We construct two-dimensional topological $\sigma$ models with loop groups as the target manifolds and study their properties. These models are related to four-dimensional topological Yang-Mills theory as a consequence of a theorem of Atiyah, which identifies the instanton moduli spaces of these two types of theories in the simplest cases. The field-theory implications of this identification of the moduli spaces are also discussed.

• Received 22 May 1991

DOI:https://doi.org/10.1103/PhysRevD.44.2498