Weighted trace cochains; a geometric setup for anomalies

TL;DR

This paper develops a geometric framework for analyzing anomalies in regularized traces of pseudodifferential operators, extending existing formulas to handle algebraic and geometric discrepancies using trace cochains and residues.

Contribution

It introduces an extension of regularized trace formulas to n-cochains, enabling simultaneous analysis of algebraic and geometric anomalies in pseudodifferential operator families.

Findings

Discrepancies are expressed as linear combinations of Wodzicki residues.

Constructs covariantly closed trace cochains using super connection curvature.

Geometric discrepancies vanish with Bismut-Quillen super connection curvature.

Abstract

We extend formulae which measure discrepancies for regularized traces on classical pseudodifferential operators to regularized trace cochains, regularized traces corresponding to 0-regularized trace cochains. This extension from 0-cochains to $n$-cochains is appropriate to handle simultaneously algebraic and geometric discrepancies/anomalies. Algebraic anomalies are Hochschild coboundaries of regularized trace cochains on a fixed algebra of pseudodifferential operators weighted by a fixed classical pseudodifferential operator with positive order and positive scalar leading symbol. In contrast, geometric anomalies arise when considering families of pseudodifferential operators associated with a smooth fibration of manifolds. They correspond to covariant derivatives (and possibly their curvature) of smooth families of regularized trace cochains, the weight being here an elliptic operator valued form on the base manifold. Both types of discrepancies can be expressed as finite linear combinations of Wodzicki residues.We apply the formulae obtained in the family setting to build Chern-Weil type weighted trace cochains on one hand, and on the other hand, to show that choosing the curvature of a Bismut-Quillen type super connection as a weight, provides covariantly closed weighted trace cochains in which case the geometric discrepancies vanish.