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

1Just

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].

2Also

called modified dimensional regularisation.

3Which

amounts to cut-off regularisation.

4See

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.

6With

the work of Ray and Singer [RaSi] on analytic torsion where the zeta determinant

was first introduced in mathematics.

7E.g.

for the geometry of loop groups, see [Fr].

8Starting

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.

9We

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.

10For

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].

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