  One may trace the idea that spectral flow should be given as the integral of
a one form back to the 1974 Vancouver ICM address of I.M. Singer. Our main
theorem gives analytic formulae for the spectral flow along a norm
differentiable path of self-adjoint bounded Breuer-Fredholm operators in a
semi-finite von Neumann algebra. These formulae have a geometric interpretation
which derives from the proof. Namely we define a family of Banach submanifolds
of all bounded self-adjoint Breuer-Fredholm operators and on each submanifold
define global one forms whose integral on a norm differentiable path contained
in the submanifold calculates the spectral flow along this path. We emphasise
that our methods do not give a single globally defined one form on the self
adjoint Breuer- Fredholms whose integral along all paths is spectral flow
rather, as the choice of the plural `forms' in the title suggests, we need a
family of such one forms in order to confirm Singer's idea. The original
context for this result concerned paths of unbounded self-adjoint Fredholm
operators. We therefore prove analogous formulae for spectral flow in the
unbounded case as well. The proof is a synthesis of key contributions by
previous authors, whom we acknowledge in detail in the introduction, combined
with an additional important recent advance in the differential calculus of
functions of non-commuting operators.
