An L 2-index Theorem for Dirac Operators on S 1 × R 3

An expression is found for the L 2-index of a Dirac operator coupled to a connection on a Un vector bundle over S 1 × R 3. Boundary conditions for the connection are given which ensure the coupled Dirac operator is Fred-holm. Callias' index theorem is used to calculate the index when the connection is independent of the coordinate on S 1. An excision theorem due to Gromov, Lawson, and Anghel reduces the index theorem to this special case. The index formula can be expressed using the adiabatic limit of the η-invariant of a Dirac operator canonically associated to the boundary. An application of the theorem is to count the zero modes of the Dirac operator in the background of a caloron (periodic instanton).