Joanna Kania-Bartoszynska

3-manifold invariants and periodicity of homology spheres.

We show how the periodicity of a homology sphere is reflected in the Reshetikhin-Turaev-Witten invariants of the manifold. These yield a criterion for the periodicity of a homology sphere.

TOPOLOGICAL INTERPRETATIONS OF LATTICE GAUGE FIELD THEORY

Abstract:We construct lattice gauge field theory based on a quantum group on a lattice of dimension one. Innovations include a coalgebra structure on the connections and an investigation of connections that are not distinguishable by observables. We prove that when the quantum group is a deformation of a connected algebraic group G (over the complex numbers), then the algebra of observables forms a deformation quantization of the ring of

A quantum obstruction to embedding

G

The Kauffman bracket skein as an algebra of observables

-characters of the fundamental group of the lattice. Finally, we investigate lattice gauge field theory based on quantum SL2ℂ, and conclude that the algebra of observables is the Kauffman bracket skein module of a cylinder over a surface associated to the lattice.

THE QUANTUM CONTENT OF THE NORMAL SURFACES IN A THREE-MANIFOLD

An invariant of a three-manifold with boundary is derived from topological quantum field theory. This invariant is then used as an obstruction to embedding one 3-manifold into another.

Shadow world evaluation of the Yang-Mills measure

We prove that the Kauffman bracket skein algebra of a cylinder over a surface with boundary, defined over complex numbers, is isomorphic to the observables of an appropriate lattice gauge field theory.

Unicity for Representations of the Kauffman bracket Skein Algebra

The formula for the Turaev–Viro invariant of a three-manifold depends on a complex parameter t. When t is not a root of unity, the formula becomes an infinite sum. This paper analyzes convergence of this sum when t does not lie on the unit circle, in the presence of an efficient triangulation of the three-manifold. The terms of the sum can be indexed by surfaces lying in the three-manifold. The contribution of a surface is largest when the surface is normal and when its genus is the lowest.

Skein quantization and lattice gauge field theory

A new state-sum formula for the evaluation of the Yang-Mills measure in the Kauffman bracket skein algebra of a closed surface is derived. The formula extends the Kauffman bracket to diagrams that lie in surfaces other than the plane. It also extends Turaevs shadow world invariant of links in a circle bundle over a surface away from roots of unity. The limiting behavior of the Yang-Mills measure when the complex parameter approaches −1 is studied. The formula is applied to compute integrals of simple closed curves over the character variety of the surface against Goldmans symplectic measure. AMS Classification 57M27; 57R56, 81T13

Dehn’s algorithm for simple diagrams

This paper resolves the unicity conjecture of Bonahon and Wong for the Kauffman bracket skein algebras of all oriented finite type surfaces at all roots of unity. The proof is a consequence of a general unicity theorem that says that the irreducible representations of a prime affine k-algebra over an algebraically closed field k, that is finitely generated as a module over its center, are generically classified by their central characters. The center of the Kauffman bracket skein algebra of any orientable surface at any root of unity is characterized, and it is proved that the skein algebra is finitely generated as a module over its center. It is shown that for any orientable surface the center of the skein algebra at any root of unity is the coordinate ring of an affine algebraic variety.

The Yang-Mills measure in the Kauffman bracket skein module

This is a survey article describing the various ways in which the Kauffman bracket skein module is a quantization of surface group characters. These include a purely heuristic sense of deformation of a presentation, a Poisson quantization, and a lattice gauge field theory quantization.