Quantization of the nonlinear sigma model revisited (Q2820906)

This paper discusses mathematical aspects of perturbative quantization of non-linear sigma models based on fields that are functions with two dimensional domain and some Riemannian manifold as codomain. It reviews the theory of graded manifolds, jet bundles, and lie algebra cohomology in appendices. The first section reviews Batalin-Vilkovisky geometry. Basically, when a lie group acts on a manifold, it is natural to add terms on a real valued function defined on the manifold that encode this group action. The extended action \(S\) will then satisfy what is known as the master equation. When the action is included in the Gaussian measure the requirement that the result be invariant under the group action adds a term to the master equation resulting in the quantum master equation. When one applies this to a space of functions one must regularize to make terms in the expansion converge. This regularization process is the topic of the next section of the paper. Solutions to the quantum master equation are understood via a cohomology theory. The fourth section of the paper explains this and computes the relevant cohomology groups. The main result of the paper is that there is no so-called anomoly for non-linear sigma models taing values in homogeneous spaces that meet certain conditions. Similarly, there is no obstruction to quantization invariant under the full diffeomorphism group. The paper also includes a discussion of the renormalization group and flow. This paper includes many references and would be a good starting point for someone wishing to learn a bout a mathematical theory of this piece of quantum field theory.