Minkowski Polynomials and Mutations

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.

This was formerly IPMU preprint number 12-0120. Appendices are available at http://dx.doi.org/10.3842/SIGMA.2012.094

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Cited by in corpus (38)

• Quantum Periods for 3-Dimensional Fano Manifolds
• Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians
• Mirror Symmetry and the Classification of Orbifold del Pezzo Surfaces
• Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds
• Landau-Ginzburg models -- old and new
• Laurent Inversion
• Mutations of fake weighted projective planes
• Scattering diagrams, stability conditions, and coherent sheaves on ${\mathbb{P}}^2$
• Open Gromov-Witten invariants and SYZ under local conifold transitions
• String topology with gravitational descendants, and periods of Landau-Ginzburg potentials
• On deformations of toric Fano varieties
• On toric geometry and K-stability of Fano varieties
• Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras
• On the quantum periods of del Pezzo surfaces with $\frac{1}{3}(1,1)$ singularities
• Mirror Symmetry and smoothing Gorenstein toric affine 3-folds
• Upper Bounds for Mutations of Potentials
• Cracked Polytopes and Fano Toric Complete Intersections
• Two poset polytopes are mutation-equivalent
• Minimality and mutation-equivalence of polygons
• Lagrangian torus fibration models of Fano threefolds
• Projecting Fanos in the mirror
• Toric degenerations of partial flag varieties and combinatorial mutations of matching field polytopes
• Homogeneous deformations of toric pairs
• Exotic Lagrangian tori in Grassmannians
• Algebraic and symplectic viewpoint on compactifications of two-dimensional cluster varieties of finite type
• Deformations of Dimer Models
• Laurent polynomials in Mirror Symmetry: why and how?
• Newton-Okounkov bodies of flag varieties and combinatorial mutations
• Mutations of Fake Weighted Projective Spaces
• Reconstruction of singularities on orbifold del Pezzo surfaces from their Hilbert series
• Birational Transformations on Dimer Integrable Systems
• Some examples of non-smoothable Gorenstein Fano toric threefolds
• Restricted Birkhoff polytopes and Ehrhart period collapse
• Birational Transformations and 2d (0,2) Quiver Gauge Theories beyond Toric Fano 3-folds
• Classification and Birational Equivalence of Dimer Integrable Systems for Reflexive Polygons
• Resolution of degenerate mirror families via toric morphisms
• Mutation equivalence of toric Landau-Ginzburg models
• Degenerations, transitions and quantum cohomology