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15
Classification of local conformal nets. Case c < 1
"... We completely classify diffeomorphism covariant local nets of von Neumann algebras on the circle with central charge c less than 1. The irreducible ones are in bijective correspondence with the pairs of AD2nE6,8 Dynkin diagrams such that the difference of their Coxeter numbers is equal to 1. We f ..."
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Cited by 27 (13 self)
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We completely classify diffeomorphism covariant local nets of von Neumann algebras on the circle with central charge c less than 1. The irreducible ones are in bijective correspondence with the pairs of AD2nE6,8 Dynkin diagrams such that the difference of their Coxeter numbers is equal to 1. We first identify the nets generated by irreducible representations of the Virasoro algebra for c<1 with certain coset nets. Then, by using the classification of modular invariants for the minimal models by CappelliItzyksonZuber and the method of αinduction in subfactor theory, we classify all local irreducible extensions of the Virasoro nets for c<1 and infer our main classification result. As an application, we identify in our classification list certain concrete coset nets studied in the literature.
Topological sectors and a dichotomy in conformal field theory
 Commun. Math. Phys
"... Let A be a local conformal net of factors on S 1 with the split property. We provide a topological construction of soliton representations of the nfold tensor product A ⊗ · · · ⊗ A, that restrict to true representations of the cyclic orbifold (A ⊗ · · · ⊗ A) Zn. We prove a quantum index the ..."
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Cited by 27 (14 self)
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Let A be a local conformal net of factors on S 1 with the split property. We provide a topological construction of soliton representations of the nfold tensor product A ⊗ · · · ⊗ A, that restrict to true representations of the cyclic orbifold (A ⊗ · · · ⊗ A) Zn. We prove a quantum index theorem for our sectors relating the Jones index to a topological degree. Then A is not completely rational iff the symmetrized tensor product (A ⊗ A) flip has an irreducible representation with infinite index. This implies the following dichotomy: if all irreducible sectors of A have a conjugate sector then either A is completely rational or A has uncountably many different irreducible sectors. Thus A is rational iff A is completely rational. In particular, if the µindex of A is finite then A turns out to be strongly additive. By [31], if A is rational then the tensor category of representations of A is automatically modular, namely the braiding symmetry is nondegenerate. In interesting cases, we compute the fusion rules of the topological solitons and show that they determine all twisted sectors of the cyclic orbifold.
The role of type III factors in quantum field theory
 Reports in Mathematical Physics 55
, 2005
"... One of von Neumann’s motivations for developing the theory of operator algebras and his and Murray’s 1936 classification of factors was the question of possible decompositions of quantum systems into independent parts. For quantum systems with a finite number of degrees of freedom the simplest possi ..."
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Cited by 11 (0 self)
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One of von Neumann’s motivations for developing the theory of operator algebras and his and Murray’s 1936 classification of factors was the question of possible decompositions of quantum systems into independent parts. For quantum systems with a finite number of degrees of freedom the simplest possibility, i.e., factors of type I in the terminology of Murray and von Neumann, are perfectly adequate. In relativistic quantum field theory (RQFT), on the other hand, factors of type III occur naturally. The same holds true in quantum statistical mechanics of infinite systems. In this brief review some physical consequences of the type III property of the von Neumann algebras corresponding to localized observables in RQFT and their difference from the type I case will be discussed. The cumulative effort of many people over more than 30 years has established a remarkable uniqueness result: The local algebras in RQFT are generically isomorphic to the unique, hyperfinite type III1 factor in Connes ’ classification of 1973. Specific theories are characterized by the net structure of the collection of these isomorphic algebras for different spacetime regions, i.e., the way they are embedded into each other. John von Neumann was the father of the Hilbert space formulation of quantum mechanics [1] that has been the basis of almost all mathematically rigorous investigations of the theory to this day. We start by recalling the main concepts and explaining some notations.
