Classification of Positive Energy Representations of the Virasoro Group

Neeb, Karl-Hermann; Salmasian, Hadi

doi:10.1093/imrn/rnu197

Cited by 15 publications

(14 citation statements)

References 22 publications

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“…For example, in AdS 3 gravity, each BTZ black hole [8] lies in a unique Virasoro coadjoint orbit [2,3] of the type Diff(S 1 )/S 1 , and one can consider the observables on this symplectic manifold and attempt to give a unitary representation of them on some Hilbert space, thus quantizing this "BTZ sector" of AdS 3 gravity. It also has a mathematical interest on its own, and the complete classification of unitary positive-energy representations of Virasoro group has been recently given by Neeb and Salmasian in [9]. Their work can be regarded, among other things, as a (kind of) geometric quantization of the orbit Diff(S 1 )/S 1 , since their Hilbert space is given by certain holomorphic sections on a line bundle over the orbit.…”

Section: Introductionmentioning

confidence: 99%

Quantization of BMS 3 orbits: A perturbative approach

Garbarz

Leston

2016

Nuclear Physics B

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We compute characters of the BMS group in three dimensions. The approach is the same as that performed by Witten in the case of coadjoint orbits of the Virasoro group in the eighties, within the large central charge approximation. The procedure involves finding a Poisson bracket between classical variables and the corresponding commutator of observables in a Hilbert space, explaining why we call this a quantization. We provide first a pedagogical warm up by applying the method to both SL(2, R) and Poincaré 3 groups. As for BMS 3 , our results coincide with the characters of induced representations recently studied in the literature. Moreover, we relate the 'coadjoint representations' with the induced representations. *

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Section: Introductionmentioning

confidence: 99%

Quantization of BMS 3 orbits: A perturbative approach

Garbarz

Leston

2016

Nuclear Physics B

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“…is dense in H. Then, using the arguments in [72, Chapter 1], se also [18, Appendix A], one can prove that there is a unique positive-energy unitary representation π of the Virasoro algebra on H f in such that U = U π , see also [85] for related results. We collect the results discussed above in the following theorem.…”

Section: Diff + (S 1 ) and Its Subgroup Möbmentioning

confidence: 99%

From Vertex Operator Algebras to Conformal Nets and Back

et al. 2018

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We consider unitary simple vertex operator algebras whose vertex operators satisfy certain energy bounds and a strong form of locality and call them strongly local. We present a general procedure which associates to every strongly local vertex operator algebra V a conformal net A V acting on the Hilbert space completion of V and prove that the isomorphism class of A V does not depend on the choice of the scalar product on V . We show that the class of strongly local vertex operator algebras is closed under taking tensor products and unitary subalgebras and that, for every strongly local vertex operator algebra V , the map W → A W gives a one-to-one correspondence between the unitary subalgebras W of V and the covariant subnets of A V . Many known examples of vertex operator algebras such as the unitary Virasoro vertex operator algebras, the unitary affine Lie algebras vertex operator algebras, the known c = 1 unitary vertex operator algebras, the moonshine vertex operator algebra, together with their coset and orbifold subalgebras, turn out to be strongly local. We give various applications of our results. In particular we show that the even shorter Moonshine vertex operator algebra is strongly local and that the automorphism group of the corresponding conformal net is the Baby Monster group. We prove that a construction of Fredenhagen and Jörß gives back the strongly local vertex operator algebra V from the conformal net A V and give conditions on a conformal net A implying that A = A V for some strongly local vertex operator algebra V .if S ⊂ B(H) is self-adjoint then S ′ is a self-adjoint subalgebra of B(H) which is also unital, i.e. 1 H ∈ S ′ .A self-adjoint subalgebra M ⊂ B(H) is called a von Neumann algebra if M = M ′′ . Accordingly, (S ∪ S * ) ′ is a von Neumann algebra for all subsets S ⊂ B(H) and W * (S) ≡ (S ∪ S * ) ′′ is the smallest von Neumann algebra containing S.A von Neumann algebra M is said to be a factor if M ′ ∩ M = C1 H , i.e. M has a trivial center. B(H) is a factor for any Hilbert space H. Its isomorphism class as an abstract complex * -algebra only depends on the Hilbertian dimension of H. A von Neumann algebra M isomorphic to some B(H) (here H is not necessarily the same Hilbert space on which M acts) is called a type I factor. If H has dimension n ∈ Z >0 then M is called a type I n factor while if H is infinite-dimensional then M is called a type I ∞ factor.There exist factors which are not of type I. They are divided in two families: the type II factors (type II 1 or type II ∞ ) and type III factors (type III λ , λ ∈ [0, 1], cf.[23]).If M and N are von Neumann algebras and N ⊂ M then N is called a von Neumann subalgebra of M. If M is a factor then a von Neumann subalgebra N ⊂ M which is also a factor is called a subfactor. The theory of subfactors is a central topic in the theory of operator algebras and in its applications to quantum field theory. Subfactor theory was initiated in the seminal work [56] where V. Jones introduced and studied an index [M : N] for type II 1 factors. Subfac...

