Enumerative geometry of Calabi-Yau 4-folds

Details Enumerative geometry of Calabi-Yau 4-folds Enumerative geometry of Calabi-Yau 4-folds

2007-02-07

arxiv.org/abs/math/0702189v1

Commun.Math.Phys.281:621-653,2008

Mathematics

Algebraic Geometry

44 pages

Scientific paper

10.1007/s00220-008-0490-9

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 CP5, 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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