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
But we also have in mind
methods used in number
also comprise zeta regularisation used in physics in the form of zeta determinants
to compute effective
and particularly in the context of
and index theory as a substitute for the equivalent
Regularised integrals, discrete sums, and
obtained by means of a regu-
larisation procedure present many discrepancies responsible for various
In contrast, the underlying canonical integrals, discrete sums, and traces are well
behaved. Canonical integrals are indeed covariant, translation invariant, and obey
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].
called modified dimensional regularisation.
amounts to cut-off regularisation.
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
the work of Ray and Singer [RaSi] on analytic torsion where the zeta determinant
was first introduced in mathematics.
for the geometry of loop groups, see [Fr].
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.
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
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