Spectral flow is the integral of one forms on the Banach manifold of self adjoint Fredholm operators

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 semifinite 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.