# Topological Mass Generation Mechanism

1. What is the topological mass generation mechanism?

2. And what is its relation with the Higgs mechanism?

3. Can we say that after the discovery of Higgs boson, the topological mass generation mechanism is ruled out?

Topological mass generation is a phenomenon in 2+1 dimensions discovered by: Deser, Jackiw and Templeton, in which Yang Mills fields acquire a mass, without breaking the gauge invariance, in 2+1 dimensions upon the inclusion of a Chern Simons term in the action. (Please see a more recent review by Roman Jackiw). For the Maxwell theory, the action has the form

$$S = S_{M}+S_{CS} = \int d^3x \frac{1}{4}F_{\mu\nu}F^{\mu\nu} - m \int d^3x \frac{1}{4}\epsilon^{\mu\nu\rho}A_{\mu}F_{\nu\rho}$$

The field equations describe a massive gauge field without being gauge noninvariant.

$$\partial_{\mu} F^{\mu\nu} + m \epsilon^{\mu\nu\rho}F_{\nu\rho} =0$$

The Maxwell (or more generally Yang-Mills)-Chern Simons violates parity due to the antisymmetric tensor. The Lagrangian changes by a total derivative under a gauge transformation:

$$\delta \mathcal{L} \propto \frac{m}{4}\partial{\mu}(\epsilon^{\mu\nu\rho}F_{\nu\rho})$$

Which leads to the quantization of the topological mass in the Non-Abelian case on a compact space time manifold. (By a similar mechanism as the Dirac quantization condition for the magnetic charge).

At low energy, the mass term dominates and this model with sources describes the integer Hall effect.

Topological mass generation models in 3+1 dimensions were proposed, (please see for example, the following article by Savvidy, but they are not considered attractive because they require the inclusion of additional tensor fields. They do not seem directly relevant to the standard model.

However, we can add a Higgs action in 2+1 dimensions

$$S_{H}= \int d^3x \big ( D_{\mu}{\Phi}D^{\mu}{\Phi} + \frac{\lambda}{4}(|\Phi|^2-1)^2\big )$$

(to the theory without the Maxwell term). The resulting model is called: the Chern-Simons-Higgs model. The Chern-Simons-Higgs model exhibits soliton solutions (vortices), (please see Paul and Khare) with fractional electric charge and used in the explanation of the fractional quantum hall effect.

• On re-re-re-reading your answer, I see you have stated the results which are generally known about topologically massive fields. Do you know about any reference where the mass generating mechanism is explained? As in for dummies, like myself? May 27, 2018 at 7:54
• @topologically_astounded The following short review by Marina von Steinkirch may be suitable astro.sunysb.edu/steinkirch/reviews/CHERNSIMONS.pdf A more advanced and quite extensive review is given by Richard Szabo in: arxiv.org/abs/hep-th/0405289v1 (Third chapter). May 27, 2018 at 9:07