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Comparison of Frobenius algebra structures on Calabi-Yau toric hypersurfaces

Jeehoon Park, Philsang Yoo

TL;DR

The paper establishes an isomorphism between two Frobenius algebra structures on the primitive cohomology of Calabi–Yau toric hypersurfaces: a CY construction via polyvector fields and a Landau–Ginzburg (LG) construction on a toric Jacobian subalgebra. Central to the result is the CY–LG comparison theorem (Theorem mto), which identifies the two algebraic structures on $H_{\mathrm{pr}}^{m-1}(X)$ and allows the Barannikov–Kontsevich formal Frobenius manifold on $H_{\mathrm{pr}}^{0}(\mathrm{PV}(X))$ to be transported to an LG setting, producing a formal Frobenius manifold on the finite-dimensional subspace $A(f)$ even when the critical locus of $f$ is non-compact. The approach relies on the Batyrev–Cox isomorphism, toric Carlson–Griffiths residues (via Villaflor Loyola), and Čech techniques to compare products and traces. The work thus extends Frobenius-manifold structures to non-isolated singularities within toric Calabi–Yau contexts and provides explicit machinery and examples linking CY and LG perspectives in mirror-symmetry frameworks.

Abstract

We establish an isomorphism between two Frobenius algebra structures, termed CY and LG, on the primitive cohomology of a smooth Calabi--Yau hypersurface in a simplicial Gorenstein toric Fano variety. As an application of our comparison isomorphism, we observe the existence of a Frobenius manifold structure on a finite-dimensional subalgebra of the Jacobian algebra of a homogeneous polynomial which may exhibit a non-compact singularity locus.

Comparison of Frobenius algebra structures on Calabi-Yau toric hypersurfaces

TL;DR

The paper establishes an isomorphism between two Frobenius algebra structures on the primitive cohomology of Calabi–Yau toric hypersurfaces: a CY construction via polyvector fields and a Landau–Ginzburg (LG) construction on a toric Jacobian subalgebra. Central to the result is the CY–LG comparison theorem (Theorem mto), which identifies the two algebraic structures on and allows the Barannikov–Kontsevich formal Frobenius manifold on to be transported to an LG setting, producing a formal Frobenius manifold on the finite-dimensional subspace even when the critical locus of is non-compact. The approach relies on the Batyrev–Cox isomorphism, toric Carlson–Griffiths residues (via Villaflor Loyola), and Čech techniques to compare products and traces. The work thus extends Frobenius-manifold structures to non-isolated singularities within toric Calabi–Yau contexts and provides explicit machinery and examples linking CY and LG perspectives in mirror-symmetry frameworks.

Abstract

We establish an isomorphism between two Frobenius algebra structures, termed CY and LG, on the primitive cohomology of a smooth Calabi--Yau hypersurface in a simplicial Gorenstein toric Fano variety. As an application of our comparison isomorphism, we observe the existence of a Frobenius manifold structure on a finite-dimensional subalgebra of the Jacobian algebra of a homogeneous polynomial which may exhibit a non-compact singularity locus.

Paper Structure

This paper contains 9 sections, 13 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Two Frobenius algebra structures
  3. CY Frobenius algebra
  4. LG Frobenius algebra
  5. Main theorem and application
  6. Comparison of two Frobenius algebras
  7. Proofs of main results
  8. Transport of formal Frobenius manifold structures
  9. Comparison with Isolated Singularities

Key Result

Theorem 1.1

Let $\mathbf{P}$ be an $m$-dimensional simplicial Gorenstein toric Fano variety with a toric homogeneous coordinate ring $S = \mathbb{C}[z_1, \ldots, z_r]$. Let $f \in S$ be a polynomial of degree given by the anti-canonical divisor $\beta$ of $\mathbf{P}$, defining a smooth Calabi--Yau hypersurface is an isomorphism, where $[X] \in H^2(\mathbf{P})$ denotes the cohomology class of $X = X_f(\mathbb

Theorems & Definitions (28)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Theorem 2.5: Batyrev--Cox
  • Lemma 2.6
  • proof
  • Corollary 2.7
  • ...and 18 more