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Minkowski polynomials and mutations

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


Mohammad Akhtar

Tom Coates

Sergey Galkin

Alexander M. Kasprzyk


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.

Journal Article Type Article
Publication Date Jan 1, 2012
Journal Symmetry, Integrability and Geometry: Methods and Applications
Electronic ISSN 1815-0659
Peer Reviewed Peer Reviewed
Volume 8
Article Number 094, pp. 707
APA6 Citation Akhtar, M., Coates, T., Galkin, S., & Kasprzyk, A. M. (2012). Minkowski polynomials and mutations. Symmetry, Integrability and Geometry: Methods and Applications, 8, doi: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
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