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1.
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 相似文献
2.
We construct local, boost covariant boundary QFT nets of von Neumann algebras on the interior of the Lorentz hyperboloid \({\mathfrak{H}_R}\), x 2 ? t 2 > R 2, x > 0, in the two-dimensional Minkowski spacetime. Our first construction is canonical, starting with a local conformal net on \({\mathbb{R}}\), and is analogous to our previous construction of local boundary CFT nets on the Minkowski half-space. This net is in a thermal state at Hawking temperature. Then, inspired by a recent construction by E. Witten and one of us, we consider a unitary semigroup that we use to build up infinitely many nets. Surprisingly, the one-particle semigroup is again isomorphic to the semigroup of symmetric inner functions of the disk. In particular, by considering the U(1)-current net, we can associate with any given symmetric inner function a local, boundary QFT net on \({\mathfrak{H}_R}\). By considering different states, we shall also have nets in a ground state, rather than in a KMS state. 相似文献
3.
F. Xu 《Communications in Mathematical Physics》2007,270(3):835-847
In this paper we prove a general theorem on the extensions of local nets which was inspired by recent examples of exotic extensions
for Virasoro nets with central charge less than one and earlier work on cosets and conformal inclusions. When applying the
theorem to conformal inclusions and diagonal inclusions, we obtain infinite series of new examples of completely rational
nets.
Supported in part by NSF. 相似文献
4.
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. 相似文献
5.
Paolo Camassa Roberto Longo Yoh Tanimoto Mihály Weiner 《Communications in Mathematical Physics》2012,309(3):703-735
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. 相似文献
6.
We define and study two-dimensional, 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 III1 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 Tomita-Takesaki 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 background-independent formulation of conformal field theories. 相似文献
7.
The present paper deals with the question of representability of nets of C*-algebras whose underlying poset, indexing the
net, is not upward directed. A particular class of nets, called C*-net bundles, is classified in terms of C*-dynamical systems
having as group the fundamental group of the poset. Any net of C*-algebras has a canonical morphism into a C*-net bundle,
the enveloping net bundle, which generalizes the notion of universal C*-algebra given by Fredenhagen to nonsimply connected
posets. This allows a classification of nets; in particular, we call injective those nets such that the canonical morphism
is faithful. Injectivity turns out to be equivalent to the existence of faithful representations. We further relate injectivity
to a generalized Čech cocycle of the net, and this allows us to give examples of nets exhausting the above classification. 相似文献
8.
Benjamin Doyon 《Letters in Mathematical Physics》2013,103(3):233-284
We give a construction of the stress–energy tensor of conformal field theory (CFT) as a local “object” in conformal loop ensembles CLE κ , for all values of κ in the dilute regime 8/3 < κ ≤ 4 (corresponding to the central charges 0 < c ≤ 1 and including all CFT minimal models). We provide a quick introduction to CLE, a mathematical theory for random loops in simply connected domains with properties of conformal invariance, developed by Sheffield and Werner (Ann Math 176, 1827–1917, 2012). We consider its extension to more general regions of definition and make various hypotheses that are needed for our construction and expected to hold for CLE in the dilute regime. Using this, we identify the stress–energy tensor in the context of CLE. This is done by deriving its associated conformal Ward identities for single insertions in CLE probability functions, along with the appropriate boundary conditions on simply connected domains; its properties under conformal maps, involving the Schwarzian derivative; and its one-point average in terms of the “relative partition function”. Part of the construction is in the same spirit as, but widely generalizes, that found in the context of SLE8/3 by the author, Riva and Cardy (Commun Math Phys 268, 687–716, 2006), which only dealt with the case of zero central charge in simply connected hyperbolic regions. We do not use the explicit construction of the CLE probability measure, but only its defining and expected general properties. 相似文献
9.
Investigated is a variant of the Wess-Zumino-Witten model called a twisted WZW model, which is associated to a certain Lie
group bundle on a family of elliptic curves. The Lie group bundle is a non-trivial bundle with flat connection and related
to the classical elliptic r-matrix. (The usual (non-twisted) WZW model is associated to a trivial group bundle with trivial connection on a family of
compact Riemann surfaces and a family of its principal bundles.) The twisted WZW model on a fixed elliptic curve at the
critical level describes the XYZ Gaudin model. The elliptic Knizhnik-Zamolodchikov equations associated to the classical elliptic
r-matrix appear as flat connections on the sheaves of conformal blocks in the twisted WZW model.
Received: 21 January 1997 / Accepted: 1 April 1997 相似文献
10.
Paolo Bertozzini Roberto Conti Roberto Longo 《Communications in Mathematical Physics》1998,193(2):471-492
Let ? be a local conformal net of von Neumann algebras on S
1 and ρ a M?bius covariant representation of ?, possibly with infinite dimension. If ρ has finite index, ρ has automatically
positive energy. If ρ has infinite index, we show the spectrum of the energy always to contain the positive real line, but,
as seen by an example, it may contain negative values. We then consider nets with Haag duality on ℝ, or equivalently sectors
with non-solitonic extension to the dual net; we give a criterion for irreducible sectors to have positive energy, namely
this is the case iff there exists an unbounded M?bius covariant left inverse. As a consequence the class of sectors with positive
energy is stable under composition, conjugation and direct integral decomposition.
Received: 21 April 1997 / Accepted: 23 September 1997 相似文献
11.
Using twistor methods, we explicitly construct all local forms of four–dimensional real analytic neutral signature anti–self–dual
conformal structures (M, [g]) with a null conformal Killing vector. We show that M is foliated by anti-self-dual null surfaces, and the two-dimensional leaf space inherits a natural projective structure.