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“…These host algebras can be used to obtained direct integral decompositions of semibounded, resp., positive energy representations into irreducible ones. This provides an alternative to the complex analytic methods used in [NS14] for the same purpose. Since G does not carry an analytic Lie group structure, it would be meaningless to strengthen (S) in Theorem 5.15 to an analytic version.…”

Section: The Translation Action On a = C B (R)mentioning

confidence: 99%

“…The analysis of positive covariant representations for one-parameter automorphism groups of groups (rather than algebras) has been well-studied in mathematics (cf. [Ne14,NS14]), and we consider how the current analysis connects with that area in Section 5. Here one would choose for the algebra with the singular action the discrete group algebra C * (G d ).…”

Section: Introductionmentioning

confidence: 99%

Crossed products ofC⁎-algebras for singular actions

Grundling

Neeb

2014

Journal of Functional Analysis

Self Cite

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We analyze existence of crossed product constructions of Lie group actions on C *algebras which are singular. These are actions where the group need not be locally compact, or the action need not be strongly continuous. In particular, we consider the case where spectrum conditions are required for the implementing unitary group in covariant representations of such actions. The existence of a crossed product construction is guaranteed by the existence of "cross representations". For one-parameter automorphism groups, we prove that the existence of cross representations is stable with respect to a large set of perturbations of the action, and we fully analyze the structure of cross representations of inner actions on von Neumann algebras. For one-parameter automorphism groups we study the cross property for covariant representations, where the generator of the implementing unitary group is positive. In particular, we find that if the Borchers-Arveson minimal implementing group is cross, then so are all other implementing groups. We study a smoothing phenomenon for one-parameter actions on Lie groups, and display the usefulness of cross representations for this context. For higher dimensional Lie group actions, we consider a class of spectral conditions which include the ones occurring in physics, and is sensible also for non-abelian or for infinite dimensional Lie groups. We prove that the cross property of a covariant representation is fully determined by the cross property of a certain one-parameter subsystem. This greatly simplifies the analysis of the existence of cross representations, and it allows us to prove the cross property for several examples of interest to physics. We also consider non-abelian extensions of the Borchers-Arveson theorem. There is a full extension in the presence of a cyclic invariant vector, but otherwise one needs to determine the vanishing of lifting obstructions.

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Classification of Positive Energy Representations of the Virasoro Group

Abstract: We give a complete classification of all positive energy unitary representations of the Virasoro group. More precisely, we prove that every such representation can be expressed in an essentially unique way as a direct integral of irreducible highest weight representations.

Cited by 15 publications

References 22 publications

Quantization of BMS 3 orbits: A perturbative approach

Quantization of BMS 3 orbits: A perturbative approach

From Vertex Operator Algebras to Conformal Nets and Back

Crossed products ofC⁎-algebras for singular actions

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