Complex Counterpart of Chern-Simons-Witten Theory and Holomorphic Linking

Details Complex Counterpart of Chern-Simons-Witten Theory and Holomorphic Linking Complex Counterpart of Chern-Simons-Witten Theory and Holomorphic Linking

2005-02-08

arxiv.org/abs/math/0502169v2

Mathematics

Algebraic Geometry

a typo was corrected

Scientific paper

In this paper we are begining to explore the complex counterpart of the Chern-Simon-Witten theory. We define the complex analogue of the Gauss linking number for complex curves embedded in a Calabi-Yau threefold using the formal path integral that leads to a rigorous mathematical expression. We give an analytic and geometric interpretation of our holomorphic linking following the parallel with the real case. We show in particular that the Green kernel that appears in the explicit integral for the Gauss linking number is replaced by the Bochner-Martinelli kernel. We also find canonical expressions of the holomorphic linking using the Grothendieck-Serre duality in local cohomology, the latter admits a generalization for an arbitrary field.

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