Thermal States in Conformal QFT. I

We analyze the set of locally normal KMS states w.r.t. the translation group for a local conformal net A{{\mathcal A}} of von Neumann algebras on \mathbb R{\mathbb R} . In this first part, we focus on the completely rational net A{{\mathcal A}} . Our main result here states that, if A{{\mathcal{A}}} is completely rational, there exists exactly one locally normal KMS state j{\varphi} . Moreover, j{\varphi} is canonically constructed by a geometric procedure. A crucial r?le is played by the analysis of the “thermal completion net” associated with a locally normal KMS state. A similar uniqueness result holds for KMS states of two-dimensional local conformal nets w.r.t. the time-translation one-parameter group.     

Conformal Subnets and Intermediate Subfactors

 Given an irreducible local conformal net 𝒜 of von Neumann algebras on S ¹ and a finite-index conformal subnet ℬ⊂𝒜, we show that 𝒜 is completely rational iff ℬ is completely rational. In particular this extends a result of F. Xu for the orbifold construction. By applying previous results of Xu, many coset models turn out to be completely rational and the structure results in [27] hold. Our proofs are based on an analysis of the net inclusion ℬ⊂𝒜; among other things we show that, for a fixed interval I, every von Neumann algebra  intermediate between ℬ(I) and 𝒜(I) comes from an intermediate conformal net ℒ between ℬ and 𝒜 with ℒ(I)=. We make use of a theorem of Watatani (type II case) and Teruya and Watatani (type III case) on the finiteness of the set ℑ(𝒩,ℳ) of intermediate subfactors in an irreducible inclusion of factors 𝒩⊂ℳ with finite Jones index [ℳ:𝒩]. We provide a unified proof of this result that gives in particular an explicit bound for the cardinality of ℑ(𝒩,ℳ) which depends only on [ℳ:𝒩]. Received: 21 December 2001 / Accepted: 28 February 2002 Published online: 14 March 2003 RID="⋆" ID="⋆" Supported in part by MIUR and INDAM-GNAMPA. Communicated by K. Fredenhagen     

Generalized Longo–Rehren Subfactors and α-Induction

We study the recent construction of subfactors by Rehren which generalizes the Longo–Rehren subfactors. We prove that if we apply this construction to a non-degenerately braided subfactor N⊂M and α^±-induction, then the resulting subfactor is dual to the Longo–Rehren subfactor M⊗M ^opp⊂R arising from the entire system of irreducible endomorphisms of M resulting from α^plusmn;-induction. As a corollary, we solve a problem on existence of braiding raised by Rehren negatively. Furthermore, we generalize our previous study with Longo and Müger on multi-interval subfactors arising from a completely rational conformal net of factors on S ¹ to a net of subfactors and show that the (generalized) Longo–Rehren subfactors and α-induction naturally appear in this context. Received: 11 September 2001 / Accepted: 7 October 2001     

Thermal States in Conformal QFT. II

We continue the analysis of the set of locally normal KMS states w.r.t. the translation group for a local conformal net of von Neumann algebras on . In the first part we have proved the uniqueness of the KMS state on every completely rational net. In this second part, we exhibit several (non-rational) conformal nets which admit continuously many primary KMS states. We give a complete classification of the KMS states on the U(1)-current net and on the Virasoro net Vir₁ with the central charge c = 1, whilst for the Virasoro net Vir _c with c > 1 we exhibit a (possibly incomplete) list of continuously many primary KMS states. To this end, we provide a variation of the Araki-Haag-Kastler-Takesaki theorem within the locally normal system framework: if there is an inclusion of split nets and is the fixed point of w.r.t. a compact gauge group, then any locally normal, primary KMS state on extends to a locally normal, primary state on , KMS w.r.t. a perturbed translation. Concerning the non-local case, we show that the free Fermi model admits a unique KMS state.     

A Remark on CFT Realization of Quantum Doubles of Subfactors: Case Index { < 4}

It is well known that the quantum double \({D(N\subset M)}\) of a finite depth subfactor \({N\subset M}\), or equivalently the Drinfeld center of the even part fusion category, is a unitary modular tensor category. It is big open conjecture that all (unitary) modular tensor categories arise from conformal field theory. We show that for every subfactor \({N\subset M}\) with index \({[M:N] < 4}\) the quantum double \({D(N\subset M)}\) is realized as the representation category of a completely rational conformal net. In particular, the quantum double of \({E_6}\) can be realized as a \({\mathbb{Z}_2}\)-simple current extension of \({{{\rm SU}(2)}_{10}\times {{\rm Spin}(11)}_1}\) and thus is not exotic in any sense. As a byproduct, we obtain a vertex operator algebra for every such subfactor. We obtain the result by showing that if a subfactor \({N\subset M }\) arises from \({\alpha}\)-induction of completely rational nets \({\mathcal{A}\subset \mathcal{B}}\) and there is a net \({\tilde{\mathcal{A}}}\) with the opposite braiding, then the quantum \({D(N\subset M)}\) is realized by completely rational net. We construct completely rational nets with the opposite braiding of \({{{\rm SU}(2)}_k}\) and use the well-known fact that all subfactors with index \({[M:N] < 4}\) arise by \({\alpha}\)-induction from \({{{\rm SU}(2)}_k}\).     