The twistor space of this projective structure is the quotient of the twistor space of (M, [g]) by the group action induced by the conformal Killing vector.
We obtain a local classification which branches according to whether or not the conformal Killing vector is hyper-surface
orthogonal in (M, [g]). We give examples of conformal classes which contain Ricci–flat metrics on compact complex surfaces and discuss other conformal
classes with no Ricci–flat metrics.
Dedicated to the memory of Jerzy Plebański 相似文献
12.
Yasuyuki Kawahigashi Roberto Longo Michael Müger 《Communications in Mathematical Physics》2001,219(3):631-669
We describe the structure of the inclusions of factors ?(E)⊂?(E′)′ associated with multi-intervals E⊂ℝ for a local irreducible net ? 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 Longo–Rehren inclusion, which provides a quantum double construction of the tensor category of superselection
sectors of ?. As a consequence, the index of ?(E)⊂?(E′)′ coincides with the global index associated with all irreducible sectors, the braiding symmetry associated with all sectors
is non-degenerate, namely the representations of ? 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.
Received: 7 July 1999 / Accepted: 13 January 2001 相似文献
13.
Antonio di Teodoro Víctor M. Villalba 《International Journal of Theoretical Physics》2008,47(7):1885-1890
In the present article we study asymptotic isometries and asymptotic conformal Killing motions of some anisotropic Bianchi
cosmological models. We show that asymptotically isotropic homogeneous cosmological spaces can be covariantly studied and
characterized using scalars constructed with the help the Killing equations. We exhibit some illustrative examples of space-times
possessing asymptotic Killing collineations. We apply our results in order to discuss asymptotic symmetries associated with
scalar cosmological perturbations. 相似文献
14.
We classify two-dimensional local conformal nets with parity symmetry and central charge less than 1, up to isomorphism. The maximal ones are in a bijective correspondence with the pairs of A-D-E Dynkin diagrams with the difference of their Coxeter numbers equal to 1. In our previous classification of one-dimensional local conformal nets, Dynkin diagrams D
2n +1
and E
7
do not appear, but now they do appear in this classification of two-dimensional local conformal nets. Such nets are also characterized as two-dimensional local conformal nets with -index equal to 1 and central charge less than 1. Our main tool, in addition to our previous classification results for one-dimensional nets, is 2-cohomology vanishing for certain tensor categories related to the Virasoro tensor categories with central charge less than 1.Supported in part by JSPS.Supported in part by GNAMPA and MIUR. 相似文献
15.
Sebastiano Carpi Yasuyuki Kawahigashi Roberto Longo 《Communications in Mathematical Physics》2013,322(1):149-166
Given a conformal QFT local net of von Neumann algebras ${\mathcal {B}_2}$ on the two-dimensional Minkowski spacetime with irreducible subnet ${\mathcal {A} \otimes \mathcal {A}}$ , where ${\mathcal {A}}$ is a completely rational net on the left/right light-ray, we show how to consistently add a boundary to ${\mathcal {B}_2}$ : we provide a procedure to construct a Boundary CFT net ${\mathcal {B}}$ of von Neumann algebras on the half-plane x > 0, associated with ${\mathcal {A}}$ , and locally isomorphic to ${\mathcal {B}_2}$ . All such locally isomorphic Boundary CFT nets arise in this way. There are only finitely many locally isomorphic Boundary CFT nets and we get them all together. In essence, we show how to directly redefine the C* representation of the restriction of ${\mathcal {B}_2}$ to the half-plane by means of subfactors and local conformal nets of von Neumann algebras on S 1. 相似文献
16.
We consider a relation between the conformal factor in the stationary axisymmetric (SAS) Einstein-Maxwell field equations and a central extension of a formal loop group which is described by a group 2-cocycle on the formal loop group. The corresponding 2-cocycle on the Lie algebra of the formal loop group is the one which describes an affine Lie algebra. As a result, we see that the space of formal solutions with conformal factors is a homogeneous space of a central extension of the Hauser group. 相似文献
17.
18.
The modular structure of the von Neumann algebra of local observables associated with a double cone in the vacuum representation of the free massless scalar field theory of any number of dimensions is described. The modular automorphism group is induced by the unitary implementation of a family of generalized fractional linear transformations on Minkowski space and is a subgroup of the conformal group. The modular conjugation operator is the anti-unitary implementation of a product of time reversal and relativistic ray inversion. The group generated by the modular conjugation operators for the local algebras associated with the family of double cone regions is the group of proper conformal transformations. A theorem is presented asserting the unitary equivalence of local algebras associated with lightcones, double cones, and wedge regions. For the double cone algebras, this provides an explicit realization of spacelike duality and establishes the known typeIII
1 factor property. It is shown that the timelike duality property of the lightcone algebras does not hold for the double cone algebras. A different definition of the von Neumann algebras associated with a region is introduced which agrees with the standard one for a lightcone or a double cone region but which allows the timelike duality property for the double cone algebras. In the case of one spatial dimension, the standard local algebras associated with the double cone regions satisfy both spacelike and timelike duality.Supported by the National Science Foundation under Grant No. PHY-79-23251Supported in part by C. N. R. 相似文献
20.
Jakob Yngvason 《Letters in Mathematical Physics》1994,31(2):127-141
By considering some simple models, it is shown that the essential duality condition for local nets of von Neumann algebras associated with Wightman fields need not be fulfilled if Lorentz covariance is dropped. These models illustrate a point made by Borchers in the proof of his two-dimensional CPT theorem for local nets: The Lorentz covariant net constructed from the wedge algebras of a given two-dimensional net may not be unique. It is also shown that in higher dimensions, the Lorentz boosts constructed by means of the modular groups of wedge algebras may act nonlocally in the directions parallel to the edge of the wedge. 相似文献