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SIGMA, 2012, Volume 8, 094, 707 pages (Mi sigma771)  

This article is cited in 15 scientific papers (total in 15 papers)

Minkowski Polynomials and Mutations

Mohammad Akhtara, Tom Coatesa, Sergey Galkinb, Alexander M. Kasprzyka

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

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.

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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ArXiv: 1212.1785
MSC: 52B20; 16S34; 14J33
Received: June 14, 2012; in final form December 1, 2012; Published online December 8, 2012

Citation: Mohammad Akhtar, Tom Coates, Sergey Galkin, Alexander M. Kasprzyk, “Minkowski Polynomials and Mutations”, SIGMA, 8 (2012), 094, 707 pp.

Citation in format AMSBIB
\by Mohammad~Akhtar, Tom~Coates, Sergey~Galkin, Alexander~M.~Kasprzyk
\paper Minkowski Polynomials and Mutations
\jour SIGMA
\yr 2012
\vol 8
\papernumber 094
\totalpages 707

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    This publication is cited in the following articles:
    1. John Alexander Cruz Morales, Sergey Galkin, “Upper Bounds for Mutations of Potentials”, SIGMA, 9 (2013), 005, 13 pp.  mathnet  crossref  mathscinet
    2. Coates T., Gonshaw S., Kasprzyk A., Nabijou N., “Mutations of Fake Weighted Projective Spaces”, Electron. J. Comb., 21:4 (2014)  mathscinet  zmath  isi
    3. Lau S.-Ch., “Open Gromov Witten Invariants and Syz Under Local Conifold Transitions”, J. Lond. Math. Soc.-Second Ser., 90:2 (2014), 413–435  crossref  mathscinet  zmath  adsnasa  isi  scopus
    4. Coates T., Kasprzyk A., Prince T., “Four-Dimensional Fano Toric Complete Intersections”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 471:2175 (2015), 20140704  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    5. Akhtar M., Coates T., Corti A., Heuberger L., Kasprzyk A., Oneto A., Petracci A., Prince T., Tveiten K., “Mirror Symmetry and the Classification of Orbifold Del Pezzo Surfaces”, Proc. Amer. Math. Soc., 144:2 (2016), 513–527  crossref  mathscinet  zmath  isi  scopus
    6. Doran Ch.F., Harder A., “Toric Degenerations and Laurent Polynomials Related to Givental's Landau-Ginzburg Models”, Can. J. Math.-J. Can. Math., 68:4 (2016), 784–815  crossref  mathscinet  zmath  isi  scopus
    7. Akhtar M.E., Kasprzyk A.M., “Mutations of Fake Weighted Projective Planes”, Proc. Edinb. Math. Soc., 59:2 (2016), 271–285  crossref  mathscinet  zmath  isi  elib  scopus
    8. Coates T., Corti A., Galkin S., Kasprzyk A., “Quantum periods for 3–dimensional Fano manifolds”, Geom. Topol., 20:1 (2016), 103–256  crossref  mathscinet  zmath  isi  elib  scopus
    9. V. V. Przyjalkowski, “Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds”, Sb. Math., 208:7 (2017), 992–1013  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. A. Kasprzyk, B. Nill, T. Prince, “Minimality and mutation-equivalence of polygons”, Forum Math. Sigma, 5 (2017), 1–48  crossref  mathscinet  isi
    11. T. Prince, “Smoothing toric fano surfaces using the gross-siebert algorithm”, Proc. London Math. Soc., 117:3 (2018), 617–660  crossref  isi  scopus
    12. Tveiten K., “Period Integrals and Mutation”, Trans. Am. Math. Soc., 370:12 (2018), 8377–8401  crossref  mathscinet  zmath  isi
    13. V. V. Przyjalkowski, “Toric Landau–Ginzburg models”, Russian Math. Surveys, 73:6 (2018), 1033–1118  mathnet  crossref  crossref  adsnasa  isi  elib
    14. V. M. Bukhshtaber, A. P. Veselov, “Topograf Konveya, $\mathrm{PGL}_2(\mathbb Z)$-dinamika i dvuznachnye gruppy”, UMN, 74:3(447) (2019), 17–62  mathnet  crossref
    15. Da Silva Jr. Genival, “On the Arithmetic of Landau-Ginzburg Model of a Certain Class of Threefolds”, Commun. Number Theory Phys., 13:1 (2019), 149–163  crossref  zmath  isi
  • Symmetry, Integrability and Geometry: Methods and Applications
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