Maximally mutable Laurent polynomials

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

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.

Citation

Coates, T., Kasprzyk, A. M., Pitton, G., & Tveiten, K. (2021). Maximally mutable Laurent polynomials. Proceedings of the Royal Society B: Biological Sciences, 477(2254), Article 20210584. https://doi.org/10.1098/rspa.2021.0584

• Journal Article Type: Article
• Acceptance Date: Sep 16, 2021
• Online Publication Date: Oct 20, 2021
• Publication Date: Oct 27, 2021
• Deposit Date: Sep 16, 2021
• Publicly Available Date: Oct 20, 2021
• Journal: Proceedings of the Royal Society B: Biological Sciences
• Print ISSN: 0962-8452
• Electronic ISSN: 1471-2954
• Publisher: The Royal Society
• Peer Reviewed: Peer Reviewed
• Volume: 477
• Issue: 2254
• Article Number: 20210584
• DOI: https://doi.org/10.1098/rspa.2021.0584
• Keywords: General Physics and Astronomy; General Engineering; General Mathematics
• Public URL: https://nottingham-repository.worktribe.com/output/6188851
• Publisher URL: https://royalsocietypublishing.org/doi/10.1098/rspa.2021.0584