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38
Curve counting via stable pairs in the derived category
, 2009
"... For a nonsingular projective 3fold X, we define integer invariants virtually enumerating pairs (C,D) where C ⊂ X is an embedded curve and D ⊂ C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting in ..."
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Cited by 114 (21 self)
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For a nonsingular projective 3fold X, we define integer invariants virtually enumerating pairs (C,D) where C ⊂ X is an embedded curve and D ⊂ C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting invariants are conjecturally equivalent, after universal transformations, to both the GromovWitten and DT theories of X. For CalabiYau 3folds, the latter equivalence should be viewed as a wallcrossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric CalabiYau case, a completely new form of the topological vertex is described. The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We
GromovWitten/DonaldsonThomas correspondence for toric 3folds
, 2008
"... We prove the equivariant GromovWitten theory of a nonsingular toric 3fold X with primary insertions is equivalent to the equivariant DonaldsonThomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Mariño, and Vafa of the GromovWitten theory of local CalabiYau ..."
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Cited by 61 (17 self)
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We prove the equivariant GromovWitten theory of a nonsingular toric 3fold X with primary insertions is equivalent to the equivariant DonaldsonThomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Mariño, and Vafa of the GromovWitten theory of local CalabiYau toric 3folds are proven to be correct in the full 3leg setting.
The tropical vertex
"... Abstract. Elements of the tropical vertex group are formal families of symplectomorphisms of the 2dimensional algebraic torus. We prove ordered product factorizations in the tropical vertex group are equivalent to calculations of certain genus 0 relative GromovWitten invariants of toric surfaces. ..."
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Cited by 33 (10 self)
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Abstract. Elements of the tropical vertex group are formal families of symplectomorphisms of the 2dimensional algebraic torus. We prove ordered product factorizations in the tropical vertex group are equivalent to calculations of certain genus 0 relative GromovWitten invariants of toric surfaces. The relative invariants which arise have full tangency to a toric divisor at a single unspecified point. The method uses scattering diagrams, tropical curve counts, degeneration formulas, and exact multiple cover calculations in orbifold GromovWitten theory. Contents
Phase transitions, double–scaling limit, and topological strings
, 2007
"... Topological strings on Calabi–Yau manifolds are known to undergo phase transitions at small distances. We study this issue in the case of perturbative topological strings on local Calabi–Yau threefolds given by a bundle over a twosphere. This theory can be regarded as a q–deformation of Hurwitz the ..."
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Cited by 27 (8 self)
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Topological strings on Calabi–Yau manifolds are known to undergo phase transitions at small distances. We study this issue in the case of perturbative topological strings on local Calabi–Yau threefolds given by a bundle over a twosphere. This theory can be regarded as a q–deformation of Hurwitz theory, and it has a conjectural nonperturbative description in terms of q–deformed 2d Yang–Mills theory. We solve the planar model and find a phase transition at small radius in the universality class of 2d gravity. We give strong evidence that there is a double–scaled theory at the critical point whose all genus free energy is governed by the Painlevé I equation. We compare the critical behavior of the perturbative theory to the critical behavior of its nonperturbative description, which belongs to the universality class of 2d supergravity, and we comment on possible implications for nonperturbative 2d gravity. We also give evidence for a new open/closed duality relating these Calabi–Yau backgrounds to open strings with framing.
The 3fold vertex via stable pairs
"... Abstract. The theory of stable pairs in the derived category yields an enumerative geometry of curves in 3folds. We evaluate the equivariant vertex for stable pairs on toric 3folds in terms of weighted box counting. In the toric CalabiYau case, the result simplifies to a new form of pure box coun ..."
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Cited by 20 (5 self)
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Abstract. The theory of stable pairs in the derived category yields an enumerative geometry of curves in 3folds. We evaluate the equivariant vertex for stable pairs on toric 3folds in terms of weighted box counting. In the toric CalabiYau case, the result simplifies to a new form of pure box counting. The conjectural equivalence with the DT vertex predicts remarkable identities. The equivariant vertex governs primary insertions in the theory of stable pairs for toric varieties. We consider also the descendent vertex and conjecture the complete rationality of the descendent
GromovWitten/Pairs correspondence for the quintic 3fold
, 2012
"... We use the GromovWitten/Pairs descendent correspondence for toric 3folds and degeneration arguments to establish the GW/P correspondence for several compact CalabiYau 3folds (including all CY complete intersections in products of projective spaces). A crucial aspect of the proof is the study of ..."
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Cited by 19 (6 self)
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We use the GromovWitten/Pairs descendent correspondence for toric 3folds and degeneration arguments to establish the GW/P correspondence for several compact CalabiYau 3folds (including all CY complete intersections in products of projective spaces). A crucial aspect of the proof is the study of the GW/P correspondence for descendents in relative geometries. Projective bundles over surfaces relative to a section play a special role. The GW/P correspondence for CalabiYau complete intersections provides a structure result for the GromovWitten invariants in a fixed curve class. After change of variables, the GromovWitten series is a rational function in the variable −q = e iu invariant under q ↔ q −1.