Conformal Covariance and Positivity of Energy in Charged Sectors

Weiner, Mihály

doi:10.1007/s00220-006-1536-5

Cited by 28 publications

(34 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the local case, the automatic diffeomorphism covariance is proved in [15] and the automatic positivity of the energy in [55]. Let U λ b be the covariance unitary representation associated with Lemma 13), so we are done by replacing U λ with U λ b .…”

Section: Dhr Representationsmentioning

confidence: 99%

Structure and Classification of Superconformal Nets

Carpi

Kawahigashi

Longo

2008

Ann. Henri Poincaré

View full text Add to dashboard Cite

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 Neveu-Schwarz 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 super-Virasoro 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 super-Virasoro net in the discrete series, thus providing a classification of all superconformal nets with central charge less than 3/2.

show abstract

Section: Dhr Representationsmentioning

confidence: 99%

Structure and Classification of Superconformal Nets

Carpi

Kawahigashi

Longo

2008

Ann. Henri Poincaré

View full text Add to dashboard Cite

show abstract

“…In Quantum Field Theory there appear non-local conditions such as the ANEC (see [21,43]), namely the integrated energy density along an entire null ray is to be positive; yet the energy may locally have negative density states [19], although energy lower bounds hold true even locally. In particular, in conformal QFT, model independent local lower bounds have been obtained in [20,44].…”

Section: Introductionmentioning

confidence: 99%

Entropy Distribution of Localised States

Longo

2019

Commun. Math. Phys.

View full text Add to dashboard Cite

We study the geometric distribution of the relative entropy of a charged localised state in Quantum Field Theory. With respect to translations, the second derivative of the vacuum relative entropy is zero out of the charge localisation support and positive in mean over the support of any single charge. For a spatial strip, the asymptotic mean entropy density is πE, with E the corresponding vacuum charge energy. In a conformal QFT, for a charge in a ball of radius r, the relative entropy is non linear, the asymptotic mean radial entropy density is πE and Bekenstein's bound is satisfied. We also study the null deformation case. We construct, operator algebraically, a positive selfadjoint operator that may be interpreted as the deformation generator, we thus get a rigorous form of the Averaged Null Energy Condition that holds in full generality. In the one dimensional conformal U (1)-current model, we give a complete and explicit description of the entropy distribution of a localised charged state in all points of the real line; in particular, the second derivative of the relative entropy is strictly positive in all points where the charge density is non zero, thus the Quantum Null Energy Condition holds here for these states and is not saturated in these points. * Supported by the ERC Advanced Grant 669240 QUEST "Quantum Algebraic Structures and Models", MIUR FARE R16X5RB55W QUEST-NET and GNAMPA-INdAM. where ϕ and ψ are restricted to A(W ), E loc = (ξ, K ρ,W ξ) is the charge local energy, with K ρ,W the Rindler modular Hamiltonian in presence of the charge ρ, andis half of the conditional entropy of ρ, independent of ψ (see Sect. 3.1.2); d(ρ) is the DHR statistical dimension [17], which is equal the square root of the Jones index of ρ [33,34]. We shall also have a corresponding formula for S(ψ||ϕ).Formula (1) is the key to study the dependence of S(ϕ||ψ) on W . In particular, if W t is the shifted wedge by a null translation by t ≥ 0, ρ 1 , . . . ρ n are charges localised in W and S(t) is the relative entropy between ϕ| A(Wt) and ψ| A(Wt) we have S ′′ (t) = 0 , 1 The convexity of S is not an intrinsic concept, as it depends on re-parameterizing t. However, in Section 3.2.2, we shall have a natural deformation parameter t, the half-sided modular translation parameter.

show abstract

“…As in the case of DHR representations (see the comments after Definition 2.3) it can be shown that in various cases, as a consequence of the results in [Wei06], the positive energy condition is automatic for G-covariant general solitons, see [CKL08,Prop.12&Prop.21].…”

Section: Superconformal Netsmentioning

confidence: 99%

“…[KLM01, App.B]. If π is locally normal, then it is automatically PSL(2, R)-covariant [CKL08,DFK04] and of positive energy [Wei06]. Moreover, the representation U π can be uniquely chosen to be inner, i.e., such that U π (g) ∈ I∈I π I (A(I)), for all g ∈ PSL(2, R) (∞) (cf.…”

Section: Superconformal Netsmentioning

confidence: 99%

Superconformal nets and noncommutative geometry

Carpi¹,

Hillier²,

Longo³

2015

J. Noncommut. Geom.

View full text Add to dashboard Cite

This paper provides a further step in the program of studying superconformal nets over S 1 from the point of view of noncommutative geometry. For any such net A and any family ∆ of localized endomorphisms of the even part A γ of A, we define the locally convex differentiable algebra A ∆ with respect to a natural Dirac operator coming from supersymmetry. Having determined its structure and properties, we study the family of spectral triples and JLO entire cyclic cocycles associated to elements in ∆ and show that they are nontrivial and that the cohomology classes of the cocycles corresponding to inequivalent endomorphisms can be separated through their even or odd index pairing with K-theory in various cases. We illustrate some of those cases in detail with superconformal nets associated to well-known CFT models, namely supercurrent algebra nets and super-Virasoro nets. All in all, the result allows us to encode parts of the representation theory of the net in terms of noncommutative geometry.

show abstract

Conformal Covariance and Positivity of Energy in Charged Sectors

Cited by 28 publications

References 21 publications

Structure and Classification of Superconformal Nets

Structure and Classification of Superconformal Nets

Entropy Distribution of Localised States

Superconformal nets and noncommutative geometry

Contact Info

Product

Resources

About