Period integrals and mutation

TL;DR

This paper investigates the relationship between Laurent polynomials, their Newton polygons, and the associated differential operators, focusing on mutation operations and monodromy, with implications for mirror symmetry and Fano manifold classification.

Contribution

It provides a comprehensive analysis of how mutation affects the differential operators and monodromy of Laurent polynomial period integrals, especially for maximally mutable cases.

Findings

Complete description of monodromy around the origin

Mutation impacts on differential operators and monodromy

Applications to Fano manifold classification

Abstract

Let $f$ be a Laurent polynomial in two variables, whose Newton polygon strictly contains the origin and whose vertices are primitive lattice points, and let $L_f$ be the minimal-order differential operator that annihilates the period integral of $f$. We prove several results about $f$ and $L_f$ in terms of the Newton polygon of $f$ and the combinatorial operation of *mutation*, in particular we give an in principle complete description of the monodromy of $L_f$ around the origin. Special attention is given to the class of *maximally mutable* Laurent polynomials, which has applications to the conjectured classification of Fano manifolds via mirror symmetry.