A topological view of Gromov–Witten theory

Maulik, Davesh; Pandharipande, Rahul

doi:10.1016/j.top.2006.06.002

Cited by 131 publications

(290 citation statements)

References 23 publications

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“…They do however fulfill nontrivial constraints coming from the geometric transitions on the type II side [14]. Furthermore, a rigorous mathematical framework for computing Gromov-Witten invariants along the fiber of certain K3-fibrations has been established in [29,30]. With these techniques, one might be able to prove some of our physical predictions for Calabi-Yau manifolds of this type.…”

Section: Resultsmentioning

confidence: 99%

Topological amplitudes in heterotic strings with Wilson lines

Weiss¹

2007

J. High Energy Phys.

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We consider d=4, N = 2 compactifications of heterotic strings with an arbitrary number of Wilson lines. In particular, we focus on known chains of candidate heterotic/type II duals. We give closed expressions for the topological amplitudes F (g) in terms of automorphic forms of SO(2 + k, 2, Z), and find agreement with the geometric data of the dual K3 fibrations wherever those are known.a marlene.weiss@cern.ch

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Section: Resultsmentioning

confidence: 99%

Topological amplitudes in heterotic strings with Wilson lines

Weiss¹

2007

J. High Energy Phys.

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“…On the Gromov-Witten side, the KatzKlemm-Vafa formula is open even in the primitive case. See [24] for a discussion. to r 0,h .…”

Section: Proposition B5mentioning

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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“…result is proven 7 in [31] by a comparison of the reduced and usual deformation theories of maps of curves to the K3 fibers of π.…”

Section: 22mentioning

confidence: 89%

“…Let [31] by the intersection of ι π (C) with Noether-Lefschetz divisors in M Λ . We briefly review the definition of the Noether-Lefschetz divisors.…”

Section: 22mentioning

confidence: 99%

Noether-Lefschetz theory and the Yau-Zaslow conjecture

Klemm

Maulik

Pandharipande

et al. 2010

J. Amer. Math. Soc.

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103

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The Yau-Zaslow conjecture predicts the genus 0 curve counts of K 3 K3 surfaces in terms of the Dedekind η \eta function. The classical intersection theory of curves in the moduli of K 3 K3 surfaces with Noether-Lefschetz divisors is related to 3-fold Gromov-Witten invariants via the K 3 K3 curve counts. Results by Borcherds and Kudla-Millson determine these classical intersections in terms of vector-valued modular forms. Proven mirror transformations can often be used to calculate the 3-fold invariants which arise. Via a detailed study of the STU model (determining special curves in the moduli of K 3 K3 surfaces), we prove the Yau-Zaslow conjecture for all curve classes on K 3 K3 surfaces. Two modular form identities are required. The first, the Klemm-Lerche-Mayr identity relating hypergeometric series to modular forms after mirror transformation, is proven here. The second, the Harvey-Moore identity, is proven by D. Zagier and presented in the paper.

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A topological view of Gromov–Witten theory

Cited by 131 publications

References 23 publications

Topological amplitudes in heterotic strings with Wilson lines

Topological amplitudes in heterotic strings with Wilson lines

Stable pairs and BPS invariants

Noether-Lefschetz theory and the Yau-Zaslow conjecture

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