Topological invariants in two and three dimensions

Abstract

We consider the non-abelian SU(2) Chern-Simons field theory defined in the three-manifolds of the type Σ g × S 1, where Σ g is a Riemann surface of genus g. We define a set of topological invariants for the punctured surface Σ g in terms of invariants in three dimensions. We compute, in particular, the dimension of the physical state space associated with a generic punctured Riemann surface of arbitrary genus. We explain why these invariants are described by the Feynman diagrams of a certain φ 3 theory. We also give the expression of these invariants in terms of the S-matrix of the conformal models.