Gromov-Witten invariants via algebraic geometry

Katz, Sheldon

doi:10.1016/0920-5632(96)00012-6

“…Then we argue inductively and extend the vanishing result for I β g to all curve classes in H 2 (X, Z). In contrast, the GW invariants do not have such a nice property due to the presence of "bubbling phenomenon" [68].…”

Section: Modular Bootstrap From the Vanishing Boundmentioning

confidence: 97%

Bootstrapping ADE M-strings

Duan

¹

,

Nahmgoong

²

2021

J. High Energ. Phys.

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We study elliptic genera of ADE-type M-strings in 6d (2,0) SCFTs from their modularity and explore the relation to topological string partition functions. We find a novel kinematical constraint that elliptic genera should follow, which determines elliptic genera at low base degrees and helps us to conjecture a vanishing bound for the refined Gopakumar-Vafa invariants of related geometries. Using this, we can bootstrap the elliptic genera to arbitrary base degree, including D/E-type theories for which explicit formulas are only partially known. We utilize our results to obtain the 6d Cardy formulas and the superconformal indices for (2,0) theories.

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“…Then we argue inductively and extend the vanishing result for I β g to all curve classes in H 2 (X, Z). In contrast, the GW invariants do not have such a nice property due to the presence of "bubbling phenomenon" [69].…”

Section: Modular Bootstrap From the Vanishing Boundmentioning

confidence: 97%

Bootstrapping ADE M-strings

Duan,

Nahmgoong

2020

Preprint

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We study elliptic genera of ADE-type M-strings in 6d (2,0) SCFTs from their modularity and explore the relation to topological string partition functions. We find a novel kinematical constraint that elliptic genera should follow, which determines elliptic genera at low base degrees and helps us to conjecture a vanishing bound for the refined Gopakumar-Vafa invariants of related geometries. Using this, we can bootstrap the elliptic genera to arbitrary base degree, including D/E-type theories for which explicit formulas are only partially known. We utilize our results to obtain the 6d Cardy formulas and the superconformal indices for (2,0) theories.

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Mathematical aspects of mirror symmetry

Morrison¹

1997

IAS/Park City Mathematics Series

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IntroductionMirror symmetry is the remarkable discovery in string theory that certain "mirror pairs" of Calabi-Yau manifolds apparently produce isomorphic physical theoriesrelated by an isomorphism which reverses the sign of a certain quantum numberwhen used as backgrounds for string propagation. The sign reversal in the isomorphism has profound effects on the geometric interpretation of the pair of physical theories. It leads to startling predictions that certain geometric invariants of one Calabi-Yau manifold (essentially the numbers of rational curves of various degrees) should be related to a completely different set of geometric invariants of the mirror partner (period integrals of holomorphic forms). The period integrals are much easier to calculate than the numbers of rational curves, so this idea has been used to make very specific predictions about numbers of curves on certain Calabi-Yau manifolds; hundreds of these predictions have now been explicitly verified. Why either the pair of manifolds, or these different invariants, should have anything to do with each other is a great mathematical mystery.The focus in these lectures will be on giving a precise mathematical description of two string-theoretic quantities which play a primary rôle in mirror symmetry: the so-called A-model and B-model correlation functions on a Calabi-Yau manifold. The first of these is related to the problem of counting rational curves while the second is related to period integrals and variations of Hodge structure. A natural mathematical consequence of mirror symmetry is the assertion that Calabi-Yau manifolds often come in pairs with the property that the A-model correlation function of the first manifold coincides with the B-model correlation function of the second, and vice versa. Our goal will be to formulate this statement as a precise mathematical conjecture. There are other recent mathematical expositions of mirror symmetry, by Voisin [15] and by Cox and Katz [3], which concentrate on other aspects of the subject; the reader may wish to consult those as well in order to obtain a complete picture.

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Gromov-Witten invariants via algebraic geometry

Cited by 3 publications

References 21 publications

Bootstrapping ADE M-strings

Bootstrapping ADE M-strings

Bootstrapping ADE M-strings

Mathematical aspects of mirror symmetry

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