abstract

We develop a perturbative algorithm for constructing formal flat -manifold structures on the cohomologies of dGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebras associated with Landau-Ginzburg models. As an application, this approach provides a perturbative construction of formal flat -manifold structures on two important objects: the Jacobian algebra of a homogeneous polynomial with an isolated singularity at the origin, and the primitive cohomology of smooth projective Calabi-Yau complete intersections.
Paper Structure (6 sections, 7 theorems, 47 equations)

This paper contains 6 sections, 7 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. DGBV algebras and flat $F$-manifolds
  3. Flat $F$-manifolds and differential equations
  4. An explicit algorithm with flat coordinates
  5. A proof why the algorithm works
  6. Comparison with Li-Li-Saito's algorithm

Key Result

Proposition 2.5

If $(M, \circ, e,g)$ is a Frobenius manifold, then $(M, \circ, e, \nabla^g)$ is a flat $F$-manifold. If $(M,\circ, e, \nabla)$ is a flat $F$-manifold, then $(M, \circ, e)$ is an $F$-manifold.

Theorems & Definitions (20)

  • Example 1.1
  • Definition 2.1
  • Definition 2.2: $F$-manifolds
  • Definition 2.3: flat $F$-manifolds
  • Definition 2.4: Frobenius manifolds
  • Proposition 2.5
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Definition 4.1
  • ...and 10 more