Communications in Mathematical Physics

, Volume 281, Issue 3, pp 621–653

Enumerative Geometry of Calabi-Yau 4-Folds


DOI: 10.1007/s00220-008-0490-9

Cite this article as:
Klemm, A. & Pandharipande, R. Commun. Math. Phys. (2008) 281: 621. doi: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 \({{\mathbb{P}^5}}\), are also studied. A complete solution of the Gromov-Witten theory of the sextic is conjecturally obtained by the holomorphic anomaly equation.

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of PhysicsUniv. of WisconsinMadisonUSA
  2. 2.Department of MathematicsPrinceton UniversityPrincetonUSA