Let me state first off that in this question I am most interested in lattice gauge theories, and not necessarily with Fermion couplings. But if Fermions and continuum gauge theories can also be addressed in the same breath, I am interested in them too.
I have seen many different gauge groups for both lattice gauge theories and continuum field theories---SU(N), O(N), and Z_N being the most common---but I do not know whether there is a systematic way to generate a gauge theory for an arbitrary group, or even whether it is sensible to talk about gauge theories for arbitrary groups. So my first question is: Can any group be a gauge group for some theory? If not, why? If so, to what extent is the structure of the gauge theory determined by the group?
Related to the last point, is there a "canonical" choice of action (or Boltzmann weight) associated to a given gauge group? For example, in this paper (eqns. 5 and 7) I see that they define a Boltzmann weight for an S_N lattice gauge theory in terms of a sum of characters of the group. Is such a prescription general?
Answer
Yes, any group can be a gauge group. To each oriented edge of the lattice you assign a group element, with the opposite orientations of an edge associated to inverse group elements. The observables are conjugacy classes of products of group elements around a loop of oriented edges. The most basic such observable is the product of group elements around an elementary square (a "plaquette") of the lattice. The action is typically some function of the conjugacy class of the product around a plaquette, summed over all plaquettes in the lattice. The only "canonical" action I know of is the "delta action", where the conjugacy class of the group identity element has one value and all other conjugacy classes have another value (this works because in any group the identity element is in its own conjugacy class).
References:
Phase structure of non-Abelian lattice gauge theories
Monte Carlo study of Abelian lattice gauge theories
Phase structure of lattice gauge theories for non-abelian subgroups of SU(3)
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