(Second) Quantised resolvents and regularised traces
Let \(Q\) be a classical elliptic pseudodifferential operator acting on smooth sections of a vector bundle \(E\) over a close manifold \(M\). It is showed that if \(Q\) has scalar leading symbol then quantised weighted traces \(\text{ tr}_n^{Q+\theta}\) can be expressed as a finite linear combination of noncommutative residues, thus the traces are local for any positive integer \(n\). Moreover, when \(Q\) has positive leading symbol then heat-kernel quantised weighted traces coincide for large quantum level with the quantised weighted traces. This contrasts with a zero level. As a consequence it is show that the anomaly of a quantised regular trace of level \(n\) is a finite linear combination of quantised regularised traces of level \(n+1\). In particular, if \(Q\) has scalar leading symbol then their Hochschild coboundary is also local as a finite linear combination of noncommutative residues. Moreover, the local expressions for the variation of quantised weighted traces and the exterior differential of quantised reqularised traces are proved.
scientific article; zbMATH DE number 5147163
scientific article