From String Nets to Nonabelions

We discuss Hilbert spaces spanned by the set of string nets, i.e. trivalent graphs, on a lattice. We suggest some routes by which such a Hilbert space could be the low-energy subspace of a model of quantum spins on a lattice with short-ranged interactions. We then explain conditions which a Hamiltonian acting on this string net Hilbert space must satisfy in order for the system to be in the DFib (Doubled Fibonacci) topological phase, that is, be described at low energy by an SO(3)₃ × SO(3)₃ doubled Chern-Simons theory, with the appropriate non-abelian statistics governing the braiding of the low-lying quasiparticle excitations (nonabelions). Using the string net wavefunction, we describe the properties of this phase. Our discussion is informed by mappings of string net wavefunctions to the chromatic polynomial and the Potts model.     

Unitarity of rationalN = 2 superconformal theories

We demonstrate that all rational models of theN = 2 super Virasoro algebra are unitary. Our arguments are based on three different methods: we determine Zhu’s algebraA(H₀) (for which we give a physically motivated derivation) explicitly for certain theories, we analyse the modular properties of some of the vacuum characters, and we use the coset realisation of the algebra in terms ofsu(2) and two free fermions. Some of our arguments generalise to the Kazama-Suzuki models indicating that all rationalN = 2 supersymmetric models might be unitary.     

On the Uniqueness of Diffeomorphism Symmetry in Conformal Field Theory

A Möbius covariant net of von Neumann algebras on S¹ 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 4-regular 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⁺(S¹).Supported in part by the Italian MIUR and GNAMPA-INDAM.     

Critical region of D-dimensional spins: extension and analysis of the hierarchical reference theory

The hierarchical reference theory (HRT) is generalised to spins of dimensionality D. Then its properties are investigated by both analytical and numerical evaluations for supercritical temperatures. The HRT is closely related to the self-consistent Ornstein–Zernike approximation (SCOZA) that was developed earlier for arbitrary D. Like the D = 1 case we studied earlier, our investigation is facilitated by a situation where both HRT and SCOZA give identical results with a mean spherical model behaviour (i.e. D = ∞). However, for the more general situation we find that an additional intermediate term appears. With an interplay between leading and subleading contributions, simple rational numbers, independent of D (<∞), are found for the critical indices.     

Solitons in Affine and Permutation Orbifolds

We consider properties of solitons in general orbifolds in the algebraic quantum field theory framework and constructions of solitons in affine and permutation orbifolds. Under general conditions we show that our construction gives all the twisted representations of the fixed point subnet. This allows us to prove a number of conjectures: in the affine orbifold case we clarify the issue of fixed point resolutions; in the permutation orbifold case we determine all irreducible representations of the orbifold, and we also determine the fusion rules in a nontrivial case, which imply an integral property of chiral data for any completely rational conformal net.Supported in part by NSF.Supported in part by GNAMPA-INDAM and MIUR.Supported in part by NSF.     

OnE 10 and the DDF construction

An attempt is made to understand the root spaces of Kac Moody algebras of hyperbolic type, and in particularE ₁₀, in terms of a DDF construction appropriate to a subcritical compactified bosonic string. While the level-one root spaces can be completely characterized in terms of transversal DDF states (the level-zero elements just span the affine subalgebra), longitudinal DDF states are shown to appear beyond level one. In contrast to previous treatments of such algebras, we find it necessary to make use of a rational extension of the self-dual root lattice as an auxiliary device, and to admit non-summable operators (in the sense of the vertex algebra formalism). We demonstrate the utility of the method by completely analyzing a non-trivial level-two root space, obtaining an explicit and comparatively simple representation for it. We also emphasize the occurrence of several Virasoro algebras, whose interrelation is expected to be crucial for a better understanding of the complete structure of the Kac Moody algebra.Supported by Konrad-Adenauer-Stiftung e.V.This article was processed by the author using the Latex style filepljour1 from Springer-Verlag.     

Algebraic Geometry in Discrete Quantum Integrable Model and Baxter's T–Q Relation

We study the diagonalization problem of certain discrete quantum integrable models by the method of Baxter's T–Q relation from the algebraic geometry aspect. Among those the Hofstadter type model (with the rational magnetic flux), discrete quantum pendulum and discrete sine-Gordon model are our main concern in this report. By the quantum inverse scattering method, the Baxter's T–Q relation is formulated on the associated spectral curve, a high genus Riemann surface in general, arisen from the study of spectrum problem of the system. In the case of degenerated spectral curve where the spectral variables lie on rational curves, we obtain the complete and explicit solution of the T–Q polynomial equation associated to the model, and the intimate relation between the Baxter's T–Q relation and algebraic Bethe Ansatz is clearly revealed. The algebraic geometry of a general spectral curve attached to the model and certain qualitative properties of solutions of the Baxter's T–Q relation are discussed incorporating the physical consideration.     

