Constructing the LG/CY isomorphism between tt^* geometries

2022/10/30 by Huijun Fan, Fan, Huijun, Lan, Tian +2
Mathematics · Physics and Astronomy · #14F25(primary) #53D37 #53D45(secondary) #Advanced Differential Geometry Research #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2210.16747

openalex publication_date 2022/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a nondegenerate homogeneous polynomial f∈ℂ[z₀, …, z_n+1] with degree n+2, we can obtain a tt^* structure from the Landau-Ginzburg model (\Cⁿ⁺², f) and a (new) tt^* structure on the Calabi-Yau hypersurface defined by the zero locus of f in \C Pⁿ⁺¹. We can prove that the big residue map considered by Steenbrink gives an isomorphism between the two tt^* structures. We also build the correspondence for non-Calabi-Yau cases, and it turns out that only partial structure can be preserved. As an application, we show that the tt^* geometry structure of Landau-Ginzburg model on relavant deformation space uniquely determines the tt^* geometry structure on Calabi-Yau side. This explains the folklore conclusion in physical literature. This result is based on our early work \citeFLY.

Related

• LG/CY correspondence between tt^* geometries
• Calabi-Yau/Landau-Ginzburg Correspondence for Weil-Peterson Metrics and tt^* Structures
• Nonabelian Landau-Ginzburg orbifolds and Calabi-Yau/Landau-Ginzburg correspondence
• Laurent polynomial Landau-Ginzburg models for cominuscule homogeneous spaces
• Modularity of Landau-Ginzburg models