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Enumerative Geometry of Calabi-Yau 4-Folds

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Abstract

Gromov-Witten theory is used to define an enumerative geometry of curves in Calabi-Yau 4-folds. The main technique is to find exact solutions to moving multiple cover integrals. The resulting invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds. We conjecture the 4-fold invariants to be integers and expect a sheaf theoretic explanation.

Several local Calabi-Yau 4-folds are solved exactly. Compact cases, including the sextic Calabi-Yau in \({{\mathbb{P}^5}}\), are also studied. A complete solution of the Gromov-Witten theory of the sextic is conjecturally obtained by the holomorphic anomaly equation.

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Authors and Affiliations

  1. Department of Physics, Univ. of Wisconsin, Madison, WI, 53706, USA

    A. Klemm

  2. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    R. Pandharipande

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  1. A. Klemm
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  2. R. Pandharipande
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Correspondence to R. Pandharipande.

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Communicated by N.A. Nekrasov

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Klemm, A., Pandharipande, R. Enumerative Geometry of Calabi-Yau 4-Folds. Commun. Math. Phys. 281, 621–653 (2008). https://doi.org/10.1007/s00220-008-0490-9

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