Preface Regularisation techniques, implemented in quantum field theory, number the- ory, and geometry to make sense of divergent integrals, discrete sums, or traces, might seem very arbitrary and uncanonical at first glance. They nevertheless con- ceal canonical concepts, namely canonical integrals, sums, and traces, which we want to bring to the forefront in these lectures. Cut-off and dimensional regularisation are prototypes of regularisation tech- niques used in quantum field theory.1 But we also have in mind Riesz2 and Hadamard finite parts3 methods used in number theory.4 Regularisation techniques also comprise zeta regularisation used in physics in the form of zeta determinants to compute effective actions,5 or in geometry6 and particularly in the context of infinite dimensional manifolds7 and index theory as a substitute for the equivalent heat-kernel methods.8 Regularised integrals, discrete sums, and traces9 obtained by means of a regu- larisation procedure present many discrepancies responsible for various anomalies.10 In contrast, the underlying canonical integrals, discrete sums, and traces are well behaved. Canonical integrals are indeed covariant, translation invariant, and obey 1 Just to quote a few books amongst the vast literature on the subject, see e.g. [Col], [CMa], [D], [Sm1], [Sm2] as well as more specific references in the context of renormalisation, such as [Et], [CMa], [He], [HV], [Sp], [Zi]. 2 Also called modified dimensional regularisation. 3 Which amounts to cut-off regularisation. 4 See e.g. [Ca] for an introductory presentation. 5Starting with pioneering work by Hawkins [Haw], see other applications in [El], [EORZ], and further developments in string theory, see e.g. [D] and [AJPS] for a mathematical presentation. 6 With the work of Ray and Singer [RaSi] on analytic torsion where the zeta determinant was first introduced in mathematics. 7 E.g. for the geometry of loop groups, see [Fr]. 8 Starting with pioneering work by Atiyah and Singer [APS1, APS2, APS3] and later by Quillen [Q1] and Bismut and Freed [BF], see also more recent work by Scott [Sc1] in the context of the family index theorem. 9 We use the terminology regularised when a physicist might call this a renormalised in- tegral, discrete sum, or trace since it is the result of a regularisation procedure combined with a subtraction scheme used to extract a finite part. We choose not to use of the word “renor- malisation” because in physics this concept involves much more than merely evaluating divergent integrals, divergent discrete sums in one variable, or divergent traces that we are concerned with here. 10 For a treatment of anomalies in physics (see e.g. [D] and [N] for a mathematical presen- tation) from the point of view of discrepancies also called trace anomalies, see e.g. [CDP] and [Mi]. vii
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