Twisted Sectors in Calabi-Yau Type Fermat Polynomial Singularities and Automorphic Forms

TL;DR

This paper explores the relationship between twisted sectors in Calabi-Yau Fermat singularities and automorphic forms, revealing that Gromov-Witten series are also automorphic, using advanced geometric and mirror symmetry tools.

Contribution

It demonstrates that twisted sectors and Gromov-Witten series in Calabi-Yau Fermat varieties are components of automorphic forms, connecting singularity theory with automorphic representations.

Findings

Twisted sectors correspond to automorphic forms for certain groups.

Genus zero Gromov-Witten series are automorphic forms.

Uses mixed Hodge structures and mirror symmetry as main tools.

Abstract

We study one-parameter deformations of Calabi-Yau type Fermat polynomial singularities along degree-one directions. We show that twisted sectors in the vanishing cohomology are components of automorphic forms for certain triangular groups. We prove consequentially that genus zero Gromov-Witten generating series of the corresponding Fermat Calabi-Yau varieties are components of automorphic forms. The main tools we use are mixed Hodge structures for quasi-homogeneous polynomial singularities, Riemann-Hilbert correspondence, and genus zero mirror symmetry.