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Maximally mutable Laurent polynomials.

Tom Coates1, Alexander M Kasprzyk2, Giuseppe Pitton1

  • 1Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK.

Proceedings. Mathematical, Physical, and Engineering Sciences
|February 14, 2022
PubMed
Summary
This summary is machine-generated.

We introduce maximally mutable Laurent polynomials (MMLPs) and show rigid MMLPs correspond to Fano varieties. This work classifies MMLPs in two and three variables, linking them to del Pezzo surfaces and 3D Fano manifolds.

Keywords:
Fano varietymirror symmetrymutationquantum period

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Area of Science:

  • Algebraic Geometry
  • Mirror Symmetry
  • Combinatorial Geometry

Background:

  • Mirror symmetry posits dual pairs of Calabi-Yau manifolds.
  • Fano varieties are key objects in algebraic geometry with rich structures.
  • Developing a combinatorial approach to mirror symmetry for Fano varieties is an ongoing challenge.

Purpose of the Study:

  • Introduce maximally mutable Laurent polynomials (MMLPs) as candidates for mirrors to Fano varieties.
  • Classify rigid MMLPs in two and three variables.
  • Establish a concrete dictionary between MMLPs and Fano varieties.

Main Methods:

  • Definition and analysis of maximally mutable Laurent polynomials (MMLPs).
  • Combinatorial classification of rigid MMLPs based on mutation classes.
  • Computer-assisted enumeration for MMLPs in three variables with reflexive Newton polytopes.

Main Results:

  • Identified 10 mutation classes of rigid MMLPs in two variables, corresponding to 10 deformation classes of smooth del Pezzo surfaces.
  • Classified rigid MMLPs in three variables with reflexive Newton polytopes, corresponding to 98 deformation classes of 3D Fano manifolds.
  • Demonstrated that all known mirrors to Fano manifolds are rigid MMLPs.

Conclusions:

  • Maximally mutable Laurent polynomials provide a promising combinatorial framework for mirror symmetry to Fano varieties.
  • The correspondence holds for low-dimensional Fano varieties and suggests a general principle for higher dimensions.
  • This work offers a new perspective on constructing and understanding mirrors to Fano manifolds.