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

TL;DR

This paper develops a complex analogue of the Gauss linking number for curves in Calabi-Yau threefolds, using path integrals and duality theories, extending classical linking concepts into complex geometry.

Contribution

It introduces a rigorous mathematical expression for holomorphic linking, replacing the Green kernel with the Bochner-Martinelli kernel, and generalizes the framework using duality in local cohomology.

Findings

Defined the complex Gauss linking number using path integrals.

Replaced Green kernel with Bochner-Martinelli kernel in the integral.

Expressed holomorphic linking via Grothendieck-Serre duality.

Abstract

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.