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The reason for topological stability of a kink solution in scalar field theory in $1+1$ dimensions is the fact that the finite energy scalar field cannot be continuously deformed into a vacuum.

  1. How can this be related to the existence of a conserved topological current $j^\mu=\epsilon^{\mu\nu}\partial_\nu\phi$?

Hopefully an answer to that question can also answer the following:

  1. What is the origin and the intuition for this topological current? Is there a formal way to obtain it for field theories in general?
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The topological charge which is the space integral of the zeroth component of the topological curent is responsible for the stability of the kink:

A configuration with a nonvanishing topological charge cannot evolve into a vacuum solution by means of any Hamiltonian, because Hamiltonian evolution is continuous, thus cannot change the topological charge.

Topological currents are conserved although they do not correspond to a symmetry of the theory. Thus they cannot be derived by Noether's theorem.

Since no symmetry exists, then when they are coupled to gauge fields, these gauge fields cannot acquire a mass by the Anderson-Higgs mechanism, please see the elaboration of this point in this work by Vyas and Panigrahi.

Examples of topological currents are the scalar field topological current in $1+1$ dimensions and the Baryon current in the Skyrme model:

$$ j^{\mu} = \mathrm{tr} (\epsilon^{\mu \nu \rho \sigma} U^{-1}\partial{\nu} U U^{-1}\partial{\rho} U U^{-1}\partial{\sigma} U)$$

(where $U \in SU(3)$ is the meson field)

About the origin of the topological currents:

The above theories are known to be (approximately in the second case) bosonizations of fermionic theories. In the first case this is the well known bosonization in 1+1 dimensions, and in the second case it is believed that the Skyrme model is a low energy effective theory of the standard model.

Surprisingly, in the fermionic theories before the bosonization, the topological currents are mapped into ordinary Noether currents. For example in 1+1 dimensions the topological current is just the fermionic vector current. The bosonic theory does not have this vector symmetry. The same happens in 3+1 dimensions where the Baryonic current is just a conserved Noether current in the standard formulation of the standard model.

My understanding of this point is that in the bosonic theories describe only a single sector of the full theory describing a single kink or a single Baryon in the second case. It is not a field theory of kinks and baryons. Thus the conserved charge operator commutes with all other operators in the theory and can be well replaced by a number.

In the standard model, it is known that the baryon current is anomalous due to the coupling to the electroweak bosons. In the Skyrme model the Baryon current is topological thus cannot be anomalous. However after the switching of the electroweak interaction, it needs to be gauged to become locally electroweak invariant (in the fermionic theory, the Baryon current doesn't involve derivatives, thus doesn't need to be gauged). Gauging consists of replacing the derivatives in the baryon current by their covariant derivative, the resulting physical Baryon current (which is also sometimes called the equivariant current) will not be conserved and will reflect the anomaly in the fermionic formulation. This new current consists of two pieces: The original Baryon current and the gauging terms leading to its nonconservation.

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