between the particles. This is in contrast – more precisely one would say dual– to what one is used to, since usually one connects the absence of interactions with large distances. However, as already mentioned in the original paper of Franz Wegner, a solid state theorist who introduced 1971 simple gauge invariant lattice models, the high-temperature behaviour of the original model, e.g. the strong decay of correlations at large distances, corresponds to the low-temperature behaviour of the (usually ordered!) dual model, namely the asymptotic decay of non-trivial correlations, e.g. short-range deviations from almost perfect arrangements, for short distances. Here, in contrast to Wegner, we have only the dual model, which is that one described in this article. Symmetry groups The color group SU(3) corresponds to the local symmetry whose gauging gives rise to QCD. The electric charge labels a representation of the local symmetry group U(1), which is gauged to give QED: this is an abelian group. If one considers a version of QCD with Nf flavors of massless quarks, then there is a global (chiral) flavor symmetry group SUL(Nf) × SUR(Nf) × UB(1) × UA(1). The chiral symmetry is spontaneously broken by the QCD vacuum to the vector (L+R) SUV(Nf) with the formation of a chiral condensate. The vector symmetry, UB(1) corresponds to the baryon number of quarks and is an exact symmetry. The axial symmetry UA(1) is exact in the classical theory, but broken in the quantum theory, an occurrence called an anomaly. Gluon field configurations called instantons are closely related to this anomaly. There are two different types of SU(3) symmetry: there is the symmetry that acts on the different colors of quarks, and this is an exact gauge symmetry mediated by the gluons, and there is also a flavor symmetry that rotates different flavors of quarks to each other, or flavor SU(3). Flavor SU(3) is an approximate symmetry of the vacuum of QCD, and is not a fundamental symmetry at all. It is an accidental consequence of the small mass of the three lightest quarks. In the QCD vacuum there are vacuum condensates of all the quarks whose mass is less than the QCD scale. This includes the up and down quarks, and to a lesser extent the strange quark, but not any of the others. The vacuum is symmetric under SU(2) isospin rotations of up and down, and to a lesser extent under rotations of