Self-adjoint elliptic operators and manifold decompositions Part III: Determinant line bundles and Lagrangian intersection

Abstract

The theory of spectral flows developed in the series [10, 11, 12], and the present paper has a wide range of applications to important geometric operators on compact manifolds. To present our results on spectral flow and manifold decomposition, the present paper develops a theory of determinant line bundles and infinitedimensional Lagrangians associated to self-adjoint elliptic operators on compact manifolds. The trace-class properties of these infinite Lagrangians established here and the precise uniform estimates relating them to finite Lagrangians are crucial for such a determinant line bundle approach to analytical questions. As an application, we elucidate the Walker’s and other generalizations of Casson’s SU(2) representation theoretic invariant of 3-manifolds in terms of the -invariant of certain Dirac operators. This is carried out by introducing the technique of “canonical perturbations” of singular Lagrangian subvarieties in symplectic geometry. At the end of Part II of this series, we obtained a formulation of the spectral flow of a family of self-adjoint elliptic operators D(u) : L2(E)! L2(E) in terms