Abstract. Various approaches to construction of dual formulations of non-abelian lattice gauge theories are reviewed. In the case of U(N) LGT we use a theory of the Weingarten functions to construct a dual formulation. In particular, the dual representations are constructed 1) for pure gauge models in all dimensions, 2) in the strong coupling limit for the models with arbitrary number of flavours and 3) for two-dimensional U(N) QCD with staggered fermions. Applications related to the finite temperature/density QCD are discussed. Duals of lattice spin and gauge modelsDual representations proved to be a very useful concept in the context of abelian spin and gauge models. The application of dual representations ranges from the determination of the critical points in the self-dual abelian models to the proof of the confinement in the three-dimensional U(1) lattice gauge theory (LGT) [1,2] and the numerical study of the U(1) LGT both at zero [3] and at finite temperature [4]. For abelian models the dual transformations are well-defined and described in many reviews and text books [5]. The status of the dual representations of non-abelian models is very different. During decades several different approaches have been attempted to construct dual representations.• Dual representations based on the plaquette formulation [6][7][8]. Dual variables are introduced as variables conjugate to local Bianchi identities [6,9,10]. The dual model appears to be non-local due to the presence of connectors in the Bianchi identities for gauge models. An analogue of the plaquette formulation for the principal chiral model is so-called link representation [11,12]. In this case one can construct a local dual theory for all U(N) and S U(N) principal chiral models [13].• Dual representations based on 1) the character expansion of the Boltzmann weight and 2) the integration over link variables using Clebsch-Gordan expansion [14,15]. This approach is not very useful in the context of principal chiral models as the summation over group indices cannot be performed locally. But in the case of LGT, due to the gauge invariance the summation over group (colour) indices can be done and this results in the local formulation in terms of invariant 6 j symbols. This dual form can be sudied using Monte-Carlo simulations [16,17].• In the strong coupling limit the S U (N) LGT can be mapped onto monomer-dimer-closed baryon loop model [18].
We use the method of the Weingarten functions to evaluate SU (N ) integrals of the polynomial type. As an application we calculate various one-link integrals for lattice gauge and spin SU (N ) theories.
Dual representations are constructed for non-Abelian lattice spin models with UðNÞ and SUðNÞ symmetry groups, for all N and in any dimension. These models are usually related to the effective models describing the interaction between Polyakov loops in the strong coupled QCD. The original spin degrees of freedom are explicitly integrated out and a dual theory appears to be a local theory for the dual integervalued variables. The construction is performed for the partition function and for the most general correlation function. The latter include the two-point function corresponding to quark-anti-quark free energy and the N-point function related to the free energy of a baryon. We consider both pure gauge models and models with static fermion determinant for both the staggered and Wilson fermions with an arbitrary number of flavours. While the Boltzmann weights of such models are complex in the presence of nonzero chemical potential the dual Boltzmann weights appear to be strictly positive on admissible configurations. An essential part of this work with respect to previous studies is an extension of the dual representation to the case of (1) an arbitrary value of the temporal coupling constant in the Wilson action and (2) an arbitrary number of flavors of static quark determinants. The applications and extensions of the results are discussed in detail. In particular, we outline a possible approach to Monte-Carlo simulations of the dual theory, to the large N expansion and to the development of a tensor renormalization group.
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