### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 094, 707 pages      arXiv:1212.1785      https://doi.org/10.3842/SIGMA.2012.094
Contribution to the Special Issue “Mirror Symmetry and Related Topics”

### Minkowski Polynomials and Mutations

Mohammad Akhtar a, Tom Coates a, Sergey Galkin b and Alexander M. Kasprzyk a
a) Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK
b) Universität Wien, Fakultät für Mathematik, Garnisongasse 3/14, A-1090 Wien, Austria

Received June 14, 2012, in final form December 01, 2012; Published online December 08, 2012

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.

Key words: mirror symmetry; Fano manifold; Laurent polynomial; mutation; cluster transformation; Minkowski decomposition; Minkowski polynomial; Newton polytope; Ehrhart series; quasi-period collapse.

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