Noncommutative spectral invariants and black hole entropy
 Commun. Math. Phys
, 405
"... We consider an intrinsic entropy associated with a local conformal net A by the coefficients in the expansion of the logarithm of the trace of the “heat kernel ” semigroup. In analogy with Weyl theorem on the asymptotic density distribution of the Laplacian eigenvalues, passing to a quantum system w ..."
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Cited by 10 (7 self)
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We consider an intrinsic entropy associated with a local conformal net A by the coefficients in the expansion of the logarithm of the trace of the “heat kernel ” semigroup. In analogy with Weyl theorem on the asymptotic density distribution of the Laplacian eigenvalues, passing to a quantum system with infinitely many degrees of freedom, we regard these coefficients as noncommutative geometric invariants. Under a natural modularity assumption, the leading term of the entropy (noncommutative area) is proportional to the central charge c, the first order correction (noncommutative Euler characteristic) is proportional to log µA, where µA is the global index of A, and the second spectral invariant is again proportional to c. We give a further general method to define a mean entropy by considering conformal symmetries that preserve a discretization of S1 and we get the same value proportional to c. We then make the corresponding analysis with the proper Hamiltonian associated to an interval. We find here, in complete generality, a proper mean entropy proportional to log µA with a first order correction defined by means of the relative entropy associated with canonical states.
On the representation theory of Virasoro nets
 Commun. Math. Phys
"... We discuss various aspects of the representation theory of the local nets of von Neumann algebras on the circle associated with positive energy representations of the Virasoro algebra (Virasoro nets). In particular we classify the local extensions of the c = 1 Virasoro net for which the restriction ..."
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Cited by 10 (3 self)
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We discuss various aspects of the representation theory of the local nets of von Neumann algebras on the circle associated with positive energy representations of the Virasoro algebra (Virasoro nets). In particular we classify the local extensions of the c = 1 Virasoro net for which the restriction of the vacuum representation to the Virasoro subnet is a direct sum of irreducible subrepresentations with finite statistical dimension (local extensions of compact type). Moreover we prove that if the central charge c is in a certain subset of (1, ∞), including [2, ∞), and h ≥ (c − 1)/24, the irreducible representation with lowest weight h of the corresponding Virasoro net has infinite statistical dimension. As a consequence we show that if the central charge c is in the above set and satisfies c ≤ 25 then the corresponding Virasoro net has no proper local extensions of compact type. 1
Local conformal nets arising from framed vertex operator algebras
"... We construct local conformal nets of (injective type III1) factors on the circle corresponding to various lattice vertex operator algebras and perform the twisted orbifold construction, using an idea of framed vertex operator algebras. This construction fills a major gap in correspondence between al ..."
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Cited by 10 (2 self)
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We construct local conformal nets of (injective type III1) factors on the circle corresponding to various lattice vertex operator algebras and perform the twisted orbifold construction, using an idea of framed vertex operator algebras. This construction fills a major gap in correspondence between algebraic conformal quantum field theory and theory of vertex operator algebras, and in particular, we obtain a local conformal net corresponding to the moonshine vertex operator algebras of FrenkelLepowskyMeurman. Its central charge is 24, it has a trivial representation theory in the sense that the vacuum sector is the only irreducible DHR sector, and its vacuum character is the modular invariant Jfunction. We construct various order 2 automorphisms of this net and conjecture that the entire automorphism group is the Monster group. We use our previous tools such as αinduction and complete rationality to study extension of local conformal nets.
Structure and classification of superconformal nets
, 2007
"... We study the general structure of Fermi conformal nets of von Neumann algebras on S 1, consider a class of topological representations, the general representations, that we characterize as NeveuSchwarz or Ramond representations, in particular a Jones index can be associated with each of them. We th ..."
