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24
Higherdimensional algebra and topological quantum field theory
 Jour. Math. Phys
, 1995
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Cited by 140 (14 self)
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For a copy with the handdrawn figures please email
Higherdimensional algebra II: 2Hilbert spaces
"... A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also ..."
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Cited by 43 (13 self)
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A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also define monoidal, braided monoidal, and symmetric monoidal versions of 2Hilbert spaces, which we call 2H*algebras, braided 2H*algebras, and symmetric 2H*algebras, and we describe the relation between these and tangles in 2, 3, and 4 dimensions, respectively. We prove a generalized DoplicherRoberts theorem stating that every symmetric 2H*algebra is equivalent to the category Rep(G) of continuous unitary finitedimensional representations of some compact supergroupoid G. The equivalence is given by a categorified version of the Gelfand transform; we also construct a categorified version of the Fourier transform when G is a compact abelian group. Finally, we characterize Rep(G) by its universal properties when G is a compact classical group. For example, Rep(U(n)) is the free connected symmetric 2H*algebra on one even object of dimension n. 1
An introduction to ncategories
 In 7th Conference on Category Theory and Computer Science
, 1997
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Symmetry, gravity and noncommutativity
"... We review some aspects of the implementation of spacetime symmetries in noncommutative field theories, emphasizing their origin in string theory and how they may be used to construct theories of gravitation. The geometry of canonical noncommutative gauge transformations is analysed in detail and it ..."
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Cited by 18 (2 self)
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We review some aspects of the implementation of spacetime symmetries in noncommutative field theories, emphasizing their origin in string theory and how they may be used to construct theories of gravitation. The geometry of canonical noncommutative gauge transformations is analysed in detail and it is shown how noncommutative YangMills theory can be related to a gravity theory. The construction of twisted spacetime symmetries and their role in constructing a noncommutative extension of general relativity is described. We also analyse certain generic features of noncommutative gauge theories on Dbranes in curved spaces, treating several explicit examples of superstring backgrounds.
Quantum Lie Algebras of Type A n
, 1998
"... It is shown that the quantised enveloping algebra of sl(n) contains a quantum Lie algebra, defined by means of axioms similar to Woronowicz's. This gives rise to Lie algebralike generators and relations for the locally finite part of the quantised enveloping algebra, and suggests a canonical Poinca ..."
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Cited by 11 (2 self)
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It is shown that the quantised enveloping algebra of sl(n) contains a quantum Lie algebra, defined by means of axioms similar to Woronowicz's. This gives rise to Lie algebralike generators and relations for the locally finite part of the quantised enveloping algebra, and suggests a canonical PoincaréBirkhoffWitt basis.
Drinfeld coproduct, quantum fusion tensor category and applications
, 2006
"... The class of quantum affinizations (or quantum loop algebras, see [Dr2, CP3, GKV, VV2, Mi1, N1, Jin, H3]) includes quantum affine algebras and quantum toroidal algebras. In general they have no Hopf algebra structure, but have a “coproduct ” (the Drinfeld coproduct) which does not produce tensor p ..."
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Cited by 11 (5 self)
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The class of quantum affinizations (or quantum loop algebras, see [Dr2, CP3, GKV, VV2, Mi1, N1, Jin, H3]) includes quantum affine algebras and quantum toroidal algebras. In general they have no Hopf algebra structure, but have a “coproduct ” (the Drinfeld coproduct) which does not produce tensor products of modules in the usual way because it is defined in a completion. In this paper we propose a new process to produce quantum fusion modules from it: for all quantum affinizations, we construct by deformation and renormalization a new (non semisimple) tensor category Mod. For quantum affine algebras this process is new and different from the usual tensor product. For general quantum affinizations, for example for toroidal algebras, so far, no process to produce fusion modules was known. We derive several applications from it: we construct the fusion of (finitely many) arbitrary lhighest weight modules, and prove that it is always cyclic. We establish exact sequences involving fusion of KirillovReshetikhin modules related to new Tsystems extending results of [N4, N3, H5]. Eventually for a large class of quantum affinizations we prove that the subcategory of finite length modules of Mod is stable
Noncommutative field theory on homogeneous gravitational waves
 J. Phys. A39
"... We describe an algebraic approach to the timedependent noncommutative geometry of a sixdimensional CahenWallach ppwave string background supported by a constant NeveuSchwarz flux, and develop a general formalism to construct and analyse quantum field theories defined thereon. Various starprodu ..."
