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24
Higherdimensional algebra and topological quantum field theory
 Jour. Math. Phys
, 1995
"... For a copy with the handdrawn figures please email ..."
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Cited by 140 (14 self)
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For a copy with the handdrawn figures please email
A rational noncommutative invariant of boundary links
 GEOM. AND TOPOLOGY
, 2003
"... 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 pr ..."
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Cited by 42 (12 self)
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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 11. 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 socalled 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.
Representation Theory of ChernSimons Observables
, 1995
"... 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 ChernSimons theory. This algebra provides the quantization of the algebra of functions on the moduli space of flat connections on a 2dimensional surfac ..."
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Cited by 33 (0 self)
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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 ChernSimons theory. This algebra provides the quantization of the algebra of functions on the moduli space of flat connections on a 2dimensional 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.
Tree level invariants of threemanifolds, massey products and the Johnson homomorphism
, 1999
"... We show that the treelevel part of a theory of finite type invariants of 3manifolds (based on surgery on objects called claspers, Ygraphs or clovers) is essentially given by classical algebraic topology in terms of the Johnson homomorphism and Massey products, for arbitrary 3manifolds. A key r ..."
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Cited by 16 (0 self)
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We show that the treelevel part of a theory of finite type invariants of 3manifolds (based on surgery on objects called claspers, Ygraphs or clovers) is essentially given by classical algebraic topology in terms of the Johnson homomorphism and Massey products, for arbitrary 3manifolds. 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.
Topological field theory interpretation of string topology
 Comm. Math. Phys
"... 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 gene ..."
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Cited by 14 (3 self)
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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 1equivariant 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
Trace functionals on noncommutative deformations of moduli spaces of flat connections
, 8
"... 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 ..."
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Cited by 5 (1 self)
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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
Skein quantization and lattice gauge field theory
"... 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 t ..."
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Cited by 4 (2 self)
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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 noncommutative 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
YangMills Theory And Invariants Of Links
 Math. Phys. Studies
, 1997
"... . In these notes we discuss some aspects of 2D YangMills 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 partiti ..."
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Cited by 4 (1 self)
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. In these notes we discuss some aspects of 2D YangMills 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 YangMills 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 YangMills theory i...
QuasiHamiltonian Geometry of Meromorphic Connections
"... Abstract. For each connected complex reductive group G, we find a family of new examples of complex quasiHamiltonian Gspaces with Gvalued moment maps. These spaces arise naturally as moduli spaces of (suitably framed) meromorphic connections on principal Gbundles over a disc, and they generalise ..."
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Cited by 4 (0 self)
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Abstract. For each connected complex reductive group G, we find a family of new examples of complex quasiHamiltonian Gspaces with Gvalued moment maps. These spaces arise naturally as moduli spaces of (suitably framed) meromorphic connections on principal Gbundles 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.
The necklace Lie coalgebra and renormalization algebras
 J. Noncommut. Geom
"... We give a natural monomorphism from the necklace Lie coalgebra, defined for any quiver, to Connes and Kreimer’s Lie coalgebra of trees, and extend this to a map from a certain quivertheoretic Hopf algebra to Connes and Kreimer’s renormalization Hopf algebra, as well as to preLie versions. These re ..."
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Cited by 2 (1 self)
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We give a natural monomorphism from the necklace Lie coalgebra, defined for any quiver, to Connes and Kreimer’s Lie coalgebra of trees, and extend this to a map from a certain quivertheoretic Hopf algebra to Connes and Kreimer’s renormalization Hopf algebra, as well as to preLie versions. These results are direct analogues of Turaev’s results in 2004, by replacing algebras of loops on surfaces with algebras of paths on quivers. We also factor the morphism through an algebra of chord diagrams and explain the geometric version. We then explain how all of the Hopf algebras are uniquely determined by the preLie structures, and discuss noncommutative versions of the Hopf algebras. 1