Higher dimensional knot spaces for manifolds with vector cross products

Jae Hyouk Lee; Naichung Conan Leung

doi:10.1016/j.aim.2006.12.003

Higher dimensional knot spaces for manifolds with vector cross products

Jae Hyouk Lee, Naichung Conan Leung

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

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 language	English
Pages (from-to)	140-164
Number of pages	25
Journal	Advances in Mathematics
Volume	213
Issue number	1
DOIs	https://doi.org/10.1016/j.aim.2006.12.003
State	Published - 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.

Keywords

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

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10.1016/j.aim.2006.12.003

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title = "Higher dimensional knot spaces for manifolds with vector cross products",

abstract = "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.",

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AB - 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.

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KW - Instantons

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Higher dimensional knot spaces for manifolds with vector cross products

Abstract

Bibliographical note

Keywords

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