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104
Multiinterval subfactors and modularity of representations in conformal field theory
 Commun. Math. Phys
"... Dedicated to John E. Roberts on the occasion of his sixtieth birthday We describe the structure of the inclusions of factors A(E) ⊂A(E ′ ) ′ associated with multiintervals E ⊂ R for a local irreducible net A of von Neumann algebras on the real line satisfying the split property and Haag duality. I ..."
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Cited by 63 (26 self)
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Dedicated to John E. Roberts on the occasion of his sixtieth birthday We describe the structure of the inclusions of factors A(E) ⊂A(E ′ ) ′ associated with multiintervals E ⊂ R for a local irreducible net A of von Neumann algebras on the real line satisfying the split property and Haag duality. In particular, if the net is conformal and the subfactor has finite index, the inclusion associated with two separated intervals is isomorphic to the LongoRehren inclusion, which provides a quantum double construction of the tensor category of superselection sectors of A. As a consequence, the index of A(E) ⊂A(E ′ ) ′ coincides with the global index associated with all irreducible sectors, the braiding symmetry associated with all sectors is nondegenerate, namely the representations of A form a modular tensor category, and every sector is a direct sum of sectors with finite dimension. The superselection structure is generated by local data. The same results hold true if conformal invariance is replaced by strong additivity and there exists a modular PCT symmetry.
Operator Algebras and Conformal Field Theory
 COMMUNICATIONS MATHEMATICAL PHYSICS
, 1993
"... We define and study twodimensional, chiral conformal field theory by the methods of algebraic field theory. We start by characterizing the vacuum sectors of such theories and show that, under very general hypotheses, their algebras of local observables are isomorphic to the unique hyperfinite typ ..."
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Cited by 55 (2 self)
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We define and study twodimensional, chiral conformal field theory by the methods of algebraic field theory. We start by characterizing the vacuum sectors of such theories and show that, under very general hypotheses, their algebras of local observables are isomorphic to the unique hyperfinite type III 1 factor. The conformal net determined by the algebras of local observables is proven to satisfy Haag duality. The representation of the Moebius group (and presumably of the entire Virasoro algebra) on the vacuum sector of a conformal field theory is uniquely determined by the TomitaTakesaki modular operators associated with its vacuum state and its conformal net. We then develop the theory of Moebius covariant representations of a conformal net, using methods of Doplicher, Haag and Roberts. We apply our results to the representation theory of loop groups. Our analysis is motivated by the desire to find a "backgroundindependent" formulation of conformal field theories.
Nets of subfactors
, 1994
"... A subtheory of a quantum field theory specifies von Neumann subalgebras A(O) (the ‘observables’ in the spacetime region O) of the von Neumann algebras B(O) (the ‘fields ’ localized in O). Every local algebra being a (type III1) factor, the inclusion A(O) ⊂ B(O) is a subfactor. The assignment of th ..."
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Cited by 51 (4 self)
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A subtheory of a quantum field theory specifies von Neumann subalgebras A(O) (the ‘observables’ in the spacetime region O) of the von Neumann algebras B(O) (the ‘fields ’ localized in O). Every local algebra being a (type III1) factor, the inclusion A(O) ⊂ B(O) is a subfactor. The assignment of these local subfactors to the spacetime regions is called a ‘net of subfactors’. The theory of subfactors is applied to such nets. In order to characterize the ‘relative position ’ of the subtheory, and in particular to control the restriction and induction of superselection sectors, the canonical endomorphism is studied. The crucial observation is this: the canonical endomorphism of a local subfactor extends to an endomorphism of the field net, which in turn restricts to a localized endomorphism of the observable net. The method allows to characterize, and reconstruct, local extensions B of a given theory A in terms of the observables. Various nontrivial examples are given. Several results go beyond the quantum field theoretical application.
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
Scaling algebras and renormalization group in algebraic quantum field theory
 Rev. Math. Phys
, 1995
"... Abstract: The concept of scaling algebra provides a novel framework for the general structural analysis and classification of the short distance properties of algebras of local observables in relativistic quantum field theory. In the present article this method is applied to the simple example of ma ..."
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Cited by 42 (7 self)
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Abstract: The concept of scaling algebra provides a novel framework for the general structural analysis and classification of the short distance properties of algebras of local observables in relativistic quantum field theory. In the present article this method is applied to the simple example of massive free field theory in s = 1,2 and 3 spatial dimensions. Not quite unexpectedly, one obtains for s = 2,3 in the scaling (short distance) limit the algebra of local observables in massless free field theory. The case s = 1 offers, however, some surprises. There the algebra of observables acquires in the scaling limit a nontrivial center and describes charged physical states satisfying Gauss ’ law. The latter result is of relevance for the interpretation of the Schwinger model at short distances and illustrates the conceptual and computational virtues of the method. 1
An algebraic spin and statistics theorem
 Commun. Math. Phys
, 1995
"... Dedicated to Hans Borchers on the occasion of his seventieth birthday ..."
