a cosmic Galois group

By lieven

non-commutative geometry

Are there hidden relations between mathematical and physical constants such as

$\frac{e^2}{4 \pi \epsilon_0 h c} \sim \frac{1}{137}$

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 $n_1 > n_2 > ! > n_r \geq 1$ of the form

$\zeta(k_1,\dots,k_r) = \sum n_1^{-k_1} \dots n_r^{-k_r}$

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

$\zeta(a)\zeta(b)=\zeta(a+b)+\zeta(a,b)+\zeta(b,a)$

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 $\zeta(3),\zeta(5),\dots$ are all trancendental!

Drinfeld introduced the Grothendieck-Teichmuller group-scheme over $\mathbb{Q}$ whose Lie algebra $\mathfrak{grt}_1$ is conjectured to be the free Lie algebra on infinitely many generators which correspond in a natural way to the numbers $\zeta(3),\zeta(5),\dots$ . 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 $\mathbb{Q}$ -algebra whose degree -term are the linear combinations of the numbers $\zeta(k_1,\dots,k_r)$ with rational coefficients and such that $k_1+\dots+k_r=d$ .

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.

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arxiv, Connes, Galois, geometry, Grothendieck, Kontsevich, Marcolli, non-commutative, symmetry

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Posted in geometry, groups
Written on Wed, 06 October 2004 at 8:26 am
Tags: arxiv, Connes, Galois, geometry, Grothendieck, Kontsevich, Marcolli, non-commutative, symmetry
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a cosmic Galois group

non-commutative geometry

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