An L^2-Index Theorem for Dirac Operators on S^1 * R^3

Details An L^2-Index Theorem for Dirac Operators on S^1 * R^3 An L^2-Index Theorem for Dirac Operators on S^1 * R^3

2000-09-14

arxiv.org/abs/math/0009144v1

Mathematics

Differential Geometry

14 pages, Latex, to appear in the Journal of Functional Analysis

Scientific paper

An expression is found for the $L^2$-index of a Dirac operator coupled to a connection on a $U_n$ vector bundle over $S^1\times{\mathbb R}^3$. Boundary conditions for the connection are given which ensure the coupled Dirac operator is Fredholm. 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 $\eta$-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).

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