Weakly gauge a symmetry? 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.
 A: 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

