The Maslov index, the spectral flow, and decompositions of manifolds (Q1913365)

Suppose that a compact oriented Riemann manifold without boundary \(M\) is divided into two manifolds with boundary \(M_1\) and \(M_2\) by an oriented hypersurface \(\Sigma \subset M\). Consider a Clifford bundle \(E\) over \(M\) and a smooth path of selfadjoint Dirac operators \(D^t : C^\infty (E) \to C^\infty (E)\). Set \(E_0 = E |\Sigma\) and denote by \(D^t_i\) the restriction of \(D^t\) to \(M_i\). Restriction to \(\Sigma\) gives the infinite-dimensional Cauchy-data spaces (CD-spaces) \(\Lambda_i (t) = \text{Ker} (D^t_i |\Sigma)\) in \(L^2 (E_0)\) with finite-dimensional intersection \(\Lambda_1 (t) \cap \Lambda_2 (t)\). \(L^2 (E_0)\) has a symplectic structure and the CD-spaces \(\Lambda_i (t)\) are infinite-dimensional Lagrangian subspaces of \(L^2 (E_0)\) that vary smoothly with \(t\), and the pair \((\Lambda_1 (t), \Lambda_2 (t))\) is a special ``Fredholm'' pair. As in the finite-dimensional case, one can associate to a path of Fredholm pairs of Lagrangians an integer called the Maslov index. The spectral flow of the family \(D^t\) is the integer obtained by counting, with sign, the number of eigenvalues of \(D^t\) that cross 0 as \(t\) varies. The aim of this paper is to prove that this Maslov index equals the spectral flow of the family \(D^t\).