vi CONTENTS
5.6. Regularised discrete sums on symbols 95
5.7. Two special zeta functions 98
Chapter 6. Traces on pseudodifferential operators 105
6.1. From symbols to operators 105
6.2. Basic properties of pseudodifferential operators 109
6.3. Pseudodifferential operators on manifolds 114
6.4. From closed linear forms on symbols to traces on operators 118
6.5. A first characterisation of the noncommutative residue 126
Chapter 7. Weighted traces 131
7.1. Complex powers 131
7.2. A fundamental formula 133
7.3. Zeta regularised traces 135
7.4. Logarithms of admissible operators 136
7.5. Discrepancies 139
7.6. Characterising traces on operators 141
Chapter 8. Logarithmic residues 145
8.1. Regularised traces—locality versus nonlocality 145
8.2. Application to ζ-regularised traces 147
8.3. Traces of differentiable families 149
8.4. The residue determinant 155
8.5. The index as a superresidue 157
8.6. Application to the Atiyah-Singer index theorem—a toy model 160
Chapter 9. Anomalies of regularised determinants 167
9.1. Weighted and ζ-determinants 167
9.2. Multiplicative anomaly of a weighted determinant 170
9.3. Multiplicative anomaly of the ζ-determinant 173
9.4. Conformally covariant operators 174
9.5. Conformal anomalies 176
9.6. Conformal anomaly of the ζ-determinant 181
Bibliography 183
Index 189
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