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247
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 92 (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
Topological Mtheory as Unification of Form Theories of Gravity, arXiv:hepth/0411073
"... 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 59 (7 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
and Category: is quantum gravity algebraic
 Journal of Mathematical Physics
, 1995
"... ABSTRACT: We investigate the possibility that the quantum theory of gravity could be constructed discretely using algebraic methods. The algebraic tools are similar to ones used in constructing Topological Quantum Field theories. The algebraic structures are related to ideas about the reinterpretati ..."
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Cited by 51 (3 self)
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ABSTRACT: We investigate the possibility that the quantum theory of gravity could be constructed discretely using algebraic methods. The algebraic tools are similar to ones used in constructing Topological Quantum Field theories. The algebraic structures are related to ideas about the reinterpretation of quantum mechanics in a general relativistic context. I.
On algebraic structures implicit in topological quantum field theories
 J. Knot Theory Ramifications
, 1999
"... In the course of the development of our understanding of topological quantum field theory (TQFT) [1,2], it has emerged that the structures of generators and relations for the construction of low dimensional TQFTs by various combinatorial methods are equivalent to the structures of various fundamenta ..."
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Cited by 48 (2 self)
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In the course of the development of our understanding of topological quantum field theory (TQFT) [1,2], it has emerged that the structures of generators and relations for the construction of low dimensional TQFTs by various combinatorial methods are equivalent to the structures of various fundamental objects in abstract algebra.
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 44 (5 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].
Orbifold subfactors from Hecke algebras
 Comm. Math. Phys
, 1994
"... A. Ocneanu has observed that a mysterious orbifold phenomenon occurs in the system of the M∞M ∞ bimodules of the asymptotic inclusion, a subfactor analogue of the quantum double, of the Jones subfactor of type A2n+1. We show that this is a general phenomenon and identify some of his orbifolds with ..."
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Cited by 39 (23 self)
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A. Ocneanu has observed that a mysterious orbifold phenomenon occurs in the system of the M∞M ∞ bimodules of the asymptotic inclusion, a subfactor analogue of the quantum double, of the Jones subfactor of type A2n+1. We show that this is a general phenomenon and identify some of his orbifolds with the ones in our sense as subfactors given as simultaneous fixed point algebras by working on the Hecke algebra subfactors of type A of Wenzl. That is, we work on their asymptotic inclusions and show that the M∞M ∞ bimodules are described by certain orbifolds (with ghosts) for SU(3)3k. We actually compute several examples of the (dual) principal graphs of the asymptotic inclusions. As a corollary of the identification of Ocneanu’s orbifolds with ours, we show that a nondegenerate braiding exists on the even vertices of D2n, n>2. 1
Involutory Hopf algebras and 3manifold invariants
 Intern. J. Math
, 1991
"... We establish a 3manifold invariant for each finitedimensional, involutory Hopf algebra. If the Hopf algebra is the group algebra of a group G, the invariant counts homomorphisms from the fundamental group of the manifold to G. The invariant can be viewed as a state model on a Heegaard diagram or a ..."
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Cited by 39 (4 self)
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We establish a 3manifold invariant for each finitedimensional, involutory Hopf algebra. If the Hopf algebra is the group algebra of a group G, the invariant counts homomorphisms from the fundamental group of the manifold to G. The invariant can be viewed as a state model on a Heegaard diagram or a triangulation of the manifold. The computation of the invariant involves tensor products and contractions of the structure tensors of the algebra. We show that every formal expression involving these tensors corresponds to a unique 3manifold modulo a wellunderstood equivalence. This raises the possibility of an algorithm which can determine whether two given 3manifolds are homeomorphic. 1
Modular forms and quantum invariants of 3manifolds
 Asian J. Math
, 1999
"... 1. Introduction. The WittenReshetikhinTuraev (WRT) invariant of a compact connected oriented 3manifold M may be formally defined by [16] ..."
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Cited by 35 (1 self)
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1. Introduction. The WittenReshetikhinTuraev (WRT) invariant of a compact connected oriented 3manifold M may be formally defined by [16]
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.
Hecke algebras, modular categories and 3manifolds quantum invariants
 Topology
, 2000
"... Abstract. We construct modular categories from Hecke algebras at roots of unity. For a special choice of the framing parameter, we recover the ReshetikhinTuraev invariants of closed 3manifolds constructed from the quantum groups Uqsl(N) by ReshetikhinTuraev and TuraevWenzl, and from skein theory ..."
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Cited by 33 (3 self)
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Abstract. We construct modular categories from Hecke algebras at roots of unity. For a special choice of the framing parameter, we recover the ReshetikhinTuraev invariants of closed 3manifolds constructed from the quantum groups Uqsl(N) by ReshetikhinTuraev and TuraevWenzl, and from skein theory by Yokota. The possibility of such a construction was suggested by Turaev, as a consequence of SchurWeil duality. We then discuss the choice of the framing parameter. This leads, for any rank N and level K, to a modular category ˜ H N,K and a reduced invariant ˜τN,K. If N and K are coprime, then this invariant coincides with the known invariant τ PSU(N) at level K. If gcd(N, K) = d> 1, then we show that the reduced invariant admits spin or cohomological refinements, with a nice decomposition formula which extends a theorem of H. Murakami.