Skip to content →

a cosmic Galois group

Are
there hidden relations between mathematical and physical constants such
as

e24πϵ0hc1137

or are these numerical relations mere accidents? A couple of years
ago, Pierre Cartier proposed in his paper A mad day’s work : from Grothendieck to Connes and
Kontsevich : the evolution of concepts of space and symmetry
that
there are many reasons to believe in a cosmic Galois group acting on the
fundamental constants of physical theories and responsible for relations
such as the one above.

The Euler-Zagier numbers are infinite
sums over n1>n2>!>nr1 of the form

ζ(k1,,kr)=n1k1nrkr

and there are polynomial relations with rational coefficients between
these such as the product relation

ζ(a)ζ(b)=ζ(a+b)+ζ(a,b)+ζ(b,a)

It is
conjectured that all polynomial relations among Euler-Zagier numbers are
consequences of these product relations and similar explicitly known
formulas. A consequence of this conjecture would be that
ζ(3),ζ(5), are all trancendental!

Drinfeld
introduced the Grothendieck-Teichmuller group-scheme over Q
whose Lie algebra grt1 is conjectured to be the free Lie
algebra on infinitely many generators which correspond in a natural way
to the numbers ζ(3),ζ(5),. The Grothendieck-Teichmuller
group itself plays the role of the Galois group for the Euler-Zagier
numbers as it is conjectured to act by automorphisms on the graded
Q-algebra whose degree d-term are the linear combinations
of the numbers ζ(k1,,kr) with rational coefficients and
such that k1++kr=d.

The Grothendieck-Teichmuller
group also appears mysteriously in non-commutative geometry. For
example, the set of all Kontsevich deformation quantizations has a
symmetry group which Kontsevich conjectures to be isomorphic to the
Grothendieck-Teichmuller group. See section 4 of his paper Operads and motives in
deformation quantzation
for more details.

It also appears
in the renormalization results of Alain Connes and Dirk Kreimer. A very
readable introduction to this is given by Alain Connes himself in Symmetries Galoisiennes
et renormalisation
. Perhaps the latest news on Cartier’s dream of a
cosmic Galois group is the paper by Alain Connes and Matilde Marcolli posted
last month on the arXiv : Renormalization and
motivic Galois theory
. A good web-page on all of this, including
references, can be found here.

Published in featured