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17
Curve counting via stable pairs in the derived category
"... Abstract. 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 resu ..."
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Cited by 45 (10 self)
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Abstract. 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 21 (11 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.
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 17 (5 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 10 (4 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
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 9 (4 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
An obstruction bundle relating GromovWitten invariants of curves and Kähler Surfaces
, 2008
"... In [LP] the authors defined symplectic “Local GromovWitten invariants ” associated to spin curves and showed that the GW invariants of a Kähler surface X with pg> 0 are a sum of such local GW invariants. This paper describes how the local GW invariants arise from an obstruction bundle (in the se ..."
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Cited by 6 (4 self)
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In [LP] the authors defined symplectic “Local GromovWitten invariants ” associated to spin curves and showed that the GW invariants of a Kähler surface X with pg> 0 are a sum of such local GW invariants. This paper describes how the local GW invariants arise from an obstruction bundle (in the sense of Taubes) over the space of stable maps into curves. For the analysis, we introduce the useful notion of a “quotient structure ” on the space of maps. For a compact Kähler surface X, a holomorphic 2form α is a section of the canonical bundle with divisor Dα. Several years ago, the first author observed that each such 2form α naturally induces an almost complex structure Jα that satisfies a remarkable property: Image Localization Property: The image of every Jαholomorphic maps representing a (1,1) class lies in the support of Dα. After further perturbing to a generic J near Jα, the images of all Jholomorphic maps cluster in εneighborhoods of the components of the divisor D. This implies that the GromovWitten invariant of X is a sum GWg,n(X,A) = ∑
GromovWitten theory of Anresolutions
, 2008
"... We give a complete solution for the reduced GromovWitten theory of resolved surface singularities of type An, for any genus, with arbitrary descendent insertions. We also present a partial evaluation of the Tequivariant relative GromovWitten theory of the threefold An × P 1 which, under a nondege ..."
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Cited by 4 (0 self)
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We give a complete solution for the reduced GromovWitten theory of resolved surface singularities of type An, for any genus, with arbitrary descendent insertions. We also present a partial evaluation of the Tequivariant relative GromovWitten theory of the threefold An × P 1 which, under a nondegeneracy hypothesis, yields a complete solution for the theory. The results given here allow comparison of this theory with the quantum cohomology of the Hilbert scheme of points on the An surfaces. We discuss generalizations to linear Hodge insertions and to surface resolutions of type D,E. As a corollary, we present a new derivation of the stationary GromovWitten theory of P 1.
Descendents on local curves: stationary theory
"... Abstract. The stable pairs theory of local curves in 3folds (equivariant with respect to the scaling 2torus) is studied with stationary descendent insertions. Reduction rules are found to lower descendents when higher than the degree. Factorization then yields a simple proof of rationality in the ..."
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Cited by 3 (3 self)
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Abstract. The stable pairs theory of local curves in 3folds (equivariant with respect to the scaling 2torus) is studied with stationary descendent insertions. Reduction rules are found to lower descendents when higher than the degree. Factorization then yields a simple proof of rationality in the stationary case and a proof of the functional equation related to inverting q. The method yields an effective determination of stationary descendent integrals. The series Z cap d,(d) (τd(p)) plays a special role and is calculated exactly using the stable pairs vertex and an analysis of the solution of the quantum differential equation for the Hilbert scheme of points of
QUIVERS, CURVES, AND THE TROPICAL VERTEX
"... Abstract. Elements of the tropical vertex group are formal families of symplectomorphisms of the 2dimensional algebraic torus. Commutators in the group are related to Euler characteristics of the moduli spaces of quiver representations and the GromovWitten theory of toric surfaces. After a short ..."
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Abstract. Elements of the tropical vertex group are formal families of symplectomorphisms of the 2dimensional algebraic torus. Commutators in the group are related to Euler characteristics of the moduli spaces of quiver representations and the GromovWitten theory of toric surfaces. After a short survey of the subject (based on lectures of Pandharipande at the 2009 Geometry summer school in Lisbon), we prove new results about the rays and symmetries of scattering diagrams of commutators (including previous conjectures by GrossSiebert and Kontsevich). Where possible, we present both the quiver and GromovWitten perspectives.