2002/05/31 by Hendryk Pfeiffer · 9 citations
Mathematics · Physics and Astronomy · #Abelian group #Black Holes and Theoretical Physics #Coset #Discrete mathematics #Duality (order theory) #Gauge theory #Geometry #Global symmetry #Homogeneous space #Lie group #Mathematical physics #Mathematics #Nonlinear system #Partition function (quantum field theory) #Physics #Pure mathematics #Quantum Chromodynamics and Particle Interactions #Quantum mechanics #Sigma model #Spontaneous symmetry breaking #Symmetry breaking #Symmetry group #Theoretical and Computational Physics #cond-mat.stat-mech #hep-lat
paper · pdf · doi:10.1063/1.1580071
published in Journal of Mathematical Physics 44(7), 2891-2938 (American Institute of Physics) · 57 pages, 15 figures, LaTeX; v2: references updated
arxiv created 2003/04/01 · openalex publication_date 2003/06/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We present an exact duality transformation in the framework of statistical mechanics for various lattice models with non-Abelian global or local symmetries. The transformation applies to sigma models with variables in a compact Lie group G with global G×G-symmetry (the chiral model) and with variables in coset spaces G/H and a global G-symmetry [for example, the nonlinear O(N) or RPN models] in any dimension d⩾1. It is also available for lattice gauge theories with local gauge symmetry in dimensions d⩾2 and for the models obtained from minimally coupling a sigma model of the type mentioned above to a gauge theory. The duality transformation maps the strong coupling regime of the original model to the weak coupling regime of the dual model. Transformations are available for the partition function, for expectation values of fundamental variables (correlators and generalized Wilson loops) and for expectation values in the dual model which correspond in the original formulation to certain ratios of partition functions (free energies of dislocations, vortices or monopoles). Whereas the original models are formulated in terms of compact Lie groups G and H, coset spaces G/H and integrals over them, the configurations of the dual model are given in terms of representations and intertwiners of G and H. They are spin networks and spin foams. The partition function of the dual model describes the group theoretic aspects of the strong coupling expansion in a closed form.