Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum spectrum and Gamma structures for quasi-homogeneous polynomials of general type

Yefeng Shen, Ming Zhang

TL;DR

This work develops Gamma-structural and asymptotic tools for Fan-Jarvis-Ruan-Witten theory of general-type Landau–Ginzburg pairs $(W,⟨J⟩)$. It introduces a quantum-spectrum framework based on the operator $\frac{\nu}{d}\tau'\star_{\tau}$ and proves the spectrum conjecture for mirror ADE and Fermat-type polynomials, while establishing a detailed link between algebraic (matrix factorizations and Orlov decompositions) and analytic (Stokes, Meijer $G$, Barnes hypergeometric) structures. The paper constructs Gamma maps and Gamma classes that connect the category of $G$-equivariant matrix factorizations to FJRW theory, and proves Gamma conjectures in several general-type cases, notably Fermat polynomials. By relating Orlov’s semiorthogonal decompositions to Stokes data, it provides a unified picture of how categorical and analytic invariants encode the massive vacuum structure in LG models, with potential implications for CY/Fano correspondences and birational invariants in LG settings.

Abstract

Let $W$ be a quasi-homogeneous polynomial of general type and $<J>$ be the cyclic symmetry group of $W$ generated by the exponential grading element $J$. We study the quantum spectrum and asymptotic behavior in Fan-Jarvis-Ruan-Witten theory of the Landau-Ginzburg pair $(W, <J>)$. Inspired by Galkin-Golyshev-Iritani's Gamma conjectures for quantum cohomology of Fano manifolds, we propose Gamma conjectures for Fan-Jarvis-Ruan-Witten theory of general type. We prove the quantum spectrum conjecture and the Gamma conjectures for Fermat homogeneous polynomials and the mirror simple singularities. The Gamma structures in Fan-Jarvis-Ruan-Witten theory also provide a bridge from the category of matrix factorizations of the Landau-Ginzburg pair (the algebraic aspect) to its analytic aspect. We will explain the relationship among the Gamma structures, Orlov's semiorthogonal decompositions, and the Stokes phenomenon.

Quantum spectrum and Gamma structures for quasi-homogeneous polynomials of general type

TL;DR

This work develops Gamma-structural and asymptotic tools for Fan-Jarvis-Ruan-Witten theory of general-type Landau–Ginzburg pairs . It introduces a quantum-spectrum framework based on the operator and proves the spectrum conjecture for mirror ADE and Fermat-type polynomials, while establishing a detailed link between algebraic (matrix factorizations and Orlov decompositions) and analytic (Stokes, Meijer , Barnes hypergeometric) structures. The paper constructs Gamma maps and Gamma classes that connect the category of -equivariant matrix factorizations to FJRW theory, and proves Gamma conjectures in several general-type cases, notably Fermat polynomials. By relating Orlov’s semiorthogonal decompositions to Stokes data, it provides a unified picture of how categorical and analytic invariants encode the massive vacuum structure in LG models, with potential implications for CY/Fano correspondences and birational invariants in LG settings.

Abstract

Let be a quasi-homogeneous polynomial of general type and be the cyclic symmetry group of generated by the exponential grading element . We study the quantum spectrum and asymptotic behavior in Fan-Jarvis-Ruan-Witten theory of the Landau-Ginzburg pair . Inspired by Galkin-Golyshev-Iritani's Gamma conjectures for quantum cohomology of Fano manifolds, we propose Gamma conjectures for Fan-Jarvis-Ruan-Witten theory of general type. We prove the quantum spectrum conjecture and the Gamma conjectures for Fermat homogeneous polynomials and the mirror simple singularities. The Gamma structures in Fan-Jarvis-Ruan-Witten theory also provide a bridge from the category of matrix factorizations of the Landau-Ginzburg pair (the algebraic aspect) to its analytic aspect. We will explain the relationship among the Gamma structures, Orlov's semiorthogonal decompositions, and the Stokes phenomenon.
Paper Structure (104 sections, 75 theorems, 366 equations, 1 table)

This paper contains 104 sections, 75 theorems, 366 equations, 1 table.

Table of Contents

  1. Introduction
  2. FJRW theory for admissible Landau-Ginzburg pairs
  3. A Landau-Ginzburg pair
  4. Fan-Jarvis-Ruan-Witten theory
  5. A quantum product
  6. Quantum spectrum for FJRW theory of general type
  7. The element $\tau(t)$
  8. Quantum spectrum conjecture for $(W, \langle J\rangle)$ of general type
  9. Main results on quantum spectrum conjecture \ref{['conjecture-C']}
  10. Asymptotic expansion in FJRW theory
  11. A Dubrovin connection in FJRW theory
  12. Weak asymptotic classes
  13. A mirror conjecture
  14. Calculation of weak asymptotic classes via a Barnes' formula
  15. Strong asymptotic classes
  16. ...and 89 more sections

Key Result

Theorem 1.6

Quantum spectrum conjecture conjecture-C holds for the following singularities:

Theorems & Definitions (142)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Conjecture 1.4: Quantum spectrum conjecture for an admissible LG pair $(W, \langle J\rangle)$ of general type
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 1.8
  • Conjecture 1.9
  • Proposition 1.10: Proposition \ref{['theorem-asymptotic-classes']}
  • ...and 132 more