Huygens' Principle,Dirac Operators,and Rational Solutions of the AKNS Hierarchy

We prove that rational solutions of the AKNS hierarchy of the form q=σ/τ and r=ρ/τ, where (σ,τ,ρ) are certain Schur functions, naturally yield Dirac operators of strict Huygens' type, i.e., the support of their fundamental solutions is the surface of the light-cone. This strengthens the connection between the theory of completely integrable systems and Huygens' principle by extending to the Dirac operators and the rational solutions of the AKNS hierarchy a classical result of Lagnese and Stellmacher concerning perturbations of wave operators. Mathematics Subject Classifications (2000) 37K10, 35Qxx, 35B40.     

Quantum Dynamical R-Matrices and Quantum Frobenius Group

We propose an algebraic scheme for quantizing the rational Ruijsenaars-Schneider model in the R-matrix formalism. We introduce a special parametrization of the cotangent bundle over . In new variables the standard symplectic structure is described by a classical (Frobenius) r-matrix and by a new dynamical -matrix. Quantizing both of them we find the quantum L-operator algebra and construct its particular representation corresponding to the rational Ruijsenaars-Schneider system. Using the dual parametrization of the cotangent bundle we also derive the algebra for the L-operator of the hyperbolic Calogero-Moser system. Received: 24 January 1997 / Accepted: 17 March 1997     

Extensions of Conformal Nets¶and Superselection Structures

Starting with a conformal Quantum Field Theory on the real line, we show that the dual net is still conformal with respect to a new representation of the M?bius group. We infer from this that every conformal net is normal and conormal, namely the local von Neumann algebra associated with an interval coincides with its double relative commutant inside the local von Neumann algebra associated with any larger interval. The net and the dual net give together rise to an infinite dimensional symmetry group, of which we study a class of positive energy irreducible representations. We mention how superselection sectors extend to the dual net and we illustrate by examples how, in general, this process generates solitonic sectors. We describe the free theories associated with the lowest weight n representations of , showing that they violate 3-regularity for $n > 2. When n≥ 2, we obtain examples of non M?bius-covariant sectors of a 3-regular (non 4-regular) net. Received: 19 March 1997 / Accepted: 1 July 1997     

Survival,Extinction and Approximation of Discrete-time Branching Random Walks

We consider a general discrete-time branching random walk on a countable set X. We relate local, strong local and global survival with suitable inequalities involving the first-moment matrix M of the process. In particular we prove that, while the local behavior is characterized by M, the global behavior cannot be completely described in terms of properties involving M alone. Moreover we show that locally surviving branching random walks can be approximated by sequences of spatially confined and stochastically dominated branching random walks which eventually survive locally if the (possibly finite) state space is large enough. An analogous result can be achieved by approximating a branching random walk by a sequence of multitype contact processes and allowing a sufficiently large number of particles per site. We compare these results with the ones obtained in the continuous-time case and we give some examples and counterexamples.     

Boundary Scattering,Symmetric Spaces and the Principal Chiral Model on the Half-Line

We investigate integrable boundary conditions (BCs) for the principal chiral model on the half-line, and rational solutions of the boundary Yang-Baxter equation (BYBE). In each case we find a connection with (type I, Riemannian, globally) symmetric spaces G/H: there is a class of integrable BCs in which the boundary field is restricted to lie in a coset of H; these BCs are parametrized by G/H×G/H; there are rational solutions of the BYBE in the defining representations of all classical G parametrized by G/H; and using these we propose boundary S-matrices for the principal chiral model, parametrized by G/H×G/H, which correspond to our boundary conditions.An erratum to this article can be found at     

Schlesinger Transformations and Quantum R-Matrices

 Schlesinger transformations are discrete monodromy preserving symmetry transformations of a meromorphic connection which shift by integers the eigenvalues of its residues. We study Schlesinger transformations for twisted -valued connections on the torus. A universal construction is presented which gives the elementary two-point transformations in terms of Belavin's elliptic quantum R-matrix. In particular, the role of the quantum deformation parameter is taken by the difference of the two poles whose residue eigenvalues are shifted. Elementary one-point transformations (acting on the residue eigenvalues at a single pole) are constructed in terms of the classical elliptic r-matrix. The action of these transformations on the τ-function of the system may completely be integrated and we obtain explicit expressions in terms of the parameters of the connection. In the limit of a rational R-matrix, our construction and the τ-quotients reduce to the classical results of Jimbo and Miwa in the complex plane. Received: 19 December 2001 / Accepted: 20 May 2002 Published online: 14 October 2002     

嵌入La和Gd原子的Si24笼团簇的稳定性

用梯度修正自旋极化密度泛函(DFT)电子结构计算,研究了具有T_h和D_2d对称性包裹La和Gd原子的Si₂₄富勒烯的稳定性.结果表明Gd@Si₂₄具有很高的磁性而La@Si₂₄的磁性完全猝灭.这些结果有可能导致Si基富勒烯团簇新的结构类型. 关键词： 24和Gd@Si₂₄包裹团簇')" href="#">La@Si₂₄和Gd@Si₂₄包裹团簇 自旋极化密度泛函 稳定性 电磁性质