Analytic aspects of periodic instantons

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 131-134) and index.The main result is a computation of the Nahm transform of a SU(2)-instanton over R x T³, called spatially-periodic instanton. It is a singular monopole over T³, a solution to the Bogomolny equation, whose rank is computed and behavior at the singular points is understood under certain conditions. A full description of the Riemannian ADHMN construction of instantons on R⁴ is given, preceding a description of the heuristic behind the theory of instantons on quotients of R⁴. The Fredholm theory of twisted Dirac operators on cylindrical manifolds is derived, the spectra of spin Dirac operators on spheres and on product manifolds are computed. A brief discussion on the decay of spatially-periodic and doubly-periodic instantons is included.by Benoit Charbonneau.Ph.D