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...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Published: |
National Academy of Science of Ukraine
2012
|
| Subjects: | |
| Online Access: | https://eprints.nottingham.ac.uk/30728/ |
| _version_ | 1848794045930799104 |
|---|---|
| author | Akhtar, Mohammad Coates, Tom Galkin, Sergey Kasprzyk, Alexander M. |
| author_facet | Akhtar, Mohammad Coates, Tom Galkin, Sergey Kasprzyk, Alexander M. |
| author_sort | Akhtar, Mohammad |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| 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. |
| first_indexed | 2025-11-14T19:09:57Z |
| format | Article |
| id | nottingham-30728 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:09:57Z |
| publishDate | 2012 |
| publisher | National Academy of Science of Ukraine |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-307282020-05-04T20:21:59Z https://eprints.nottingham.ac.uk/30728/ Minkowski polynomials and mutations Akhtar, Mohammad Coates, Tom Galkin, Sergey Kasprzyk, Alexander M. 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. National Academy of Science of Ukraine 2012 Article PeerReviewed Akhtar, Mohammad, Coates, Tom, Galkin, Sergey and Kasprzyk, Alexander M. (2012) Minkowski polynomials and mutations. Symmetry, Integrability and Geometry: Methods and Applications, 8 . 094, pp. 707. ISSN 1815-0659 Mirror Symmetry Fano Manifold Laurent Polynomial Mutation Cluster Transformation Minkowski Decomposition Minkowski Polynomial Newton Polytope Ehrhart Series Quasi-Period Collapse http://dx.doi.org/10.3842/SIGMA.2012.094 doi:10.3842/SIGMA.2012.094 doi:10.3842/SIGMA.2012.094 |
| spellingShingle | Mirror Symmetry Fano Manifold Laurent Polynomial Mutation Cluster Transformation Minkowski Decomposition Minkowski Polynomial Newton Polytope Ehrhart Series Quasi-Period Collapse Akhtar, Mohammad Coates, Tom Galkin, Sergey Kasprzyk, Alexander M. Minkowski polynomials and mutations |
| title | Minkowski polynomials and mutations |
| title_full | Minkowski polynomials and mutations |
| title_fullStr | Minkowski polynomials and mutations |
| title_full_unstemmed | Minkowski polynomials and mutations |
| title_short | Minkowski polynomials and mutations |
| title_sort | minkowski polynomials and mutations |
| topic | Mirror Symmetry Fano Manifold Laurent Polynomial Mutation Cluster Transformation Minkowski Decomposition Minkowski Polynomial Newton Polytope Ehrhart Series Quasi-Period Collapse |
| url | https://eprints.nottingham.ac.uk/30728/ https://eprints.nottingham.ac.uk/30728/ https://eprints.nottingham.ac.uk/30728/ |