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557
Notes on superconformal ChernSimonsmatter theories
 JHEP 0708, 056 (2007) [arXiv:0704.3740 [hepth
"... The three dimensional N = 2 supersymmetric ChernSimons theory coupled to matter fields, possibly deformed by a superpotential, give rise to a large class of exactly conformal theories with Lagrangian descriptions. These theories can be arbitrarily weakly coupled, and hence can be studied perturbati ..."
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Cited by 56 (3 self)
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The three dimensional N = 2 supersymmetric ChernSimons theory coupled to matter fields, possibly deformed by a superpotential, give rise to a large class of exactly conformal theories with Lagrangian descriptions. These theories can be arbitrarily weakly coupled, and hence can be studied perturbatively. We study the theories in the large N limit, and compute the twoloop anomalous dimension of certain long operators. Our result suggests that various N = 2 U(N) ChernSimons theories coupled to suitable matter fields are dual to open or closed string theories in AdS4, which are not yet constructed.
An invariant of integral homology 3spheres which is universal for all finite type invariants, preprint
, 1996
"... Abstract. In [LMO] a 3manifold invariant Ω(M) is constructed using a modification of the Kontsevich integral and the Kirby calculus. The invariant Ω takes values in a graded Hopf algebra of Feynman 3valent graphs. Here we show that for homology 3spheres the invariant Ω is universal for all finite ..."
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Cited by 54 (4 self)
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Abstract. In [LMO] a 3manifold invariant Ω(M) is constructed using a modification of the Kontsevich integral and the Kirby calculus. The invariant Ω takes values in a graded Hopf algebra of Feynman 3valent graphs. Here we show that for homology 3spheres the invariant Ω is universal for all finite type invariants, i.e. Ωn is an invariant order 3n which dominates all other invariants of the same order. Some corollaries are discussed. 1.
Quantum gravity with a positive cosmological constant
, 2002
"... A quantum theory of gravity is described in the case of a positive cosmological constant in 3 + 1 dimensions. Both old and new results are described, which support the case that loop quantum gravity provides a satisfactory quantum theory of gravity. These include the existence of a ground state, dis ..."
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Cited by 51 (9 self)
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A quantum theory of gravity is described in the case of a positive cosmological constant in 3 + 1 dimensions. Both old and new results are described, which support the case that loop quantum gravity provides a satisfactory quantum theory of gravity. These include the existence of a ground state, discoverd by Kodama, which both is an exact solution to the constraints of quantum gravity and has a semiclassical limit which is deSitter spacetime. The long wavelength excitations of this state are studied and are shown to reproduce both gravitons and, when matter is included, quantum field theory on deSitter spacetime. Furthermore, one may derive directly from the WheelerdeWitt equation corrections to the energymomentum relations for matter fields of the form E 2 = p 2 +m 2 +αlPlE 3 +... where α is a computable dimensionless constant. This may lead in the next few years to experimental tests of the theory. To study the excitations of the Kodama state exactly requires the use of the spin network representation, which is quantum deformed due to the cosmological constant. The theory may be developed within a single horizon, and the boundary states described exactly in terms of a boundary ChernSimons theory. The Bekenstein bound is recovered and the N bound of Banks is given a background independent explanation. The paper is written as an introduction to loop quantum gravity, requiring no prior knowledge of the subject. The deep relationship between quantum gravity and topological field theory is stressed throughout.
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 48 (7 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
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].
Quantization of Teichmuller spaces and the quantum dilogarithm, available as qalg/9705021
"... The Teichmüller space of punctured surfaces with the Weil–Petersson symplectic structure and the action of the mapping class group is realized as the Hamiltonian reduction of a finite dimensional symplectic space where the mapping class group acts by symplectic rational transformations. Upon quantiz ..."
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Cited by 44 (4 self)
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The Teichmüller space of punctured surfaces with the Weil–Petersson symplectic structure and the action of the mapping class group is realized as the Hamiltonian reduction of a finite dimensional symplectic space where the mapping class group acts by symplectic rational transformations. Upon quantization the corresponding (projective) representation of the mapping class group is generated by the quantum dilogarithms. ∗ On leave of absence from St. Petersburg Branch of the Steklov Mathematical Institute 1
The Verlinde algebra is twisted equivariant Ktheory
"... Ktheory in various forms has recently received much attention in 10dimensional superstring theory. Our raised consciousness about twisted Ktheory led to the serendipitous discovery that it enters in a different way into 3dimensional topological field theories, in particular ChernSimons theory. ..."
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Cited by 43 (4 self)
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Ktheory in various forms has recently received much attention in 10dimensional superstring theory. Our raised consciousness about twisted Ktheory led to the serendipitous discovery that it enters in a different way into 3dimensional topological field theories, in particular ChernSimons theory. Namely, as the title of the paper reports, the Verlinde algebra is a certain twisted Ktheory group. This assertion, and its proof, is joint work with Michael Hopkins and Constantin Teleman. The general theorem and proof will be presented elsewhere [FHT]; our goal here is to explain some background, demonstrate the theorem in a simple nontrivial case, and motivate it through the connection with topological field theory. From a mathematical point of view the Verlinde algebra is defined in the theory of loop groups. Let G be a compact Lie group. There is a version of the theorem for any compact group G, but here for the most part we focus on connected, simply connected, and simple groups—G = SU2 is the simplest example. In this case a central extension of the free loop group LG is determined by the level, which is a positive integer k. There is a finite set of equivalence classes of positive energy representations of this central extension; let Vk(G) denote the free abelian group they generate. One of the influences of 2dimensional conformal field theory on the theory of loop groups is the construction of an algebra structure on Vk(G), the fusion product. This is the Verlinde algebra [V]. Let G act on itself by conjugation. Then with our assumptions the equivariant cohomology group H3 G (G) is free of rank one. Let h(G) be the dual Coxeter number of G, and define ζ(k) ∈ H3 G (G) to be k + h(G) times a generator. We will see that elements of H3 may be used to twist Ktheory, and so elements of equivariant H 3 twist equivariant Ktheory. Theorem (FreedHopkinsTeleman). There is an isomorphism of algebras