Cooking pasta with Lie groups

Abstract

We extend the (gauged) Skyrme model to the case in which the global isospin group (which usually is taken to be SU (N )) is a generic compact connected Lie group G. We analyze the corresponding field equations in (3+1) dimensions from a group theory point of view. Several solutions can be constructed analytically and are determined by the embeddings of three dimensional simple Lie groups into G, in a generic irreducible representation. These solutions represent the so-called nuclear pasta state configurations of nuclear matter at low energy. We employ the Dynkin explicit classification of all three dimensional Lie subgroups of exceptional Lie group to classify all such solutions in the case G is an exceptional simple Lie group, and give all ingredients to construct them explicitly. As an example, we construct the explicit solutions for G = G2. We then extend our ansatz to include the minimal coupling of the Skyrme field to a U (1) gauge field. We extend the definition of the topological charge to this case and then concentrate our attention to the electromagnetic case. After imposing a "free force condition" on the gauge field, the complete set of coupled field equations corresponding to the gauged Skyrme model minimally coupled to an Abelian gauge field is reduced to just one linear ODE keeping alive the topological charge. We discuss the cases in which such ODE belongs to the (Whittaker-)Hill and Mathieu types.