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In 1999, Rozansky conjectured the existence of a rational presentation of the Kontsevich integral of a knot. Roughly speaking, this rational presentation of the Kontsevich integral would sum formal power series into rational functions with prescribed denominators. Rozansky’s conjecture was soon proven by the first author. We begin our paper by reviewing Rozansky’s conjecture and the main ideas that lead to its proof. The natural question of extending this conjecture to links leads to the class of boundary links, and a proof of Rozansky’s conjecture in this case. A subtle issue is the fact that a “hair ” map which replaces beads by the exponential of hair is not 1-1. This raises the question of whether a rational invariant of boundary links exists in an appropriate space of trivalent graphs whose edges are decorated by rational functions in noncommuting variables. A main result of the paper is to construct such an invariant, using the so-called surgery view of bounadry links and after developing a formal diagrammatic Gaussian integration. Since our invariant is one of many rational forms of the Kontsevich integral, one may ask if our invariant is in some sense canonical. We prove that this is indeed the case, by axiomatically characterizing our invariant as a universal finite type invariant of boundary links with respect to the null move. Finally, we discuss relations between our rational invariant and homology surgery, and give some applications to low dimensional topology.

In [2], [3] we suggested a new quantum algebra, the moduli algebra, which is conjectured to be a quantum algebra of observables of the Hamiltonian Chern-Simons theory. This algebra provides the quantization of the algebra of functions on the moduli space of flat connections on a 2-dimensional surface. In this paper we classify unitary representations of this new algebra and identify the corresponding representation spaces with the spaces of conformal blocks of the WZW model. The mapping class group of the surface is proved to act on the moduli algebra by inner automorphisms. The generators of these automorphisms are unitary elements of the moduli algebra. They are constructed explicitly and proved to satisfy the relations of the (unique) central extension of the mapping class group.

We show that the tree-level part of a theory of finite type invariants of 3-manifolds (based on surgery on objects called claspers, Y-graphs or clovers) is essentially given by classical algebraic topology in terms of the Johnson homomorphism and Massey products, for arbitrary 3-manifolds. A key role of our proof is played by the notion of a homology cylinder, viewed as an enlargement of the mapping class group, and an apparently new Lie algebra of graphs colored by H1(Σ) of a closed surface Σ, closely related to deformation quantization on a surface [AMR1, AMR2, Ko3] as well as to a Lie algebra that encodes the symmetries of Massey products and the Johnson homomorphism. In addition, we give a realization theorem for Massey products and the Johnson homomorphism by homology cylinders.

The string bracket introduced by Chas and Sullivan is reinterpreted from the point of view of topological field theories in the Batalin–Vilkovisky or BRST formalisms. Namely, topological action functionals for gauge fields (generalizing Chern–Simons and BF theories) are considered together with generalized Wilson loops. The latter generate a (Poisson or Gerstenhaber) algebra of functionals with values in the S 1-equivariant cohomology of the loop space of the manifold on which the theory is defined. It is proved that, in the case of GL(n, C) with standard representation, the (Poisson or BV) bracket of two generalized Wilson loops applied to two cycles is the same as the generalized Wilson loop applied to the string bracket of the cycles. Generalizations to other groups are briefly described. 1 1

The Kauffman bracket skein module is a deceptively simple construction, occurring naturally in several fields of mathematics and physics. This paper is a survey of the various ways in which it is a quantization of a classical object. Przytycki [38] and Turaev [46] introduced skein modules. Shortly thereafter, Turaev [47] discovered that they formed quantizations of loop algebras; further work in this direction was done by Hoste and Przytycki [25]. We will look at some of the heuristic reasons for treating skein modules as deformations, and then realize the Kauffman bracket module as a precise quantization in two different ways. Traditionally, this is done by locating a non-commutative algebra that deforms a commutative algebra in a manner coherent with a Poisson structure. The importance of the Kauffman bracket skein module began to emerge from its relationship with SL2(C) invariant theory. It is well known that the SL2(C)-characters of a surface group form a Poisson algebra [6, 22]. The skein module is the appropriate deformation. The idea of a lattice gauge field theory quantization of surface group characters

Let G be a compact connected and simply connected Lie group, and Σ be a compact topological Riemann surface with a point p marked on it. One can associate to this data the moduli space of flat G connections on the punctured Riemann surface Σ denoted by M G = M G [Σp]. This

Abstract. For each connected complex reductive group G, we find a family of new examples of complex quasi-Hamiltonian G-spaces with G-valued moment maps. These spaces arise naturally as moduli spaces of (suitably framed) meromorphic connections on principal G-bundles over a disc, and they generalise the conjugacy class example of Alekseev–Malkin–Meinrenken (which appears in the simple pole case). Using the ‘fusion product ’ in the theory this gives a finite dimensional construction of the natural symplectic structures on the spaces of monodromy/Stokes data of meromorphic connections over arbitrary genus Riemann surfaces, together with a new proof of the symplectic nature of isomonodromic deformations of such connections. 1.

. In these notes we discuss some aspects of 2D Yang-Mills theory and its relation to invariants of knots in circle bundles over surfaces. We show how it is related to the quantization of the algebra of functions on the moduli space flat bundles over a surface. An easy proof is given that the partition function of the Y M 2 with Wilson loops can be regarded as the integral of a function over the moduli space of flat bundles over surface with respect to the symplectic volume form. It is also shown that there is an interesting relation between Y M 2 and invariants of plain curves, in particular with Arnold's J + - invariant. 1. Introduction Two dimensional Yang-Mills theory (Y M 2 ) has been extensively studied in physical literature as one of the simplest models of quantum field theory and the simplest model with gauge invariance. Then there was a revival of interest to Y M 2 , related mostly to its interpretation as a string theory [13], [14], [19]. In dimension 2 Yang-Mills theory i...

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