Minkowski Polynomials and Mutations
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...
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National Academy of Science of Ukraine
2012-12-01
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Online Access: | http://dx.doi.org/10.3842/SIGMA.2012.094 |
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doaj-575c27f46e724f4ab17fa6bf5b9ca9732020-11-24T23:04:43ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592012-12-018094Minkowski Polynomials and MutationsMohammad AkhtarTom CoatesSergey GalkinAlexander M. KasprzykGiven 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.http://dx.doi.org/10.3842/SIGMA.2012.094mirror symmetryFano manifoldLaurent polynomialmutationcluster transformationMinkowski decompositionMinkowski polynomialNewton polytopeEhrhart seriesquasi-period collapse |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Mohammad Akhtar Tom Coates Sergey Galkin Alexander M. Kasprzyk |
spellingShingle |
Mohammad Akhtar Tom Coates Sergey Galkin Alexander M. Kasprzyk Minkowski Polynomials and Mutations Symmetry, Integrability and Geometry: Methods and Applications mirror symmetry Fano manifold Laurent polynomial mutation cluster transformation Minkowski decomposition Minkowski polynomial Newton polytope Ehrhart series quasi-period collapse |
author_facet |
Mohammad Akhtar Tom Coates Sergey Galkin Alexander M. Kasprzyk |
author_sort |
Mohammad Akhtar |
title |
Minkowski Polynomials and Mutations |
title_short |
Minkowski Polynomials and Mutations |
title_full |
Minkowski Polynomials and Mutations |
title_fullStr |
Minkowski Polynomials and Mutations |
title_full_unstemmed |
Minkowski Polynomials and Mutations |
title_sort |
minkowski polynomials and mutations |
publisher |
National Academy of Science of Ukraine |
series |
Symmetry, Integrability and Geometry: Methods and Applications |
issn |
1815-0659 |
publishDate |
2012-12-01 |
description |
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. |
topic |
mirror symmetry Fano manifold Laurent polynomial mutation cluster transformation Minkowski decomposition Minkowski polynomial Newton polytope Ehrhart series quasi-period collapse |
url |
http://dx.doi.org/10.3842/SIGMA.2012.094 |
work_keys_str_mv |
AT mohammadakhtar minkowskipolynomialsandmutations AT tomcoates minkowskipolynomialsandmutations AT sergeygalkin minkowskipolynomialsandmutations AT alexandermkasprzyk minkowskipolynomialsandmutations |
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