Using Representation Theoretic Methods to Find Renormalization Group Invariant Functions in a Supersymmetric Nonlinear Sigma Model

The highest weight vectors of a Lie algebra or superalgebra g are key in describing the structure of g, but are also important objects in the link between mathematics and physics, as they are the eigenvectors of the Laplace-Casimir operator. Here, we classify renormalization group invariant functions in a supersymmetric nonlinear sigma model whose target space has Hermitian structure using a one-to-one mapping to highest weight vectors of a particular representation of g. Specifically, we are interested in supersymmetric nonlinear sigma models, as the supersymmetric approach is more mathematically well founded, and more general. We present a method for determining some families of highest weight vectors in Sym(g) for the orthosymplectic Lie superalgebras g=osp(2n,2n) and find a computationally efficient algorithm for explicitly determining invariant functions of the sigma model target space. We ultimately aim to provide a classification of the highest weight vectors of g.