In this paper, the
-functions
introduced by M. Sato, M. Miwa, and T. Jimbo in their study of monodromy
preserving deformations of the Dirac equation are rigorously identified as
determinants of singular Dirac operators. The singular Dirac operators
have branched functions in their domains that reflect the monodromy in
the deformation theory. The principal result is a new formula for the
-function,
obtained by trivializing a suitable determinant bundle, that can be simply
related to the deformation theory and which may also be computed in the
transfer matrix formalism. These two different ways of understanding the
-function
provide the link between the deformation theory and the quantum field theory significance
of
-function
as a correlation function. This connection is the central result of the Sato-Miwa-Jimbo
theory of Holonomic Fields.
