Higher dimensional abelian Chern-Simons theories and their link invariants

TL;DR

This paper explores higher-dimensional abelian Chern-Simons theories using Deligne-Beilinson cohomology, establishing their connection to link invariants in dimensions beyond three, with novel geometric and field-theoretic computations.

Contribution

It introduces a framework for higher-dimensional abelian Chern-Simons theories and computes link invariants using Deligne-Beilinson cohomology, extending known results to dimensions $4l+3$.

Findings

Defined a natural abelian Chern-Simons action in higher dimensions.

Computed link invariants on closed manifolds and $ ext{R}^{4l+3}$.

Established quantization conditions for parameters and charges.

Abstract

The role played by Deligne-Beilinson cohomology in establishing the relation between Chern-Simons theory and link invariants in dimensions higher than three is investigated. Deligne-Beilinson cohomology classes provide a natural abelian Chern-Simons action, non trivial only in dimensions $4l+3$, whose parameter $k$ is quantized. The generalized Wilson $(2l+1)$-loops are observables of the theory and their charges are quantized. The Chern-Simons action is then used to compute invariants for links of $(2l+1)$-loops, first on closed $(4l+3)$-manifolds through a novel geometric computation, then on $\mathbb{R}^{4l+3}$ through an unconventional field theoretic computation.