# Browsing by Subject "Mirror symmetry"

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Item Admissible groups and cluster structures of families of log Calabi-Yau surfaces(2019-07-24) Zhou, Yan, Ph. D.; Keel, Seán; Neitzke, Andrew; Perutz, Timothy; Williams, LaurenShow more We describe the action of the admissible groups on the scattering diagrams for cluster varieties associated to positive log Calabi-Yau surfaces. By introducing the technique of folding to scattering diagrams, we give a functorial construction of locally closed sub-cluster varieties. In cases of universal families of positive log Calabi-Yau surfaces, the folding procedure allows us to study the full theta bases and scattering diagrams of degenerate subfamilies, overcoming the technical difficulty of infinite wall-crossings that appear when we restrict to degenerate loci. As a byproduct, we construct examples of cluster varieties with non-equivalent cluster structures.Show more Item Compactification of moduli spaces and mirror symmetry(2015-05) Zhu, Yuecheng; Keel, Seán; Hacking, Paul; Schedler, Travis; Perutz, Timothy; Neitzke, AndrewShow more Olsson gives modular compactifications of the moduli of toric pairs and the moduli of polarized abelian varieties A [subscript g,δ] in (Ols08). We give alternative constructions of these compactifications by using mirror symmetry. Our constructions are toroidal compactifications. The data needed for a toroidal compactification is a collection of fans. We obtain the collection of fans from the Mori fans of the minimal models of the mirror families. Moreover, we reinterpretate the compactification of A [subscript g,δ] in terms of KSBA stable pairs. We find that there is a canonical set of divisors S(K₂) associated with each cusp. Near the cusp, a polarized semiabelic scheme (X, G, L) is the canonical degeneration given by the compactification if and only if (X, G, Θ) is an object in A P [subscript g,d] for any Θ ∈ S(K₂). The two compactifications presented here are a part of a general program of applying mirror symmetry to the compactification problem of the moduli of Calabi–Yau manifolds. This thesis contains the results in (Zhu14b) and (Zhu14a).Show more Item Compactified mirror families for positive log Calabi-Yau surfaces(2020-08-14) Lai, Jonathan En; Keel, Seán; Allcock, Daniel; Ben-Zvi, David; Hacking, PaulShow more This work develops a method to canonically compactify mirror families for positive pairs (Y,D), where Y is a projective rational surface and D is an anti-canonical cycle of rational curves. The compactified families can be identified with universal families associated to generalized marked pairs, a notion that is also developed here. In order to make this identification, period integrals for the mirror family are required. To carry this out, recent techniques from mirror symmetry and tropical geometry are used.Show more Item A criterion for toric varieties(2013-08) Yao, Yuan, active 2013; Keel, SeánShow more We consider the pair of a smooth complex projective variety together with an anti-canonical simple normal crossing divisor (we call it "log Calabi- Yau"). Standard examples are toric varieties together with their toric boundaries (we call them "toric pairs"). We provide a numerical criterion for a general log Calabi-Yau to be toric by an inequality between its dimension, Picard number and the number of boundary components. The problem originates in birational geometry and our proof is constructive, motivated by mirror symmetry.Show more Item Dimer models and Hochschild cohomology(2018-08-15) Wong, Michael Andrew; Ben-Zvi, David, 1974-; Schedler, Travis; Neitzke, Andrew; Perutz, TimothyShow more Dimer models have appeared in the context of noncommutative crepant resolutions and homological mirror symmetry for punctured Riemann surfaces. For a zigzag consistent dimer embedded in a torus, we explicitly describe the Hochschild cohomology of its Jacobi algebra in terms of dimer combinatorics. We then compute the compactly supported Hochschild cohomology of the category of matrix factorizations for the Jacobi algebra with its canonical potential.Show more Item On the dual complexes of certain Log Calabi-Yau pairs(2019-12-04) Oldfield, Thomas George; Keel, Seán; Allcock, Daniel J.; Daniels, Mark L.; Starbird, Michael P.Show more I prove that when the dual complex of a divisorial log terminal log Calabi-Yau pair (X, ∆) is a simplicial complex with maximal intersection, then it is a pseudomanifold. Moreover, the affine variety corresponding to the Stanley-Reisner ring of this simplicial complex is Gorenstein with trivial dualising bundle and has semi-log canonical singularities.Show more Item Towards a self-dual geometric Langlands program(2018-06-15) Derryberry, Richard Thomas; Ben-Zvi, David, 1974-; Neitzke, Andrew; Blumberg, Andrew; Keel, Sean; Nadler, DavidShow more This thesis is comprised of two logically separate but conjecturally related parts. In the first part of the thesis I study theories of class S [32] via the formalism of relative quantum field theories [30]. From this physical formalism, and by analogy to the physical derivation of usual geometric Langlands [45, 86], I conjecture the existence of a self-dual version of the geometric Langlands program. In the second part of the thesis I study shifted Cartier duality for the moduli of Higgs bundles. The main results are: (1) a criteria for ramification of L-valued cameral covers, (2) a generalisation of the Langlands duality/mirror symmetry results for the moduli of Higgs bundles of [24, 37], and (3) the existence of a self-dual version of the moduli of Higgs bundles. This self-dual space is conjecturally the target space for a theory of class S compactified on a torus, and provides positive evidence for the self-dual geometric Langlands program.Show more Item Tropical theta functions and log Calabi-Yau surfaces(2014-05) Mandel, Travis Glenn; Keel, SeánShow more 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.Show more