# What classifies gaugings?

Gauging a global symmetry $$G$$ introduces several free parameters to the theory. For example,

• In $$d=4$$, gauging a simple and simply-connected Lie group introduces a coupling constant and a theta term, $$(g,\theta)\in\mathbb R\times U(1)$$.

• In $$d=3$$, gauging a simple and simply-connected Lie group introduces a coupling constant and a Chern-Simons term, $$(g,k)\in\mathbb R\times\mathbb Z$$.

Gauging semi-simple Lie groups just adds more copies of these terms. Non-simply connected groups usually have extra theta terms (which may be integral instead of continuous). Similar story for discrete symmetries.

What is the mathematical object that controls these parameters? I would expect some map $$\mu\colon\mathbf{Diff}\times \mathbf{Group}\to\mathbf{Ring}$$ or something like that (with e.g., $$\mu(M_4,SU)=\mathbb R\times U(1)$$, as above). Is this map well-defined at all? If so, does it have a name/has it been studied?

• @DanYand Good point. Yes indeed, I'm thinking of renormalizable terms only. Or relevant. Or whatever makes the problem well-defined. I'm sure there must be something -- I know people have been thinking about this kind of problems. Thank you for pointing it out and for the interest! – AccidentalFourierTransform Mar 1 '19 at 1:59
• I’m not quite what you mean here. It looks like you are confusing the notion of classifying principal bundles, the geometric language of gauge theory, with the physical notion of gauging a global symmetry group as in Yang-Mills theories. – Mozibur Ullah Mar 1 '19 at 17:48
• The examples that you gave include minimal couplings and topological terms. Is there any other object or term that you consider as gauging? A second remark: gauge symmetries appear in effective low energy limits which are not renormalizable in general. I think that this is the physically more interesting case. – David Bar Moshe Mar 3 '19 at 11:21

2. There is no dependence of the coupling constant on the group. The coupling constant really just arises because when you introduce the minimal coupling by replacing the ordinary derivative $$\partial_\mu$$ by the gauge covariant derivative $$\partial_\mu + \lambda A_\mu$$ (modulo factors of $$\mathrm{i}$$), you can put any $$\lambda\in\mathbb{R}$$ in there. This doesn't depend on the gauge group, it's simply how minimal coupling works. If you couple your gauge field to your matter sector non-minimally, you may incur more or less coupling constants. The constants are a function of the coupling prescription, not of the group.
3. The $$\theta$$ and Chern-Simons terms are topological terms. The Chern-Simons terms occur in odd dimensions and are so-called secondary characteristic classes with values in the gauge group, the $$\theta$$-terms in even dimensions are the integrals of the top-dimensional Chern classes of the principal $$G$$-bundle of the theory. That the Chern class can be expressed as an invariant polynomial form in the curvature is the content of the Chern-Weil homomorphism.
The level $$k$$ of a Chern-Simons term is usually discrete and its allowed values indeed depend on the gauge group. The physicist determines the allowed values of $$k$$ by computing for which values the Chern-Simons action is actually gauge invariant, see e.g. this answer for an explicit such computation. For some mathematicians, the level is a certain element in the integral cohomology of the classifying space of the gauge group.
The $$\theta$$ in front of the Chern class, however, is just another coupling constant, unless you promote it to a dynamical field like the axion in Peccei-Quinn theory.