In my paper "$3d$ fermion-boson map with imaginary chemical potential" I have argued that for Dirac fermions coupled to an abelian Chern-Simons field at level k in three Euclidean dimensions the presence of an imaginary chemical potential is intimatelly connected to the presence of monopole charge. I believe that the same issue happens in 5d (4+1 compact dimension) Yang-Mills theories where we have the presence of instanton configurations that behave like particles, associated with a U(1) conserved current. Are there any instanton configurations for the U(1) charge of the 5d Gross Neveu model at imaginary chemical potential and finite temperature?
There seems to be some confusion: The chemical potential is an input-one chooses to work in its presence (or not) and this, just, means that one is working not in a vacuum state, but in a state of finite density. This is for any field theory, in whatever dimensions. It's not specific to the Gross-Neveu model, or to Yang-Mills.
In 5d the 4-fermion interaction is irrelevant, however, anyway, that's why studying the Gross-Neveu model in 5d doesn't seem useful (in 3d the interaction is marginal in the large N limit, that's why it's useful to study it).
The only case where it might, perhaps, make sense is in the presence of domain walls in 5d. The problem with Yang-Mills is how to realize the dimensional reduction (Cf. https://arxiv.org/abs/1010.5281 and Preprint The layered phase of anisotropic gauge theories: A model for...
for an example).
In 5d we know that Yang-Mills gauge theories carry a U(1) conserved current associated to Yang–Mills instanton configurations, which in five-dimensional spacetime appear as particles. These particles carry a charge Q(the instanton number) and are created by acting with instanton operators on the vacuum. So instanton operators can be thought of as disorder operators (similarly to ’t Hooft lines in 4d gauge theories) obtained by imposing boundary conditions on the gauge fields at the insertion point. I believe that this 5d picture is analogous to the familiar example of monopole operators in three-dimensional gauge theories. That's why i believe that since in my paper "$3d$ fermion-boson map with imaginary chemical potential" I have showed that for Dirac fermions coupled to an abelian Chern-Simons field at level k in three Euclidean dimensions the presence of an imaginary chemical potential is connected to the presence of monopole charge, maybe in the 5d Gross Neveu model at finite T and for the themporal gauge field (plays the role of an imaginary chemical potential) we have instantons-like solutions which arise as nonperurbative configurations of the temporal gauge field. Can we say this in a more restricted scenario?
Pure Yang-Mills theory (with simple gauge group) isn’t invariant under U(1) transformations, global or local, so the first statement can’t be true (unless what is meant is that, since the interaction is irrelevant in 5d, the theory reduces to a U(1)^rank_group gauge theory, that’s free.)
Instantons are particles in Euclidian signature, in any spacetime dimension.
Once more: The crucial difference between the 3d Gross-Neveu model (in the large N limit only) and the 5d Gross-Neveu model is that the former has a non-trivial fixed point, while the latter doesn’t. The existence of a non-trivial fixed point in d=3 is a peculiarity of the large N limit of the Gross-Neveu model.
One should beware of the mean field approximation.
The temporal component of the gauge field is a ghost, incidentally, therefore can’t contribute to any physical quantity. One should be careful with ``analogies''.
The conservation of the vector current of the Gross-Neveu model, coupled to Yang-Mills fields, at the level of the classical action, is, just, a consequence of Noether's theorem and holds for any gauge field (and fermionic) configurations, that solve the equations of motion, not, just instantons and in any dimension.
Finite temperature simply means that the time-like direction is compactified. In such a situation there isn't any time-like direction, in fact, consistent with the fact that there isn't any time evolution when in equilibrium at finite temperature. It's a good idea to be precise. The corresponding conservation laws in Euclidian signature then follow straightforwardly.
While the chemical potential introduces subtleties when quantum fluctuations are taken into account, there shouldn't be any issues at the level of the classical action. It will, of course, affect, in the usual way, the conservation of the vector current.
However it should be kept in mind that the reason people are interested in imaginary values of the chemical potential at all, is in order to do numerical simulations of the quantum theory and perform analytic continuation to real chemical potential-and the Gross-Neveu model above three dimensions doesn't seem to be able to define an interesting quantum theory, since the four-fermion interaction is of dimension six, in any case.
Incidentally, it’s not true that the conserved charge of the Gross-Neveu model is the generator of a U(1) group: This is true only if the gauge field, to which it couples, is abelian. If it’s Yang-Mills this isn’t the case.
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