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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References

  1. Auroux D., Mirror symmetry and T-duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol. GGT 1 (2007), 51-91, arXiv:0706.3207.
  2. Batyrev V.V., Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994), 493-535, alg-geom/9310003.
  3. Batyrev V.V., Toric degenerations of Fano varieties and constructing mirror manifolds, in The Fano Conference, Univ. Torino, Turin, 2004, 109-122, alg-geom/9712034.
  4. Batyrev V.V., Ciocan-Fontanine I., Kim B., van Straten D., Conifold transitions and mirror symmetry for Calabi-Yau complete intersections in Grassmannians, Nuclear Phys. B 514 (1998), 640-666, alg-geom/9710022.
  5. Batyrev V.V., Ciocan-Fontanine I., Kim B., van Straten D., Mirror symmetry and toric degenerations of partial flag manifolds, Acta Math. 184 (2000), 1-39, math.AG/9803108.
  6. Batyrev V.V., Kreuzer M., Integral cohomology and mirror symmetry for Calabi-Yau 3-folds, in Mirror Symmetry. V, AMS/IP Stud. Adv. Math., Vol. 38, Amer. Math. Soc., Providence, RI, 2006, 255-270, math.AG/0505432.
  7. Beck M., Sam S.V., Woods K.M., Maximal periods of (Ehrhart) quasi-polynomials, J. Combin. Theory Ser. A 115 (2008), 517-525, math.CO/0702242.
  8. Coates T., Corti A., Galkin S., Golyshev V., Kasprzyk A.M., Fano varieties and extremal Laurent polynomials, http://www.fanosearch.net/.
  9. Coates T., Corti A., Galkin S., Golyshev V., Kasprzyk A.M., Mirror symmetry and Fano manifolds, Preprint, 2012.
  10. Cruz Morales J.A., Galkin S., Upper bounds for mutations of potentials, Preprint IPMU 12-0110, 2012.
  11. Eguchi T., Hori K., Xiong C.S., Gravitational quantum cohomology, Internat. J. Modern Phys. A 12 (1997), 1743-1782, hep-th/9605225.
  12. Fiset M.H.J., Kasprzyk A.M., A note on palindromic δ-vectors for certain rational polytopes, Electron. J. Combin. 15 (2008), Note 18, 4 pages, arXiv:0806.3942.
  13. Fomin S., Zelevinsky A., Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), 497-529, math.RT/0104151.
  14. Galkin S., Usnich A., Mutations of potentials, Preprint IPMU 10-0100, 2010.
  15. Gel'fand I.M., Kapranov M.M., Zelevinsky A.V., Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1994.
  16. Haase C., McAllister T.B., Quasi-period collapse and GLn(Z)-scissors congruence in rational polytopes, in Integer Points in Polyhedra - Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, Contemp. Math., Vol. 452, Amer. Math. Soc., Providence, RI, 2008, 115-122, arXiv:0709.4070.
  17. Hori K., Vafa C., Mirror symmetry, hep-th/0002222.
  18. Ilten N.O., Mutations of Laurent polynomials and flat families with toric fibers, SIGMA 8 (2012), 047, 7 pages, arXiv:1205.4664.
  19. Kasprzyk A.M., Canonical toric Fano threefolds, Canad. J. Math. 62 (2010), 1293-1309, arXiv:0806.2604.
  20. Kreuzer M., Skarke H., Classification of reflexive polyhedra in three dimensions, Adv. Theor. Math. Phys. 2 (1998), 853-871, hep-th/9805190.
  21. Kreuzer M., Skarke H., Complete classification of reflexive polyhedra in four dimensions, Adv. Theor. Math. Phys. 4 (2000), 1209-1230, hep-th/0002240.
  22. Lam T., Pylyavskyy P., Laurent phenomenon algebras, arXiv:1206.2611.
  23. Prokhorov Y.G., The degree of Fano threefolds with canonical Gorenstein singularities, Sb. Math. 196 (2005), 77-114, math.AG/0405347.
  24. Przyjalkowski V., On Landau-Ginzburg models for Fano varieties, Commun. Number Theory Phys. 1 (2007), 713-728, arXiv:0707.3758.
  25. Stanley R.P., Decompositions of rational convex polytopes, Ann. Discrete Math. 6 (1980), 333-342.


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