Moore and Seiberg equations, topological field theories and Galois theory

Abstract

Abstract This note is an attempt to summarize relations, partially conjectural, between Moore and Seiberg's equations, topological (projective) field theories in three dimensions and the second paragraph of Grothendieck's Esquisse d'un Programme . The first section outlines the current situation, and the second gives a summary of a review paper on the subject by the author, which could not be included in this volume because of length and scheduling problems. First of all, we recall the construction of projective topological field theories in three dimensions from solutions to Moore and Seiberg's equations. We discuss the possible relation between this result and the reconstruction conjecture of the Teichmuller tower from its first two levels. Finally, we suggest an explicit translation of the natural action of Gal into an action on a wide class of three-dimensional topological field theories arising from rational conformal field theories in two dimensions. Our aim is to point out some relationships between recent developments in Topological Field Theories, the classification program of Rational Conformal Field Theories and deep ideas expressed by A. Grothendieck in the Esquisse d'un Programme [Groth]. First of all, we would like to stress that our present knowledge does not claim to be a definitive and complete mathematical theory since most of this wonderful story remains to be discovered. We would like to point out why, in our opinion, there is a deep connection between the world of Rational Conformal Field Theory and that of Grothendieck.