From hermitian matrix model to lattice gauge theory

I consider a lattice model of a gauge field interacting with matrixvalued scalars in D dimensions. The model includes an adjustable parameter σ, which plays role of the string tension. In the limit σ = ∞ the model coincides with Kazakov-Migdal’s “induced QCD”, where Wilson loops obey a zero area law. The limit σ = 0, where Wilson loops W (C) = 1 independently of the size of the loop, corresponds to the Hermitian matrix model. For D = 2 and D = 3 I show that the model obeys the same combinatorics as the standard LGT and therefore one may expect the area law behavior. In the strong coupling expansion such a behavior is demonstrated. 1 The model. In what follows I propose a lattice gauge model interpolating between the Hermitian matrix model [1] and Kazakov-Migdal’s “induced QCD” [2]. The model is very close to standard Wilson LGT. For D = 2 and D = 3 I show that the model obeys the same combinatorics as the standard LGT and, therefore, if we generally expect area law in the latter then one may expect it in the former. Within the strong coupling expansion the area law in the model is demonstrated below. The reason for the combinatorial similarity of our model to the standard LGT is the ZN symmetry. Let us start from the Hermitian matrix model [1] in D dimensions