inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a by-product we obtain results about the Frobenius-Schur indicator in sovereign tensor categories. A braiding on C is not needed, nor is semisimplicity. We apply our results to the description of boundary conditions in twodimensional conformal field theory and present illustrative examples. We show that when the module category is tensor, then it gives rise to a NIM-rep of the fusion rules, and discuss a possible relation with the representation theory of vertex operator algebras. 1 CFT boundary conditions Boundary conditions in conformal field theory have various physical applications, ranging from the study of defects in condensed matter physics to the theory of open strings. Such boundary conditions are partially characterized by the maximal vertex operator subalgebra A of the bulk chiral algebra Abulk that they respect [43, 75]. That A is respected by a boundary condition means that the

We prove that the image of the mapping class group by the representations arising in the SU(2)-TQFT is infinite, provided that the genus g ≥ 2 and the level of the theory r ̸ = 2,3,4,6 (and r ̸ = 10 for g = 2). In particular it follows that the quotient groups Mg/N (tr) by the normalizer of the r-th power of a Dehn twist t are infinite if g ≥ 3 and r ̸ = 2,3,4,6,8,12. 1. Introduction. Witten [50] constructed a TQFT in dimension 3 using path integrals and afterwards several rigorous constructions arose, like those using the quantum group approach ([39, 25]), the Temperley-Lieb algebra ([30, 31]), the theory based on the Kauffman bracket ([4, 5]) or that obtained from the mapping

The moduli space M of semi-stable rank 2 bundles with trivial determinant over a complex curve \Sigma carries involutions naturally associated to 2-torsion points on the Jacobian of the curve. For every lift of a 2-torsion point to a 4-torsion point, we define a lift of the involution to the determinant line bundle L. We obtain an explicit presentation of the group generated by these lifts in terms of the order 4 Weil pairing. This is related to the triple intersections of the components of the fixed point sets in M , which we also determine completely using the order 4 Weil pairing. The lifted involutions act on the spaces of holomorphic sections of powers of L, whose dimensions are given by the Verlinde formula. We compute the characters of these vector spaces as representations of the group generated by our lifts, and we obtain an explicit isomorphism (as group representations) with the combinatorial-topological TQFT-vector spaces of [BHMV]. As an application, we describe a `brick...

Any closed oriented 3-manifold M can be equipped with some additional structures called complex spin structures, or Spin c-structures. Complex spin structures play a central rôle in Seiberg-Witten theory. In dimension 3, they are in canonical correspondence with Euler structures. The latter, which are classes of nonsingular vector fields on M, have been introduced by V. Turaev in order to refine Reidemeister torsions. Also, complex spin structures are related to the classical spin structures, or Spin-structures, in the sense that there exists a canonical map Spin(M) → Spin c (M), from the space of the Spin-structures on M to the space of its Spin c-structures. In this paper, we investigate the rôle played by quadratic functions in the topology of closed oriented 3-manifolds endowed with Spin c-structures. In particular, we explain how quadratic functions emerge from a Spin c-refinement of the Goussarov-Habiro theory of finite type invariants. A quadratic function on a torsion Abelian group G is a function q: G → Q/Z such that the pairing b: G × G → Q/Z, defined for any x, y ∈ G by b(x, y) =

Abstract. A version of Kirby calculus for spin and framed threemanifolds is given and is used to construct invariants of spin and framed three-manifolds in two situations. The first is ribbon ∗-categories which possess odd degenerate objects. This case includes the quantum group situations corresponding to the half-integer level Chern-Simons theories conjectured to give spin TQFTs by Dijkgraaf and Witten [10]. In particular, the spin invariants constructed by Kirby and Melvin [21] are shown to be identical to the invariants associated to SO(3). Second, an invariant of spin manifolds analogous to the Hennings invariant is constructed beginning with an arbitrary factorizable, unimodular quasitriangular Hopf algebra. In particular a framed manifold invariant is associated to every finite-dimensional Hopf algebra via its quantum double, and is conjectured to be identical to Kuperberg’s noninvolutory invariant of framed manifolds associated to that Hopf algebra.

Abstract. We construct four series of modular categories from the two-variable Kauffman polynomial, without use of the representation theory of quantum groups at roots of unity. The specializations of this polynomial corresponding to quantum groups of types B, C and D produce series of pre-modular categories. One of them turns out to be modular and three others satisfy Bruguières ’ modularization criterion. For these four series we compute the Verlinde formulas, and discuss spin and cohomological refinements.

additional algebraic features. The interest of this concept is that it provides a Topological Quantum Field Theory in dimension 3. The Verlinde formulas associated with a modular category are the dimensions of the TQFT modules. We discuss reductions and refinements of these formulas for modular categories related with SU(N). Our main result is a splitting of the Verlinde formula, corresponding to a brick decomposition of the TQFT modules whose summands are indexed by spin structures modulo an even integer. We introduce here the notion of a spin modular category, and give the proof of the decomposition theorem in this general context. Given a simple, simply connected complex Lie group G, the Verlinde formula [37] is a combinatorial function VG: (K, g) ↦ → VG(K, g) associated with G (here the integers K and g are respectively the level and the genus). In conformal field theory this formula gives the dimension

Abstract. In this article we construct a combinatorial counterpart of the action of a certain Heisenberg group on the space of conformal blocks which was studied by Andersen-Masbaum [1] and Blanchet-Habegger-Masbaum-Vogel [5]. It is a non-abelian analogue of the description of the Heisenberg action on the space of theta functions. 1.

The workshop and seminars on “Invariants of knots and 3-manifolds ” was held at

Let M be the manifold obtained by 0-framed surgery along a knot K in the 3-sphere. A Topological Quantum Field Theory assigns to a fundamental domain of the universal abelian cover of M an operator, whose non-nilpotent part is the Turaev-Viro module of K. In this paper, using surgery formulas, we give a matrix presentation for the Turaev-Viro module of any knot K, in the case of the (Vp ; Zp ) TQFT of Blanchet, Habegger, Masbaum and Vogel. We do the computation for a family of knots in the special case p = 8, and note the relation with the fibering question. Keywords: Knot, abelian cover, 3-manifold, TQFT. Introduction Given a Topological Quantum Field Theory (V; Z) over a field in dimension 2 + 1, one can associate to a knot K in the 3-sphere an invariant, the TuraevViro module of K, which is somewhat analogous to the Alexander module of K. This can be described as follows. Let K be an oriented knot in S 3 and M = S 3 (K) be the manifold obtained by 0-framed surgery along K. Let...