Maximally Mutable Laurent Polynomials

TL;DR

This paper introduces maximally mutable Laurent polynomials (MMLPs) as mirror candidates for Fano varieties, classifies rigid cases in two and three variables, and establishes their correspondence with known Fano classes.

Contribution

It defines MMLPs, classifies rigid cases in low dimensions, and links these to Fano varieties via mirror symmetry, providing a new systematic approach.

Findings

Exactly 10 mutation classes of rigid MMLPs in two variables.

Correspondence between rigid MMLPs and deformation classes of Fano surfaces.

Classification of rigid MMLPs in three variables with reflexive Newton polytope.

Abstract

We introduce a class of Laurent polynomials, called maximally mutable Laurent polynomials (MMLPs), that we believe correspond under mirror symmetry to Fano varieties. A subclass of these, called rigid, are expected to correspond to Fano varieties with terminal locally toric singularities. We prove that there are exactly 10 mutation classes of rigid MMLPs in two variables; under mirror symmetry these correspond one-to-one with the 10 deformation classes of smooth del~Pezzo surfaces. Furthermore we give a computer-assisted classification of rigid MMLPs in three variables with reflexive Newton polytope; under mirror symmetry these correspond one-to-one with the 98 deformation classes of three-dimensional Fano manifolds with very ample anticanonical bundle. We compare our proposal to previous approaches to constructing mirrors to Fano varieties, and explain why mirror symmetry in higher dimensions necessarily involves varieties with terminal singularities. Every known mirror to a Fano manifold, of any dimension, is a rigid MMLP.