An introduction to theta sectors and instantons

Whenever the fixed time configuration space of a classical theory is multiply connected, the corresponding quantum theory is not unique: there is a set of inequivalent quantum theories, all having the same classical limit, labelled by the characters of the fundamental group. When the fundamental group is the group of the integers, these inequivalent quantum theories are labelled by an angle, usually called theta. The paradigm of this phenomenon is the Aharonov-Bohm effect. The same mechanism is responsible for the theta vacua of nonlinear quantum field theories such as sigma models and Yang-Mills theory. The energy of the vacuum can be computed in the WKB approximation and is found to depend on theta in a characteristic way. In these calculations an important role is played by instantons, which are classical solutions describing the motion of the system on a noncontractible loop. (author). 11 refs