A Topological Field Theory for the triple Milnor linking coefficient

TL;DR

This paper introduces a solvable three-dimensional topological field theory with a non-semisimple gauge group that computes Milnor's triple linking invariant through explicit contour integrals, advancing the understanding of topological invariants.

Contribution

It presents the first local topological gauge field theory that is solvable and directly related to the complex triple Milnor linking coefficient.

Findings

The theory is metric independent and gauge invariant.

The exact amplitude isolates Milnor's triple linking invariant.

Scalar fields are essential for correct path ordering.

Abstract

The subject of this work is a three-dimensional topological field theory with a non-semisimple group of gauge symmetry with observables consisting in the holonomies of connections around three closed loops. The connections are a linear combination of gauge potentials with coefficients containing a set of one-dimensional scalar fields. It is checked that these observables are both metric independent and gauge invariant. The gauge invariance is achieved by requiring non-trivial gauge transformations in the scalar field sector. This topological field theory is solvable and has only a relevant amplitude which has been computed exactly. From this amplitude it is possible to isolate a topological invariant which is Milnor's triple linking invariant. The topological invariant obtained in this way is in the form of a sum of multiple contour integrals. The contours coincide with the trajectories of the three loops mentioned before. The introduction of the one-dimensional scalar field is necessary in order to reproduce correctly the particular path ordering of the integration over the contours which is present in the triple Milnor linking coefficient. This is the first example of a local topological gauge field theory that is solvable and can be associated to a topological invariant of the complexity of the triple Milnor linking coefficient.