Asymptotic spectral flow for Dirac operators

Let M denote a compact, oriented Riemannian manifold of odd dimension n >= 3. Suppose that F -->M is a principle bundle with structure group Spin(n) x({+/-1}) U(k) such that F/U(k) is the priniciple SO( n) bundle of orthonormal frames for TM. A connection on the principle bundle F/Spin(n) --> M determines a self-adjoint Dirac operator on a certain associated Clifford module. This understood, suppose that {A(s)}(s is an element of[0,1]) is a differentiable path of such connections. An estimate is given here for the spectral flow along the corresponding 1-parameter family of Dirac operators.