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Wallcrossing structures in DonaldsonThomas invariants, integrable systems and Mirror Symmetry, arXiv:1303.3253v4 [math.AG
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Floer cohomology in the mirror of the projective plane and a binodal cubic curve, preprint arXiv:1109.3255
"... Abstract. We construct a family of Lagrangian submanifolds in the Landau– Ginzburg mirror to the projective plane equipped with a binodal cubic curve as anticanonical divisor. These objects correspond under mirror symmetry to the powers of the twisting sheaf O(1), and hence their Floer cohomology gr ..."
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Abstract. We construct a family of Lagrangian submanifolds in the Landau– Ginzburg mirror to the projective plane equipped with a binodal cubic curve as anticanonical divisor. These objects correspond under mirror symmetry to the powers of the twisting sheaf O(1), and hence their Floer cohomology groups form an algebra isomorphic to the homogeneous coordinate ring. An interesting feature is the presence of a singular torus fibration on the mirror, of which the Lagrangians are sections. This gives rise to a distinguished basis of the Floer cohomology and the homogeneous coordinate ring parametrized by fractional integral points in the singular affine structure on the base of the torus fibration. The algebra structure on the Floer cohomology is computed using the symplectic techniques of Lefschetz fibrations and the TQFT counting sections of such fibrations. We also show that our results agree with the tropical analog proposed by Abouzaid–Gross–Siebert. Extensions to a restricted class of singular affine manifolds and to mirrors of the
BIRATIONAL GEOMETRY OF CLUSTER ALGEBRAS
, 1309
"... Abstract. Wegiveageometricinterpretationofclustervarietiesintermsofblowups of toric varieties. This enables us to provide, among other results, an elementary geometric proof of the Laurent phenomenon for cluster algebras (of geometric type), extend Speyer’s example [Sp13] of upper cluster algebras w ..."
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Abstract. Wegiveageometricinterpretationofclustervarietiesintermsofblowups of toric varieties. This enables us to provide, among other results, an elementary geometric proof of the Laurent phenomenon for cluster algebras (of geometric type), extend Speyer’s example [Sp13] of upper cluster algebras which are not finitely generated, and show that the FockGoncharov dual basis conjecture is usually false. Contents
AFFINE CUBIC SURFACES AND RELATIVE SL(2)CHARACTER VARIETIES OF COMPACT SURFACES
"... Abstract. A natural family of affine cubic surfaces arises from SL(2)characters of the 4holed sphere and the 1holed torus. The ideal locus is a tritangent plane which is generic in the sense that the cubic curve at infinity consists of three lines pairwise intersecting in three double points. We ..."
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Abstract. A natural family of affine cubic surfaces arises from SL(2)characters of the 4holed sphere and the 1holed torus. The ideal locus is a tritangent plane which is generic in the sense that the cubic curve at infinity consists of three lines pairwise intersecting in three double points. We show that every affine cubic surface which is smooth at infinity and whose ideal locus is a generic tritangent plane arises as a relative SL(2)character variety of the 4holed sphere. Every such affine cubic for which all the periodic automorphisms of the tritangent plane extend to automorphisms of the cubic arises as a relative SL(2)character variety of a 1holed torus.
The KSBA compactification for the moduli space of degree two K3 pairs, arXiv
, 2012
"... Abstract. Inspired by the ideas of the minimal model program, ShepherdBarron, Kollár, and Alexeev have constructed a geometric compactification for the moduli space of surfaces of log general type. In this paper, we discuss one of the simplest examples that fits into this framework: the case of pa ..."
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Abstract. Inspired by the ideas of the minimal model program, ShepherdBarron, Kollár, and Alexeev have constructed a geometric compactification for the moduli space of surfaces of log general type. In this paper, we discuss one of the simplest examples that fits into this framework: the case of pairs (X,H) consisting of a degree two K3 surface X and an ample divisor H. Specifically, we construct and describe explicitly a geometric compactification P2 for the moduli of degree two K3 pairs. This compactification has a natural forgetful map to the Baily–Borel compactification of the moduli space F2 of degree two K3 surfaces. Using this map and the modular meaning of P2, we obtain a better understanding of the geometry of the standard compactifications of F2.
Line defects, tropicalization, and multicentered quiver quantum mechanics. arXiv: 1308.6829
, 2013
"... We study BPS line defects in N = 2 supersymmetric fourdimensional field theories. We focus on theories of “quiver type, ” those for which the BPS particle spectrum can be computed using quiver quantum mechanics. For a wide class of models, the renormalization group flow between defects defined in t ..."
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We study BPS line defects in N = 2 supersymmetric fourdimensional field theories. We focus on theories of “quiver type, ” those for which the BPS particle spectrum can be computed using quiver quantum mechanics. For a wide class of models, the renormalization group flow between defects defined in the ultraviolet and in the infrared is bijective. Using this fact, we propose a way to compute the BPS Hilbert space of a defect defined in the ultraviolet, using only infrared data. In some cases our proposal reduces to studying representations of a “framed ” quiver, with one extra node representing the defect. In general, though, it is different. As applications, we derive a formula for the discontinuities in the defect renormalization group map under variations of moduli, and show that the operator product algebra of line defects contains distinguished subalgebras with universal multiplication rules. We illustrate our results in several explicit examples.