ZETA-FUNCTION REGULARIZATION AND MULTI-INSTANTON DETERMINANTS

The determinant of the covariant laplacian is discussed using the ζ-function regularization procedure. For a self-dual solution to a general euclidean Yang-Mills gauge theory, an integral expression for its variation, with respect to the parameters of the solution, is obtained in terms of the general construction of Atiyah et al. The conformal properties are analyzed and an explicit expression is obtained for a part of the determinant containing all the non-trivial behavior. The remaining conformally invariant part has not been determined. The removal of the infrared divergences by considering the problem on a sphere is discussed in some detail. © 1979.