No abstract

We show how to construct, starting from a quasi-Hopf algebra, or quasi-quantum group, invariants of knots and links. In some cases, these invariants give rise to invariants of the three-manifolds obtained by surgery along these links. This happens for a finite-dimensional quasi-quantum group, whose definition involves a finite group G, and a 3-cocycle ω, which was first studied by Dijkgraaf, Pasquier and Roche. We treat this example in more detail, and argue that in this case the invariants agree with the partition function of the topological field theory of Dijkgraaf and Witten depending on the same data G, ω.

No abstract

No abstract

No abstract

No abstract

We examine the proposal made recently that the su(3) modular invariant partition functions could be related to the geometry of the complex Fermat curves. Although a number of coincidences and similarities emerge between them and certain algebraic curves related to triangular billiards, their meaning remains obscure. In an attempt to go beyond the su(3) case, we show that any rational conformal field theory determines canonically a Riemann surface.

We compare on lens spaces the values of two topological invariants of three-manifolds, both built from a finite group G and a 3-cocycle ω, which we conjectured to be equal up to a normalization. The first invariant is defined by triangulation, it is the partition function of the Dijkgraaf-Witten topological field theory, and the second one by surgery, using a quasi-Hopf algebra. When G is a cyclic group, we show that the first invariant reduces to a Gauss sum. Some identities satisfied by 3-cocycles are derived in an appendix.