Duality in field theory and statistical systems

This paper presents a pedagogical review of duality (in the sense of Kramers and Wannier) and its application to a wide range of field theories and statistical systems. Most of the article discusses systems in arbitrary dimensions with discrete or continuous Abelian symmetry. Globally and locally symmetric interactions are treated on an equal footing. For convenience, most of the theories are formulated on a $d$-dimensional (Euclidean) lattice, although duality transformations in the continuum are briefly described. Among the familiar theories considered are the Ising model, the $x-y$ model, the vector Potts model, and the Wilson lattice gauge theory with a $Z_N$ or $U\left(1\right)\mathrm{}$ symmetry, all in various dimensions. These theories are all members of a more general heirarchy of theories with interactions which are distinguished by their geometrical character. For all these Abelian theories it is shown that the duality transformation maps the high-temperature (or, for a field theory, large coupling constant) region of the theory into the low-temperature (small coupling constant) region of the dual theory, and vice versa. The interpretation of the dual variables as disorder parameters is discussed. The formulation of the theories in terms of their topological excitations is presented, and the role of these excitations in determining the phase structure of the theories is explained. Among the other topics discussed are duality for the Abelian Higgs model and related models, duality transformations applied to random systems (such as theories of a spin glass), duality transformations in the "lattice Hamiltonian" formalism, and a description of attempts to construct duality transformations for theories with a non-Abelian symmetry, both on the lattice and in the continuum.