(Second) Quantised resolvents and regularised traces

Abstract

Regularised traces on classical pseudodifferential operators are extended to tensor products of classical pseudodifferential operators via a (second) quantisation procedure. Whereas ordinary ζ -regularised traces are not generally expected to be local, using techniques borrowed from Connes and Moscovici [A. Connes, H. Moscovici, The local index formula in noncommutative geometry, Geom. Funct. Anal. 5 (2) (1995) 174–243], Higson [N. Higson, The residue index theorem of Connes and Moscovici, in: Clay Mathematics Proceedings, 2004, http://www.math.psu.edu/higson/ResearchPapers.html], we show that if Q has scalar leading symbol, higher quantised ζ - regularised traces are local since they can be expressed as a finite linear combination of noncommutative residues. Just as ordinary ζ -regularised traces, they present anomalies (Hochschild coboundary, dependence on the weight Q ), which for quantised ζ -regularised traces of level n , are roughly speaking finite linear combinations of quantised regularised traces of level n + 1 . As a result, anomalies are local for any non negative n , which yields back as a particular case the fact that ordinary ζ -regularised traces present local anomalies. 1 1 This work is partially based on [S. Paycha, Weighted trace cochains; A geometric setup for anomalies, Max Planck Institute, 2005. Preprint] although we use other conventions here which lead to slightly different definitions.