The Wodzicki residue is an important tool in index theory, which is local and can be used to express the index of the Dirac operator. The paper under review combines Gilkey’s invariance theory, Getzler’s rescaling method and Scott’s approach to the index via Wodzicki residues to establish local index formulas for certain differential operators.
More precisely, in order to apply Getzler’s rescaling method, the authors define a special class of differential operators acting on smooth sections of a vector bundle, which are called {\em geometric} with respect to a metric \(g\) (Definition 5.8). These geometric differential operators turn out to be closed under product (Proposition 5.11). Then authors define {\em rescalable} geometric differential operators (Definition 6.1) and prove some nice properties for these operators.
Applying the terminology and techniques developed in the paper to \(E=\Lambda T^*M\) and \(E=\Sigma M\), respectively, the authors obtain their localization formulas, namely Theorem 6.7 and Corollary 6.10. Finally, the authors show that the square of a Dirac operator is a rescalable geometric differential operator, hence fits into the framework. As a consequence, a localization formula (Equation (7.1)) for the square of the Dirac operator is achieved.