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265
Feynman diagrams and lowdimensional topology, First European
 Prog. in Math. 120
, 1992
"... We shall describe a program here relating Feynman diagrams, topology of manifolds, homotopical algebra, noncommutative geometry and several kinds of “topological physics”. The text below consists of 3 parts. The first two parts (topological sigma model and ChernSimons theory) are formally independ ..."
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Cited by 163 (2 self)
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We shall describe a program here relating Feynman diagrams, topology of manifolds, homotopical algebra, noncommutative geometry and several kinds of “topological physics”. The text below consists of 3 parts. The first two parts (topological sigma model and ChernSimons theory) are formally independent and could be read separately. The third part describes the common algebraic background of both theories. Conventions Later on we shall use almost all the time the language of super linear algebra, i.e., the word vector space often means Z/2Zgraded vector space and the degree of homogeneous vector v we denote by v̄. In almost all formulas, one can replace C by any field of characteristic zero. By graph we always mean finite 1dimensional CWcomplex. For g ≥ 0 and n ≥ 1 such that 2g + n> 2, we denote by Mg,n the coarse moduli space of smooth complex algebraic curves of genus g with n unlabeled
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 146 (14 self)
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For a copy with the handdrawn figures please email
ON THE VASSILIEV KNOT INVARIANTS
, 1995
"... The theory of knot invariants of finite type (Vassiliev invariants) is described. These invariants turn out to be at least as powerful as the Jones polynomial and its numerous generalizations coming from various quantum groups, and it is conjectured that these invariants are precisely as powerful a ..."
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Cited by 144 (0 self)
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The theory of knot invariants of finite type (Vassiliev invariants) is described. These invariants turn out to be at least as powerful as the Jones polynomial and its numerous generalizations coming from various quantum groups, and it is conjectured that these invariants are precisely as powerful as those polynomials. As invariants of finite type are much easier to define and manipulate than the quantum group invariants, it is likely that in attempting to classify knots, invariants of finite type will play a more fundamental role than the various knot polynomials.
Chern–Simons Perturbation Theory
 II,” J. Diff. Geom
, 1994
"... Abstract. We study the perturbation theory for three dimensional Chern–Simons quantum field theory on a general compact three manifold without boundary. We show that after a simple change of variables, the action obtained by BRS gauge fixing in the Lorentz gauge has a superspace formulation. The bas ..."
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Cited by 133 (2 self)
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Abstract. We study the perturbation theory for three dimensional Chern–Simons quantum field theory on a general compact three manifold without boundary. We show that after a simple change of variables, the action obtained by BRS gauge fixing in the Lorentz gauge has a superspace formulation. The basic properties of the propagator and the Feynman rules are written in a precise manner in the language of differential forms. Using the explicit description of the propagator singularities, we prove that the theory is finite. Finally the anomalous metric dependence of the 2loop partition function on the Riemannian metric (which was introduced to define the gauge fixing) can be cancelled by a local counterterm as in the 1loop case [28]. In fact, the counterterm is equal to the Chern–Simons action of the metric connection, normalized precisely as one would expect based on the framing dependence of Witten’s exact solution.
On The MelvinMortonRozansky Conjecture
, 1994
"... . We prove a conjecture stated by Melvin and Morton (and elucidated further by Rozansky) saying that the AlexanderConway polynomial of a knot can be read from some of the coefficients of the Jones polynomials of cables of that knot (i.e., coefficients of the "colored" Jones polynomial) ..."
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Cited by 85 (15 self)
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. We prove a conjecture stated by Melvin and Morton (and elucidated further by Rozansky) saying that the AlexanderConway polynomial of a knot can be read from some of the coefficients of the Jones polynomials of cables of that knot (i.e., coefficients of the "colored" Jones polynomial). We first reduce the problem to the level of weight systems using a general principle, which may be of some independent interest, and which sometimes allows to deduce equality of Vassiliev invariants from the equality of their weight systems. We then prove the conjecture combinatorially on the level of weight systems. Finally, we prove a generalization of the MelvinMortonRozansky (MMR) conjecture to knot invariants coming from arbitrary semisimple Lie algebras. As side benefits we discuss a relation between the Conway polynomial and immanants and a curious formula for the weight system of the colored Jones polynomial. Contents 1. Introduction 2 1.1. The conjecture 1.2. Preliminaries 1....
