Regularized traces and the index formula for manifolds with boundary

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.