Browsing by Subject "Log Calabi-Yau"

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    GHK mirror symmetry, the Knutson-Tao hive cone, and Littlewood-Richardson coefficients
    (2017-08) Magee, Timothy Daniel; Keel, Seán; Neitzke, Andrew; Allcock, Daniel; Speyer, David
    I prove that the full Fock-Goncharov conjecture holds for Conf₃[superscript x] ([mathcal] A)-- the configuration space of triples of decorated flags in generic position. As a key ingredient of this proof, I exhibit a maximal green sequence for the quiver of the initial seed. I compute the Landau-Ginzburg potential W on Conf₃[superscript x] ([mathcal] A)[superscript vee] associated to the partial minimal model Conf₃[superscript x] ([mathcal] A) [subset] Conf₃ ([mathcal] A). The integral points of the associated "cone" [Xi] [does not equal] {W[superscript T] [less than or equal to] 0] [subset] Conf₃[superscript x] ([mathcal] A)[superscript vee] ([mathbb R][superscript T]) parametrize a basis for [mathcal O] (Conf₃[superscript x] ([mathcal] A) )= [big o plus] (V[subscript alpha] [o times] V[supscript beta] [o times] V[subscript gamma])[subscript G] and encode the Littlewood-Richardson coefficients c[superscript gamma][subscript alpha beta]. I exhibit a unimodular p[superscript *] map that identifies W with the potential of Goncharov-Shen on Conf₃[superscript x] ([mathcal] A) and Xi with the Knutson-Tao hive cone.
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    Tropical theta functions and log Calabi-Yau surfaces
    (2014-05) Mandel, Travis Glenn; Keel, Seán
    We describe combinatorial techniques for studying log Calabi-Yau surfaces. These can be viewed as generalizing the techniques for studying toric varieties in terms of their character and cocharacter lattices. These lattices are replaced by certain integral linear manifolds described in [GHK11], and monomials on toric varieties are replaced with the canonical theta functions defined in [GHK11] using ideas from mirror symmetry. We classify deformation classes of log Calabi-Yau surfaces in terms of the geometry of these integral linear manifolds. We then describe the tropicalizations of theta functions and use them to generalize the dual pairing between the character and cocharacter lattices. We use this to describe generalizations of dual cones, Newton and polar polytopes, Minkowski sums, and finite Fourier series expansions. We hope that these techniques will generalize to higher rank cluster varieties.