Antoine Coste

Quasi-quantum groups, knots, three-manifolds, and topological field theory

We show how to construct, starting from a quasi-Hopf algebra, or quasiquantum 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 groupG, 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 dataG, ω.

On the stress tensor of conformal field theories in higher dimensions

Abstract The behaviour of the stress tensor under conformal transformations of both flat and curved spaces is investigated for free theories in a classical background metric. In flat space ℝ d it is derived by the operator product expansion of two stress tensors. For Weyl transformations of curved manifolds it is given by the effective potential for the metric. In four dimensions the general form of the potential and its consistency conditions are analysed. These issues are relevant for the possible generalizations of the central charge in higher dimensions. The related subject of the Casimir effect is studied by means of closed expressions for the bosonic partition function on the manifoldsT d and S 1 ×S d −1 . The general relationship between the Casimir effect on ℝ×S d −1 and the trace anomaly is emphasized.

Zero-momentum contribution to wilson loops in periodic boxes

Abstract We compute the zero-momentum finite-size effects in non-abelian gauge theories. In some cases, these effects are to lowest-order identical for lattice and continuum theories and are of the order of a few percent for an SU(3) gauge theory in four dimensions.

Comments on the links between SU(3) modular invariants, simple factors in the Jacobian of Fermat curves, and rational triangular billiards

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. Comment: 56 pages, 4 eps figures, LaTeX, uses epsf

Finite size effects and twisted boundary conditions

We explore the possibility of reducing finite size effects in the weak coupling region of glueball correlations and Wilson loops. Our analysis indicates that twisted boundary conditions do diminish finite size effects and numerical evidence for this is given for the glueball correlation.

Calculation of the non-abelian chiral anomaly on the lattice

Abstract We check the validity of Wilson and Kogut-Susskind lattice regularizations of the Dirac fermion action by calculating perturbatively the non-abelian chiral anomaly on the lattice. We show that the result does not depend on the detailed expression of the coupling term of the lattice fermions to the chiral gauge field, taken as an external field. In two dimensions we recover the consistent expression of the anomaly, up to counterterms which we have computed. In four dimensions, we obtain the correct normalization factors of the anomaly, leaving the problem of the counterterms open.

Non-abelian anomaly from lattice fermions

Abstract The non-abelian anomaly is computed in two and four dimensions, using Wilson and Kogut-Susskind lattice regularizations of the Dirac fermion action.

Invariants of three-manifolds from finite group cohomology

Abstract We compare on lens spaces the values of two topological invariants of three-manifolds, both built from a finite group G and a three-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 three-cocycles are derived in an appendix.

Abelian Chern-Simons theories on S3

Abstract We check by explicit computations on the sphere several features of Chern-Simons theories such as scale dependence of the partition function, reduction of topologically massive QED to pure Chern-Simons theory in the strong coupling limit, and frame dependence of Wilson loops.

GENERALISED FERMAT HYPERMAPS AND GALOIS ORBITS

We consider families of quasiplatonic Riemann surfaces characterised by the fact that – as in the case of Fermat curves of exponent n – their underlying regular (Walsh) hypermap is an embedding of the complete bipartite graph K n,n , where n is an odd prime power. We show that these surfaces, regarded as algebraic curves, are all defined over abelian number fields. We determine their orbits under the action of the absolute Galois group, their minimal fields of definition and in some easier cases their defining equations. The paper relies on group – and graph – theoretic results by G. A. Jones, R. Nedela and M. Skoviera about regular embeddings of the graphs K n,n [ 7 ] and generalises the analogous results for maps obtained in [ 9 ], partly using different methods.