Gromov–Witten theory and Donaldson–Thomas theory, I

Maulik, Davesh; Nekrasov, Nikita; Okounkov, Andreĭ; Pandharipande, Rahul

doi:10.1112/s0010437x06002302

Cited by 456 publications

(863 citation statements)

References 31 publications

(51 reference statements)

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“…In mathematical terms, the DT invariants compute the dimensions of the moduli spaces of the ideal sheaves corresponding to curves and points on the Calabi-Yau. They are indeed conjectured [30,31] to contain equivalent information as the Gopakumar-Vafa invariants [22,21], which count the states of M2 branes with momentum, where the M2's are wrapped on holomorphic curves.…”

Section: Two Different Views On Dt Invariantsmentioning

confidence: 99%

“…roughly, those states with low D2-D0 charges), we will compute their BPS indices by means of Donaldson-Thomas theory, defined in [32,33,34]. The latter is conjectured [29,30,31,9] to provide an index (in some appropriate region in moduli space) for the topologically twisted gauge theory describing D6-D2-D0 systems. This will allow us to put together indices for the polar D4-D2-D0 states of the form ∼ Ω DT Ω DT in the spirit of the picture developed in [5].…”

Section: Introductionmentioning

confidence: 99%

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The elliptic genus from split flows and Donaldson-Thomas invariants

Collinucci

Wyder

2010

J. High Energ. Phys.

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We analyze a mixed ensemble of low charge D4-D2-D0 brane states on the quintic and show that these can be successfully enumerated using attractor flow tree techniques and Donaldson-Thomas invariants. In this low charge regime one needs to take into account worldsheet instanton corrections to the central charges, which is accomplished by making use of mirror symmetry. All the charges considered can be realized as fluxed D6-D2-D0 and D6-D2-D0 pairs which we enumerate using DT invariants. Our procedure uses the low charge counterpart of the picture developed Denef and Moore. By establishing the existence of flow trees numerically and refining the index factorization scheme, we reproduce and improve some results obtained by Gaiotto, Strominger and Yin. Our results provide appealing evidence that the strong split flow tree conjecture holds and allows to compute exact results for an important sector of the theory. Our refined scheme for computing indices might shed some light on how to improve index computations for systems with larger charges.arXiv:0810.4301v1 [hep-th]

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Section: Two Different Views On Dt Invariantsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The elliptic genus from split flows and Donaldson-Thomas invariants

Collinucci

Wyder

2010

J. High Energ. Phys.

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“…Arend Bayer [1] and Yukinoba Toda [33] have made the beautiful observation that (2.4) should be seen as a wall-crossing formula. In fact, the wall-crossing is much simpler than the wall-crossing conjectured in [26] to equate the invariants P n,β to the reduced DT invariants of [23]. For any…”

Section: 2mentioning

confidence: 99%

“…A virtual cycle is then obtained by [4,22]. The resulting invariants P n,β = [P n (X,β)] vir 1 are conjecturally equal to the reduced DT invariants of [23]. Let Z β (q) = n∈Z P n,β q n be the generating series.…”

Section: Introductionmentioning

confidence: 99%

Stable pairs and BPS invariants

Pandharipande

Thomas

2009

J. Amer. Math. Soc.

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We define the BPS invariants of Gopakumar-Vafa in the case of irreducible curve classes on Calabi-Yau 3-folds. The main tools are the theory of stable pairs in the derived category and Behrend's constructible function approach to the virtual class. We prove that for irreducible classes the stable pairs generating function satisfies the strong BPS rationality conjectures. We define the contribution of each curve to the BPS invariants. A curve $C$ only contributes to the BPS invariants in genera lying between the geometric genus and arithmetic genus of $C$. Complete formulae are derived for nonsingular and nodal curves. A discussion of primitive classes on K3 surfaces from the point of view of stable pairs is given in the Appendix via calculations of Kawai-Yoshioka. A proof of the Yau-Zaslow formula for rational curve counts is obtained. A connection is made to the Katz-Klemm-Vafa formula for BPS counts in all genera.Comment: Fixed typo pointed out by Filippo Vivian

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“…which preserves the canonical Calabi-Yau form [15]. We define the dimension of a sheaf by the dimension of its support.…”

Section: Equivariant Local Bps Invariantmentioning

confidence: 99%

Genus Zero BPS Invariants for Local ℙ1

Choi

2012

International Mathematics Research Notices

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Abstract. We study the equivariant version of the genus zero BPS invariants of the total space of a rank 2 bundle on P 1 whose determinant is O P 1 (−2). We define the equivariant genus zero BPS invariants by the residue integrals on the moduli space of stable sheaves of dimension one as proposed by Sheldon Katz [11]. We compute these invariants for low degrees by counting the torus fixed stable sheaves. The results agree with the prediction in local Gromov-Witten theory studied in [3].

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Gromov–Witten theory and Donaldson–Thomas theory, I

Cited by 456 publications

References 31 publications

The elliptic genus from split flows and Donaldson-Thomas invariants

The elliptic genus from split flows and Donaldson-Thomas invariants

Stable pairs and BPS invariants

Genus Zero BPS Invariants for Local ℙ1

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