Minkowski polynomials and mutations

Akhtar, Mohammad; Coates, Tom; Galkin, Sergey; Kasprzyk, Alexander M.

Abstract

Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with cluster algebras. We propose a higher-dimensional analog of mutation acting on Laurent polynomials f in n variables. In particular we give a combinatorial description of mutation acting on the Newton polytope P of f, and use this to establish many basic facts about mutations. Mutations can be understood combinatorially in terms of Minkowski rearrangements of slices of P, or in terms of piecewise-linear transformations acting on the dual polytope P* (much like cluster transformations). Mutations map Fano polytopes to Fano polytopes, preserve the Ehrhart series of the dual polytope, and preserve the period of f. Finally we use our results to show that Minkowski polynomials, which are a family of Laurent polynomials that give mirror partners to many three-dimensional Fano manifolds, are connected by a sequence of mutations if and only if they have the same period.

Citation

Akhtar, M., Coates, T., Galkin, S., & Kasprzyk, A. M. (2012). Minkowski polynomials and mutations. Symmetry, Integrability and Geometry: Methods and Applications, 8, Article 094, pp. 707. https://doi.org/10.3842/SIGMA.2012.094

• Journal Article Type: Article
• Publication Date: Jan 1, 2012
• Deposit Date: Nov 12, 2015
• Publicly Available Date: Nov 12, 2015
• Journal: Symmetry, Integrability and Geometry: Methods and Applications
• Electronic ISSN: 1815-0659
• Peer Reviewed: Peer Reviewed
• Volume: 8
• Article Number: 094, pp. 707
• DOI: https://doi.org/10.3842/SIGMA.2012.094
• Keywords: Mirror Symmetry, Fano Manifold, Laurent Polynomial, Mutation Cluster Transformation, Minkowski Decomposition, Minkowski Polynomial, Newton Polytope, Ehrhart Series, Quasi-Period Collapse
• Public URL: https://nottingham-repository.worktribe.com/output/1008195
• Publisher URL: http://dx.doi.org/10.3842/SIGMA.2012.094
• Related Public URLs: http://www.emis.de/journals/SIGMA/