Holomorphic Vector Bundles, Knots and the Rozansky-Witten Invariants

TL;DR

This paper explores the connection between holomorphic vector bundles, knots, and Rozansky-Witten invariants, focusing on the conditions for hyper-holomorphic bundles over Hyper-Kaehler manifolds and their implications for link invariants.

Contribution

It introduces the concept of hyper-holomorphic bundles necessary for defining invariants of Hyper-Kaehler manifolds within Rozansky-Witten theory and explains the conditions for their use.

Findings

Hyper-holomorphic bundles are essential for invariants of Hyper-Kaehler manifolds.

The paper derives conditions for holomorphic bundles to be hyper-holomorphic.

Results are presented for non-Hyper-Kaehler cases.

Abstract

Link invariants, for 3-manifolds, are defined in the context of the Rozansky-Witten theory. To each knot in the link one associates a holomorphic bundle over a holomorphic symplectic manifold X. The invariants are evaluated for b_{1}(M) \geq 1 and X Hyper-Kaehler. To obtain invariants of Hyper-Kaehler X one finds that the holomorphic vector bundles must be hyper-holomorphic. This condition is derived and explained. Some results for X not Hyper-Kaehler are presented.