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Cited by 10 (0 self)
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We describe an algebraic approach to the timedependent noncommutative geometry of a sixdimensional CahenWallach ppwave string background supported by a constant NeveuSchwarz flux, and develop a general formalism to construct and analyse quantum field theories defined thereon. Various starproducts are derived in closed explicit form and the Hopf algebra of twisted isometries of the plane wave is constructed. Scalar field theories are defined using explicit forms of derivative operators, traces and noncommutative frame fields for the geometry, and various physical features are described. Noncommutative worldvolume field theories of Dbranes in the ppwave background are also constructed. 1 Introduction and Summary The general construction and analysis of noncommutative gauge theories on curved spacetimes is one of the most important outstanding problems in the applications of noncommutative geometry to string theory. These nonlocal field theories arise naturally as certain decoupling limits of open string dynamics on Dbranes in curved superstring backgrounds in the presence of a nonconstant
4Dimensional BF Theory as a Topological Quantum Field Theory
 Lett. Math. Phys
, 1996
"... Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that 4dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal Gbundle. The case G ..."
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Cited by 10 (5 self)
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Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that 4dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal Gbundle. The case G = GL(4; R) is especially interesting because every 4manifold is then naturally equipped with a principal Gbundle, namely its frame bundle. In this case, the partition function of a compact oriented 4manifold is the exponential of its signature, and the resulting TQFT is isomorphic to that constructed by Crane and Yetter using a state sum model, or by Broda using a surgery presentation of 4manifolds. 1 Introduction In comparison to the situation in 3 dimensions, topological quantum field theories (TQFTs) in 4 dimensions are poorly understood. This is ironic, because the subject was initiated by an attempt to understand Donaldson theory in terms of a quantum field theory in 4 dimensions....
Exact Smatrices for supersymmetric sigma models and the Potts model
"... We study the algebraic formulation of exact factorizable Smatrices for integrable twodimensional field theories. We show that different formulations of the Smatrices for the Potts field theory are essentially equivalent, in the sense that they can be expressed in the same way as elements of the Te ..."
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Cited by 5 (3 self)
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We study the algebraic formulation of exact factorizable Smatrices for integrable twodimensional field theories. We show that different formulations of the Smatrices for the Potts field theory are essentially equivalent, in the sense that they can be expressed in the same way as elements of the TemperleyLieb algebra, in various representations. This enables us to construct the Smatrices for certain nonlinear sigma models that are invariant under the Lie “supersymmetry ” algebras sl(m + nn) (m = 1, 2, n> 0), both for the bulk and for the boundary, simply by using another representation of the same algebra. These Smatrices represent the perturbation of the conformal theory at θ = π by a small change in the topological angle θ. The m = 1, n = 1 theory has applications to the spin quantum Hall transition in disordered fermion systems. We also find Smatrices describing the flow from weak to strong coupling, both for θ = 0 and θ = π, in certain other supersymmetric sigma models. 1
Wilson Loops and AreaPreserving Diffeomorphisms in Twisted Noncommutative Gauge Theory
, 2007
"... We use twist deformation techniques to analyse the behaviour under areapreserving diffeomorphisms of quantum averages of Wilson loops in YangMills theory on the noncommutative plane. We find that while the classical gauge theory is manifestly twist covariant, the holonomy operators break the quant ..."
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Cited by 3 (2 self)
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We use twist deformation techniques to analyse the behaviour under areapreserving diffeomorphisms of quantum averages of Wilson loops in YangMills theory on the noncommutative plane. We find that while the classical gauge theory is manifestly twist covariant, the holonomy operators break the quantum implementation of the twisted symmetry in the usual formal definition of the twisted quantum field theory. These results are deduced by analysing general criteria which guarantee twist invariance of noncommutative quantum field theories. From this a number of general results are also obtained, such as the twisted symplectic invariance of noncommutative scalar quantum field theories with polynomial interactions and the existence of a large class of holonomy operators with both twisted gauge covariance and twisted symplectic invariance. 1 Introduction and Summary Noncommutative gauge theory in two dimensions possesses many interesting features that can be captured analytically and used to shed light on generic features of noncommutative field theory. It is an exactly solvable model whose partition function has been computed explicitly as a semiclassical expansion over instantons on the noncommutative torus in [39]–[41], as a sum