Abstract

Let D be a first order differential operator acting on the space of section of a finite rank vector bundle over a smooth manifold M with boundary X. In this chapter we show that the index of D, associated to Atiyah–Patodi–Singer type boundary conditions, can be expressed as aweighted (super-)trace of the identity operator, generalizing the corresponding result in the case of closed manifolds obtained in [19]. We also show that the reduced eta-invariant can be expressed as a weighted (super-)trace of an identity operator so that, actually, the index of D can be expressed as a sum of two weighted super-traces of identity operators, one giving rise to the integral term in the Atiyah–Patodi–Singer theorem and the other one corresponding to the η-term.

Introduction

Let D be a positive order differential operator acting on the space of section of a finite rank vector bundle E → M over a smooth manifold M. When the manifold is closed, it is a well-known result that the index of such an operator can be written as a weighted (super-)trace of the identity, i.e. as a regularization of the trace of the identity with respect to some positive differential operator, usually of Laplacian type (see e.g. [19], [20]). These weighted traces (or pseudo-traces) are neither independent of the reference operator used to define them, nor traces on the algebra of classical pseudo-differential operators, but the obstructions associated to these anomalies can be computed explicitly in terms of Wodzicki residues, giving rise to local terms, i.e. terms given by integrals of smooth densities on the manifold M.