This chapter describes the formal properties, and discusses the renormalization, of quantum field theories (QFT) based on homogeneous spaces: coset spaces of the form G/H, where G is a compact Lie group and H a Lie subgroup. In physics, they appear naturally in the case of spontaneous symmetry breaking, and describe the interaction between Goldstone modes. Homogeneous spaces are associated with non-linear realizations of group representations. There exist natural ways to embed these manifolds in flat Euclidean spaces, spaces in which the symmetry group acts linearly. As in the example of the non-linear σ-model, this embedding is first used, because the renormalization properties are simpler, and the physical interpretation of the more direct correlation functions. Then, in a generic parametrization, the renormalization problem is solved by the introduction of a Becchi–Rouet–Stora–Tyutin (BRST)-like symmetry with anticommuting (Grassmann) parameters, which also plays an essential role in quantized gauge theories. The more specific properties of models corresponding to a special class of homogeneous spaces, symmetric spaces (like the non-linear σ-model), are studied. These models are characterized by the uniqueness of the metric and thus, of the classical action. In two dimensions, from the classical field equations an infinite number of non-local conservation laws can be derived. The field and the unique coupling renormalization group (RG) functions are calculated at one-loop order, in two dimensions, and shown to imply asymptotic freedom.
Some nozzle flows found by the hodograph method
