Hamiltonian BF theory and projected Borromean Rings

TL;DR

This paper demonstrates how the canonical formulation of abelian BF theory in three dimensions can produce topological invariants related to curves and points, revealing a projected Milnor's link invariant that measures entanglement of Borromean rings.

Contribution

It introduces a method to derive topological invariants from abelian BF theory coupled to sources, including a novel projection of Milnor's link invariant in the plane.

Findings

Explicit calculation of a non-trivial invariant related to Borromean rings.

Identification of a projected Milnor's link invariant in 2D.

Demonstration of topological invariants from BF theory in 3D.

Abstract

It is shown that the canonical formulation of the abelian BF theory in D = 3 allows to obtain topological invariants associated to curves and points in the plane. The method consists on finding the Hamiltonian on-shell of the theory coupled to external sources with support on curves and points in the spatial plane. We explicitly calculate a non-trivial invariant that could be seen as a "projection" of the Milnor's link invariant MU(1; 2; 3), and as such, it measures the entanglement of generalized (or projected) Borromeans Rings in the Euclidean plane.