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

TL;DR

This paper investigates the quantum spectrum and Gamma structures in Fan-Jarvis-Ruan-Witten theory for quasi-homogeneous polynomials of general type, proving conjectures for specific cases and exploring their algebraic and analytic connections.

Contribution

It proposes Gamma conjectures for Landau-Ginzburg models of general type and proves them for Fermat polynomials and simple singularities, linking algebraic and analytic aspects.

Findings

Proved quantum spectrum conjecture for certain polynomials.

Established Gamma conjectures for Fermat and simple singularities.

Connected Gamma structures with matrix factorizations and Stokes phenomena.

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.