String Holonomy and Extrinsic Geometry in Four-dimensional Topological Gauge Theory

TL;DR

This paper explores a four-dimensional topological gauge theory, revealing how its observables encode complex geometric intersection data and providing explicit quantum solutions with potential applications to vortex string models.

Contribution

It introduces a novel deformation of 4D abelian BF theory, linking topological invariants to intersection indices and extending the theory's quantum solutions to arbitrary 3-manifolds.

Findings

Observables compute linking numbers and intersection indices related to Euler and Chern classes.

Explicit solutions of the Schroedinger equation with wavefunctions representing holonomies.

Potential applications to effective vortex string models in physics.

Abstract

The most general gauge-invariant marginal deformation of four-dimensional abelian BF-type topological field theory is studied. It is shown that the deformed quantum field theory is topological and that its observables compute, in addition to the usual linking numbers, smooth intersection indices of immersed surfaces which are related to the Euler and Chern characteristic classes of their normal bundles in the underlying spacetime manifold. Canonical quantization of the theory coupled to non-dynamical particle and string sources is carried out in the Hamiltonian formalism and explicit solutions of the Schroedinger equation are obtained. The wavefunctions carry a one-dimensional unitary representation of the particle-string exchange holonomies and of non-topological string-string intersection holonomies given by adiabatic limits of the worldsheet Euler numbers. They also carry a multi-dimensional projective representation of the deRham complex of the underlying spatial manifold and define a generalization of the presentation of its motion group from Euclidean space to an arbitrary 3-manifold. Some potential physical applications of the topological field theory as a dual model for effective vortex strings are discussed.