Renormalized Traces as a Geometric Tool

ζ-function regularization techniques are commonly used by physicists in Quantum Field Theory - e.g. to compute partition functions using ζ-function determinants - and by mathematicians in geometry and topology - e.g. to define invariants such as the η invariant which plays a fundamental part in the Atiyah-Patodi-Singer index or the analytic torsion involving again ζ-function regularized determinants. Using ζ-function regularization techniques we define renormalized traces of (possibly non trace-class) classical pseudo-differential operators ; these renormalized traces serve as a tool to generalize some well-known finite dimensional geometric concepts involving traces to a class of infinite dimensional vector bundles and manifolds. We express the obstructions preventing a straightforward extension in terms of Wodzicki residues of classical pseudo-differential operators and describe some cases for which these obstructions vanish.