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What does it mean to "weakly gauge" a global symmetry in a gauge theory? I have seen this term used in a number of papers, but have not seen it defined.

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it is a favorite construction by model-builders. To weakly gauge a global (continuous) symmetry $G$ means to add a gauge field $A_\mu$ in the adjoint of $G$, its kinetic term $-{\rm Tr} F_{\mu\nu}F^{\mu\nu}/2$, and the interaction potential term in the Lagrangian $$L_{\rm int} = g\int {\rm Tr} (j^\mu A_\mu)d^{d-1} x$$ where $j^\mu$ is the Noether current of the global symmetry - or its supersymmetric generalizations, depending on the amount of SUSY that is being expected - for example ${\mathcal L}_{\rm int} = 2g \int d^4\theta {\mathcal J}\, {\mathcal V}$ for ${\mathcal N}=1$ SUSY. A new gauge symmetry is added in this way: it is created out of the original global symmetry. That's where the word "gauged", describing the transformation of the character of the symmetry, comes from.

The adverb "weakly" means that it is assumed that $g\ll 1$. Because of this assumption, it is hard to observe the new force mediated by the new gauge field (it is very weak), and the new gauge symmetry is, to a large extent, physically indistinguishable from the original global symmetry. As an example, imagine that the baryon number, or at least $B-L$, may be "weakly gauged" in this way; the coupling constant has to be vastly smaller than one, otherwise the theory would predict unobserved new forces between baryons and leptons.

Just a warning: it is unnatural, marginally incompatible with ideas about the "grand unification", and possibly downright inconsistent with quantum gravity, to have gauge groups with excessively tiny values of $g$ (relatively to masses of charged objects expressed as multiples of the Planck mass); for the claim about quantum gravity problems, see the "weak gravity conjecture":

http://arxiv.org/abs/hep-th/0601001

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Excellent, thanks for the answer Lubos! I would vote it up but apparently I do not have enough reputation ;). –  cortana Mar 29 '11 at 21:58
    
No prob, cortana, I do have enough of it, so please don't feel any urge to return and upvote. ;-) By the way, by accepting the answer, you gave more "credit" than by an upvote. ... Nima Arkani-Hamed et al. would be using this technique all the time and I often listened to their discussions and talks (not to speak about papers). ... I updated and extended th answer a few times, so you may try to reload and check whether it's more informative than before. –  Luboš Motl Mar 29 '11 at 22:02
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Thanks. I will add this paper to my reading list, too! –  cortana Mar 29 '11 at 22:10
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Thanks, and please try to consider the key point of the paper, in Figure 1, a map claiming that the Czech Republic is an oasis of a relatively rational thinking inside the swampland of the European Union that promotes things such as the catastrophic man-made global warming. ;-) –  Luboš Motl Mar 29 '11 at 22:18
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The Czech republic is also one of the least religious countries and one of the leading in alcohol consumption. I am seriously considering to move there. –  MBN Mar 29 '11 at 22:42
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