Spiders for rank 2 Lie algebras
 Commun. Math. Phys
, 1996
"... Abstract. A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or grouplike object. It is also known as a spherical category, or a strict, monoidal category with a few extra properties, or by several other names. A recently useful point o ..."
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Cited by 69 (1 self)
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Abstract. A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or grouplike object. It is also known as a spherical category, or a strict, monoidal category with a few extra properties, or by several other names. A recently useful point of view, developed by other authors, of the representation theory of sl(2) has been to present it as a spider by generators and relations. That is, one has an algebraic spider, defined by invariants of linear representations, and one identifies it as isomorphic to a combinatorial spider, given by generators and relations. We generalize this approach to the rank 2 simple Lie algebras, namely A2, B2, and G2. Our combinatorial rank 2 spiders yield bases for invariant spaces which are probably related to Lusztig’s canonical bases, and they are useful for computing quantities such as generalized 6jsymbols and quantum link invariants. Their definition originates in definitions of the rank 2 quantum link invariants that were discovered independently by the author and Francois Jaeger. 1.
Models of Sharing Graphs: A Categorical Semantics of let and letrec
, 1997
"... To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sha ..."
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Cited by 63 (9 self)
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To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sharing graphs. The simplest is firstorder acyclic sharing graphs represented by letsyntax, and others are extensions with higherorder constructs (lambda calculi) and/or cyclic sharing (recursive letrec binding). For each of four settings, we provide the equational theory for representing the sharing graphs, and identify the class of categorical models which are shown to be sound and complete for the theory. The emphasis is put on the algebraic nature of sharing graphs, which leads us to the semantic account of them. We describe the models in terms of the notions of symmetric monoidal categories and functors, additionally with symmetric monoidal adjunctions and traced
Spin networks in nonperturbative quantum gravity, in The Interface of Knots and
, 1996
"... A spin network is a generalization of a knot or link: a graph embedded in space, with edges labelled by representations of a Lie group, and vertices labelled by intertwining operators. Such objects play an important role in 3dimensional topological quantum field theory, functional integration on th ..."
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Cited by 50 (8 self)
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A spin network is a generalization of a knot or link: a graph embedded in space, with edges labelled by representations of a Lie group, and vertices labelled by intertwining operators. Such objects play an important role in 3dimensional topological quantum field theory, functional integration on the space A/G of connections modulo gauge transformations, and the loop representation of quantum gravity. Here, after an introduction to the basic ideas of nonperturbative canonical quantum gravity, we review a rigorous approach to functional integration on A/G in which L 2 (A/G) is spanned by states labelled by spin networks. Then we explain the ‘new variables ’ for general relativity in 4dimensional spacetime and describe how canonical quantization of gravity in this formalism leads to interesting applications of these spin network states. 1
Categorical Construction of 4D Topological Quantum Field Theories
 in Quantum Topology, L.H. Kauffman and R.A. Baadhio, eds., World Scientific
, 1993
"... In recent years, it has been discovered that invariants of three dimensional ..."
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Cited by 49 (7 self)
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In recent years, it has been discovered that invariants of three dimensional
Recursion from Cyclic Sharing: Traced Monoidal Categories and Models of Cyclic Lambda Calculi
, 1997
"... . Cyclic sharing (cyclic graph rewriting) has been used as a practical technique for implementing recursive computation efficiently. To capture its semantic nature, we introduce categorical models for lambda calculi with cyclic sharing (cyclic lambda graphs), using notions of computation by Moggi / ..."
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Cited by 45 (5 self)
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. Cyclic sharing (cyclic graph rewriting) has been used as a practical technique for implementing recursive computation efficiently. To capture its semantic nature, we introduce categorical models for lambda calculi with cyclic sharing (cyclic lambda graphs), using notions of computation by Moggi / Power and Robinson and traced monoidal categories by Joyal, Street and Verity. The former is used for representing the notion of sharing, whereas the latter for cyclic data structures. Our new models provide a semantic framework for understanding recursion created from cyclic sharing, which includes traditional models for recursion created from fixed points as special cases. Our cyclic lambda calculus serves as a uniform language for this wider range of models of recursive computation. 1 Introduction One of the traditional methods of interpreting a recursive program in a semantic domain is to use the least fixedpoint of continuous functions. However, in the real implementations of program...