Maximally mutable Laurent polynomials.

Tom Coates,Alexander M Kasprzyk,Ketil Tveiten,Giuseppe Pitton

doi:10.1098/rspa.2021.0584

Abstract

We introduce a class of Laurent polynomials, called maximally mutable Laurent polynomials (MMLPs), which 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 anti-canonical 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.

Highlights

A new approach to the classification of Fano manifolds was proposed [1]. This centres on mirror symmetry, in the form of a conjectural relationship
If a Laurent polynomial f is a mirror partner to a Fano manifold X, it is expected that X admits a Q-Gorenstein degeneration to the singular toric variety XP associated with the Newton polytope P = Newt(f ) of f [6,7]
In dimension two, mutations of maximally mutable Laurent polynomials (MMLPs) are in one-to-one correspondence with mutations of the underlying Newton polygons

Summary

Introduction

A new approach to the classification of Fano manifolds was proposed [1]. This centres on mirror symmetry, in the form of a conjectural relationship. If a Laurent polynomial f is a mirror partner to a Fano manifold X, it is expected that X admits a Q-Gorenstein (qG-) degeneration to the singular toric variety XP associated with the Newton polytope P = Newt(f ) of f [6,7]. It is conjectured that the Laurent polynomial g is a mirror partner to X if and only if f and g are connected via a sequence of mutations Such a connection implies in particular that the corresponding toric varieties XP and XQ, P = Newt(f ), Q = Newt(g), are related via qG-deformation [8]. That the Laurent polynomial fP obtained by assigning binomial coefficients to the monomials represented by the lattice points along the edges of P is always a mirror partner to a non-singular del Pezzo surface. P is a Fano polytope if and only if Q is a Fano polytope

The mutation graph and rigid Laurent polynomials

Maximal mutability in dimension 2

Some three-dimensional results

Remarks on higher dimension