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Cited by 42 (11 self)
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Dedicated to Hans Borchers on the occasion of his seventieth birthday
Charged Sectors, Spin and Statistics in Quantum Field Theory on Curved Spacetimes
, 1999
"... The first part of this paper extends the DoplicherHaagRoberts theory of superselection sectors to quantum field theory on arbitrary globally hyperbolic spacetimes. The statistics of a superselection sector may be defined as in flat spacetime and each charge has a conjugate charge when the spacetim ..."
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Cited by 30 (9 self)
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The first part of this paper extends the DoplicherHaagRoberts theory of superselection sectors to quantum field theory on arbitrary globally hyperbolic spacetimes. The statistics of a superselection sector may be defined as in flat spacetime and each charge has a conjugate charge when the spacetime possesses noncompact Cauchy surfaces. In this case, the field net and the gauge group can be constructed as in Minkowski spacetime. The second part of this paper derives spinstatistics theorems on spacetimes with appropriate symmetries. Two situations are considered: First, if the spacetime has a bifurcate Killing horizon, as is the case in the presence of black holes, then restricting the observables to the Killing horizon together with “modular covariance” for the Killing flow yields a conformally covariant quantum field theory on the circle and a conformal spinstatistics theorem for charged sectors localizable on the Killing horizon. Secondly, if the spacetime has a rotation and PT symmetry like the SchwarzschildKruskal black holes, “geometric modular action” of the rotational symmetry leads to a spinstatistics theorem for charged covariant sectors where the spin is defined via the SU(2)covering of the spatial rotation group SO(3).
Galois theory for braided tensor categories and the modular closure
 Adv. Math
, 2000
"... Given a braided tensor ∗category C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C ⋊ S. This construction yields a tensor ∗category with conjugates and an irreducible unit. (A ∗category is a category enriched over VectC ..."
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Cited by 29 (6 self)
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Given a braided tensor ∗category C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C ⋊ S. This construction yields a tensor ∗category with conjugates and an irreducible unit. (A ∗category is a category enriched over VectC with positive ∗operation.) A Galois correspondence is established between intermediate categories sitting between C and C ⋊S and closed subgroups of the Galois group Gal(C⋊S/C) = AutC(C⋊S) of C, the latter being isomorphic to the compact group associated to S by the duality theorem of Doplicher and Roberts. Denoting by D ⊂ C the full subcategory of degenerate objects, i.e. objects which have trivial monodromy with all objects of C, the braiding of C extends to a braiding of C⋊S iff S ⊂ D. Under this condition C⋊S has no nontrivial degenerate objects iff S = D. If the original category C is rational (i.e. has only finitely many isomorphism classes of irreducible objects) then the same holds for the new one. The category C ≡ C ⋊ D is called the modular closure of C since in the rational case it is modular, i.e. gives rise to a unitary representation of the modular group SL(2, Z). (In passing we prove that every braided tensor ∗category with conjugates automatically is a ribbon category, i.e. has a twist.) If all simple objects of S have dimension one the structure of the category C ⋊ S can be clarified quite explicitly in terms of group cohomology. 1
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 27 (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.
Diagonal Crossed Products by Duals of QuasiQuantum Groups
 Rev. Math. Phys
, 1999
"... A twosided coaction δ: M → G ⊗M⊗G of a Hopf algebra (G, ∆, ǫ, S) on an associative algebra M is an algebra map of the form δ = (λ ⊗ idM) ◦ ρ = (idM ⊗ ρ) ◦ λ, where (λ, ρ) is a commuting pair of left and right Gcoactions on M, respectively. Denoting the associated commuting right and left actions ..."
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Cited by 25 (1 self)
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A twosided coaction δ: M → G ⊗M⊗G of a Hopf algebra (G, ∆, ǫ, S) on an associative algebra M is an algebra map of the form δ = (λ ⊗ idM) ◦ ρ = (idM ⊗ ρ) ◦ λ, where (λ, ρ) is a commuting pair of left and right Gcoactions on M, respectively. Denoting the associated commuting right and left actions of the dual Hopf algebra ˆ G on M by ⊳ and ⊲, respectively, we define the diagonal crossed product M ⊲ ⊳ ˆ G to be the algebra generated by M and ˆ G with relations given by ϕm = (ϕ (1) ⊲m ⊳ ˆ S −1 (ϕ (3)))ϕ (2), m ∈ M, ϕ ∈ ˆ G. We give a natural generalization of this construction to the case where G is a quasi–Hopf algebra in the sense of Drinfeld and, more generally, also in the sense of Mack and Schomerus (i.e., where the coproduct ∆ is nonunital). In these cases our diagonal crossed product will still be an associative algebra structure on M ⊗ ˆ G extending M ≡ M ⊗ ˆ1, even though the analogue of an ordinary crossed product M ⋊ ˆ G in general is not well defined as an associative algebra. Applications of our formalism include the field algebra constructions with quasiquantum