Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds

Maulik, Davesh; Oblomkov, Alexei; Okounkov, Andreĭ; Pandharipande, Rahul

doi:10.1007/s00222-011-0322-y

Cited by 134 publications

(253 citation statements)

References 39 publications

Supporting

Mentioning

245

Contrasting

Unclassified

Order By: Relevance

“…In [21,22] Lagrangian Floer theory is employed to study the case when the boundary condition is a fiber of the moment map. In the toric context, a mathematical approach [13,33,54,68] to construct operatively a virtual counting theory of open maps is via the use of localization [3,4,8,41,59], quantum knot invariants [47,62], and ordinary Gromov-Witten and DonaldsonThomas theory via "gluing along the boundary" [2,60,63]. Since Ruan's influential conjecture [69], an intensely studied problem in Gromov-Witten theory has been to determine the relation between GW invariants of target spaces related by a crepant birational transformation (CRC).…”

Section: 2mentioning

confidence: 99%

See 1 more Smart Citation

Crepant resolutions and open strings

Brini

Cavalieri

Ross

2017

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

View full text Add to dashboard Cite

Abstract. We formulate a Crepant Resolution Correspondence for open Gromov-Witten invariants (OCRC) of toric Lagrangian branes inside Calabi-Yau 3-orbifolds by encoding the open theories into sections of Givental's symplectic vector space. The correspondence can be phrased as the identification of these sections via a linear morphism of Givental spaces. We deduce from this a Bryan-Graber-type statement for disk invariants, which we extend to arbitrary topologies in the Hard Lefschetz case. Motivated by ideas of Iritani, Coates-CortiIritani-Tseng and Ruan, we furthermore propose 1) a general form of the morphism entering the OCRC, which arises from a geometric correspondence between equivariant K-groups, and 2) an all-genus version of the OCRC for Hard Lefschetz targets. We provide a complete proof of both statements in the case of minimal resolutions of threefold An-singularities; as a necessary step of the proof we establish the all-genus closed Crepant Resolution Conjecture with descendents in its strongest form for this class of examples. Our methods rely on a new description of the quantum D-modules underlying the equivariant Gromov-Witten theory of this family of targets.

show abstract

Section: 2mentioning

confidence: 99%

“…Restricting to the coordinate hyperplanes of the fundamental class insertions, the existence of the non-equivariant limits of U X ,Y ρ and the J-functions is guaranteed by the fact that we employ a torus action acting trivially on the canonical bundle of X and Y ; see e.g. [63].…”

Section: 12mentioning

confidence: 99%

Crepant resolutions and open strings

Brini

Cavalieri

Ross

2017

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

View full text Add to dashboard Cite

show abstract

“…In [17], it was shown that, for a toric CY, Z BPS is equal to Z top up to a factor which depends only on q. Here we derived the relation between Z BPS and Z top including the factor of M (q) χ/2 .…”

mentioning

confidence: 91%

“…The relation between the GV invariants and DT invariants was suggested and formulated in [13,16], and its physical explanation was given in [4] using the 4D/5D connection [8]. More recently, a mathematical proof of the GV/DT correspondence was given in [17] when X is a toric CY 3-fold.…”

mentioning

confidence: 99%

Wall Crossing and M-Theory

Aganagic¹,

Ooguri²,

Vafa³

et al. 2011

Publ. Res. Inst. Math. Sci.

123

View full text Add to dashboard Cite

We study BPS bound states of D0 and D2 branes on a single D6 brane wrapping a CalabiYau 3-fold X. When X has no compact 4-cycles, the BPS bound states are organized into a free field Fock space, whose generators correspond to BPS states of spinning M2 branes in M-theory compactified down to 5 dimensions by a Calabi-Yau 3-fold X. The generating function of the D-brane bound states is expressed as a reduction of the square of the topological string partition function, in all chambers of the Kähler moduli space.2010 Mathematics Subject Classification: 14N35. Keywords: string theory, bound state, Fock space, Calabi-Yau manifold. §1. IntroductionThe topological string theory gives solutions to a variety of counting problems in string theory and M-theory. From the worldsheet perspective, the A-model topological string partition function Z top generates the Gromov-Witten invariants, which count holomorphic curves in a Calabi-Yau (CY) 3-fold X. On the other hand, from the target space perspective, Z top computes the Gopakumar-Vafa (GV) invariants, which count BPS states of spinning black holes in 5 dimensions constructed from M2 branes in M-theory on X [9]. Moreover, the absolute-valueThis is a contribution to the special issue "The golden jubilee of algebraic analysis".Communicated by M. Kashiwara.

show abstract

“…In the forthcoming paper [44], the most general local Calabi-Yau toric geometry involving the 3-leg vertex is analysed for the Gromov-Witten/Donaldson-Thomas correspondence. It is likely the same path of argument will apply to stable pairs theory also.…”

Section: 3mentioning

confidence: 99%

Maps, Sheaves and K3 Surfaces

Pandharipande¹

2017

Oxford Scholarship Online

View full text Add to dashboard Cite

Abstract. The conjectural equivalence of curve counting on Calabi-Yau 3-folds via stable maps and stable pairs is discussed. By considering Calabi-Yau 3-folds with K3 fibrations, the correspondence naturally connects curve and sheaf counting on K3 surfaces. New results and conjectures (with D. Maulik) about descendent integration on K3 surfaces are announced. The recent proof of the Yau-Zaslow conjecture is surveyed .

show abstract

Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds

Cited by 134 publications

References 39 publications

Crepant resolutions and open strings

Crepant resolutions and open strings

Wall Crossing and M-Theory

Maps, Sheaves and K3 Surfaces

Contact Info

Product

Resources

About