Higher dimensional knot spaces for manifolds with vector cross products

Jae Hyouk Lee, Naichung Conan Leung

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


Vector cross product structures on manifolds include symplectic, volume, G2- and Spin (7)-structures. We show that the knot spaces of such manifolds have natural symplectic structures, and relate instantons and branes in these manifolds to holomorphic disks and Lagrangian submanifolds in their knot spaces. For the complex case, the holomorphic volume form on a Calabi-Yau manifold defines a complex vector cross product structure. We show that its isotropic knot space admits a natural holomorphic symplectic structure. We also relate the Calabi-Yau geometry of the manifold to the holomorphic symplectic geometry of its isotropic knot space.

Original languageEnglish
Pages (from-to)140-164
Number of pages25
JournalAdvances in Mathematics
Issue number1
StatePublished - 1 Aug 2007

Bibliographical note

Funding Information:
The research of the first author is partially supported by NSF/DMS-0103355, Direct Grant from CUHK and RGC Earmarked Grants of Hong Kong. Authors express their gratitude to the referee for useful comments to improve the presentation of this article.


  • Branes
  • Holomorphic symplectic spaces on isotropic knot spaces
  • Instantons
  • Symplectic structures on knot spaces
  • Vector cross product


Dive into the research topics of 'Higher dimensional knot spaces for manifolds with vector cross products'. Together they form a unique fingerprint.

Cite this