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429
Feynman diagrams and lowdimensional topology
, 2006
"... 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 237 (3 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.
New points of view in knot theory
 Bull. Am. Math. Soc., New Ser
, 1993
"... In this article we shall give an account of certain developments in knot theory which followed upon the discovery of the Jones polynomial [Jo3] in 1984. The focus of our account will be recent glimmerings of understanding of the topological meaning of the new invariants. A second theme will be the c ..."
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Cited by 117 (0 self)
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In this article we shall give an account of certain developments in knot theory which followed upon the discovery of the Jones polynomial [Jo3] in 1984. The focus of our account will be recent glimmerings of understanding of the topological meaning of the new invariants. A second theme will be the central role that braid
Multidimensional hypergeometric functions in conformal field theory, algebraic Ktheory, algebraic geometry
 ICM
"... Rudolf Arnheim in the book Visual Thinking (LA. 1969) writes that usually concepts tend to crystallize into simple, wellshaped forms. They are tempted by Platonic rigidity. This creates troubles when the range they are intended to cover includes relevant qualitative differences. The variations can ..."
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Cited by 96 (23 self)
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Rudolf Arnheim in the book Visual Thinking (LA. 1969) writes that usually concepts tend to crystallize into simple, wellshaped forms. They are tempted by Platonic rigidity. This creates troubles when the range they are intended to cover includes relevant qualitative differences. The variations can be so different from each other that to see them as belonging to one family of phenomena requires mature understanding. To the young mind, they look as different from each other as did the morning star from the evening star to the ancients. The notion of a general hypergeometric function was introduced by I.M. Gelfand in the mid 80s. Now it is clear that general hypergeometric functions play a major role in interesting parts of mathematics such as Conformai Field Theory, Representation Theory, Algebraic KTheory, Algebraic Geometry and provide new connections among them. The general hypergeometric functions are generalizations of the Euler betafunction. The betafunction is the integral of a product of powers of linear functions over the segment. In the generalization the segment is replaced by a polytope and the integralI I(A,fi*) = \ fp...fj»dx1...dxn is considered as a function of the polytope A a R", the linear functions {f}, and the exponents {aj c C. The simplest examples are the classical hypergeometric function, the Euler dilogarithm, the volume of a polytope. The systematic study of the general hypergeometric functions was begun only recently in works of I.M. Gelfand's school and K. Aomoto. There are three basic reasons for the appearance of general hypergeometric functions: the general hypergeometric functions satisfy remarkable differential equations, the general hypergeometric functions satisfy remarkable functional equations, the general hypergeometric functions, as analytic functions of their arguments, have remarkable monodromy groups.
Anyons in an exactly solved model and beyond
, 2005
"... A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge f ..."
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Cited by 87 (2 self)
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A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are nonAbelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and nonAbelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.
From subfactors to categories and topology III. Triangulation invariants of 3manifolds and Morita equivalence of tensor categories
 In preparation
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Threedimensional quantum gravity, ChernSimons theory, and the Apolynomial
, 2003
"... We study threedimensional ChernSimons theory with complex gauge group SL(2,C), which has many interesting connections with threedimensional quantum gravity and geometry of hyperbolic 3manifolds. We show that, in the presence of a single knotted Wilson loop in an infinitedimensional representati ..."
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Cited by 77 (10 self)
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We study threedimensional ChernSimons theory with complex gauge group SL(2,C), which has many interesting connections with threedimensional quantum gravity and geometry of hyperbolic 3manifolds. We show that, in the presence of a single knotted Wilson loop in an infinitedimensional representation of the gauge group, the classical and quantum properties of such theory are described by an algebraic curve called the Apolynomial of a knot. Using this approach, we find some new and rather surprising relations between the Apolynomial, the colored Jones polynomial, and other invariants of hyperbolic 3manifolds. These relations generalize the volume conjecture and the MelvinMortonRozansky conjecture, and suggest an intriguing connection between the SL(2,C) partition function and the colored Jones polynomial.
Invariants of piecewiselinear 3manifolds
 Trans. Amer. Math. Soc
, 1996
"... Abstract. This paper presents an algebraic framework for constructing invariants of closed oriented 3manifolds by taking a state sum model on a triangulation. This algebraic framework consists of a tensor category with a condition on the duals which we have called a spherical category. A signican ..."
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Cited by 72 (5 self)
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Abstract. This paper presents an algebraic framework for constructing invariants of closed oriented 3manifolds by taking a state sum model on a triangulation. This algebraic framework consists of a tensor category with a condition on the duals which we have called a spherical category. A signicant feature is that the tensor category is not required to be braided. The main examples are constructed from the categories of representations of involutive Hopf algebras and of quantised enveloping algebras at a root of unity. The purpose of this paper is to present an algebraic framework for constructing invariants of closed oriented 3manifolds. The construction is in the spirit of topological quantum eld theory and the invariant is calculated from a triangulation of the 3manifold. The data for the construction of the invariant is a tensor category
Topological Mtheory as Unification of Form Theories of Gravity
, 2004
"... We introduce a notion of topological Mtheory and argue that it provides a unification of form theories of gravity in various dimensions. Its classical solutions involve G2 holonomy metrics on 7manifolds, obtained from a topological action for a 3form gauge field introduced by Hitchin. We show tha ..."
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Cited by 70 (6 self)
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We introduce a notion of topological Mtheory and argue that it provides a unification of form theories of gravity in various dimensions. Its classical solutions involve G2 holonomy metrics on 7manifolds, obtained from a topological action for a 3form gauge field introduced by Hitchin. We show that by reductions of this 7dimensional theory one can classically obtain 6dimensional topological A and B models, the topological sector of loop quantum gravity in 4 dimensions, and ChernSimons gravity in 3 dimensions. We also find that the 7dimensional Mtheory perspective sheds some light on the fact that the topological string partition function is a wavefunction, as well as on Sduality between the A and B models. The degrees of freedom of the A and B models appear as conjugate variables in the 7dimensional theory. Finally, from the topological Mtheory perspective we find hints of an intriguing holographic link between nonsupersymmetric YangMills in
Tensor products of modules for a vertex operator algebras and vertex tensor categories
 in: Lie Theory and Geometry, in honor of Bertram Kostant
, 1994
"... In this paper, we present a theory of tensor products of classes of modules for a vertex operator algebra. We focus on motivating and explaining new structures and results in this theory, rather than on proofs, which are being presented in a series of papers beginning with [HL4] and [HL5]. An announ ..."
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Cited by 70 (11 self)
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In this paper, we present a theory of tensor products of classes of modules for a vertex operator algebra. We focus on motivating and explaining new structures and results in this theory, rather than on proofs, which are being presented in a series of papers beginning with [HL4] and [HL5]. An announcement has also appeared [HL1].