A topological field theory for Milnor's triple linking number

Abstract

The subject of this work is a topological field theory with a non-semisimple gauge group and local observables that are both metric independent and gauge invariant. The observables consist of the holonomies of connections around three arbitrary loops and contain not only three-dimensional vector fields, but also a set of one-dimensional scalar fields. The gauge invariance is achieved by requiring non-trivial gauge transformations in the scalar field sector. The importance of theories of this kind in the statistical physics of linked polymer rings is discussed. The presence of fields of different dimensions is the novel feature of the proposed model, which retains many properties of the non-abelian Chern–Simons field theories and has the advantage to be exactly solvable. Its only one relevant amplitude has been computed here in closed form. From its expression it is possible to isolate the Milnor's triple linking number , a link invariant which has physical applications in the dynamics of fluids and solar magnetohydrodynamics.