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Cited by 7 (6 self)
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We study the general structure of Fermi conformal nets of von Neumann algebras on S 1, consider a class of topological representations, the general representations, that we characterize as NeveuSchwarz or Ramond representations, in particular a Jones index can be associated with each of them. We then consider a supersymmetric general representation associated with a Fermi modular net and give a formula involving the Fredholm index of the supercharge operator and the Jones index. We then consider the net associated with the superVirasoro algebra and discuss its structure. If the central charge c belongs to the discrete series, this net is modular by the work of F. Xu and we get an example where our setting is verified by considering the Ramond irreducible representation with lowest weight c/24. We classify all the irreducible Fermi extensions of any superVirasoro net in the discrete series, thus providing a classification of all superconformal nets with central charge less than 3/2. Supported by MIUR, GNAMPAINDAM and EU network “Noncommutative Geometry ” MRTNCT
On the uniqueness of diffeomorphism symmetry in conformal field theory
"... A Möbius covariant net of von Neumann algebras on S 1 is diffeomorphism covariant if its Möbius symmetry extends to diffeomorphism symmetry. We prove that in case the net is either a Virasoro net or any at least 4regular net such an extension is unique: the local algebras together with the Möbius s ..."
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Cited by 6 (1 self)
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A Möbius covariant net of von Neumann algebras on S 1 is diffeomorphism covariant if its Möbius symmetry extends to diffeomorphism symmetry. We prove that in case the net is either a Virasoro net or any at least 4regular net such an extension is unique: the local algebras together with the Möbius symmetry (equivalently: the local algebras together with the vacuum vector) completely determine it. We draw the two following conclusions for such theories. (1) The value of the central charge c is an invariant and hence the Virasoro nets for different values of c are not isomorphic as Möbius covariant nets. (2) A vacuum preserving internal symmetry always commutes with the diffeomorphism symmetries. We further use our result to give a large class of new examples of nets (even strongly additive ones), which are not diffeomorphism covariant; i.e. which do not admit an extension of the symmetry to Diff + (S 1). Supported in part by the Italian MIUR and GNAMPAINDAM.
AQFT from nfunctorial QFT
"... There are essentially two different approaches to the axiomatization of quantum field theory (QFT): algebraic QFT, going back to Haag and Kastler, and functorial QFT, going back to Atiyah and Segal. More recently, based on ideas by Baez and Dolan, the latter is being refined to “extended ” functoria ..."
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Cited by 3 (1 self)
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There are essentially two different approaches to the axiomatization of quantum field theory (QFT): algebraic QFT, going back to Haag and Kastler, and functorial QFT, going back to Atiyah and Segal. More recently, based on ideas by Baez and Dolan, the latter is being refined to “extended ” functorial QFT by Freed, Hopkins, Lurie and others. The first approach uses local nets of operator algebras which assign to each patch an algebra “of observables”, the latter uses nfunctors which assign to each patch a “propagator of states”. In this note we present an observation about how these two axiom systems are naturally related: we demonstrate under mild assumptions that every 2dimensional extended Minkowskian QFT 2functor (”parallel surface transport”) naturally yields a local net. This is obtained by postcomposing the propagation 2functor with an operation that mimics the passage from the Schrödinger picture to the Heisenberg picture in quantum mechanics. The argument has a straightforward generalization to general pseudo
Massless scalar free Field in 1+1 dimensions II: Net Cohomology and . . .
, 2008
"... As an application of Roberts’ net cohomology theory, we positively answer about the completeness of the known set of DHR sectors of the observables of the model in the title, detailed in [7]. This result is achieved via the triviality of the 1cohomology of the underlying causal index poset with val ..."
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Cited by 3 (0 self)
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As an application of Roberts’ net cohomology theory, we positively answer about the completeness of the known set of DHR sectors of the observables of the model in the title, detailed in [7]. This result is achieved via the triviality of the 1cohomology of the underlying causal index poset with values in the field net, enhancing this tool for the case of Weyl nets not satisfying the split property and with anyonic commutation rules. We also take advantage of different causal index posets for the nets involved, and obtain the description of the twisted and untwisted sectors of the