Operator Algebras and Noncommutative Geometric Aspects in Conformal - PowerPoint PPT Presentation

Operator Algebras and Noncommutative Geometric Aspects in Conformal Field Theory Roberto Longo University of Rome “Tor Vergata” Vietri, September 2009 Recent work based on joint papers with S. Carpi, Y. Kawahigashi and R. Hillier

Things to discuss ◮ Getting inspired by black hole entropy ◮ Symmetry and supersymmetry ◮ Local conformal nets ◮ Modularity and asymptotic formulae ◮ Fermi and superconformal nets ◮ Neveu-Schwarz and Ramond representations ◮ Fredholm index and Jones index ◮ Noncommutative geometrization ◮ Model analysis (in progress)

Prelude. Black hole entropy Bekenstein: The entropy S of a black hole is proportional to the area A of its horizon S = A / 4 ◮ S is geometric ◮ S is proportional to the area , not to the volume as a naive microscopic interpretation of entropy would suggest (logarithmic counting of possible states). ◮ This dimensional reduction has led to the holographic principle by t’Hooft, Susskind, . . . ◮ The horizon is not a physical boundary, but a submanifold where coordinates pick critical values → conformal symmetries ◮ The proportionality factor 1 / 4 is fixed by Hawking temperature ( quantum effect).

Black hole entropy Discretization of the horizon (Bekenstein): horizon is made of cells or area ℓ 2 and k degrees of freedom ( ℓ = Planck length): A = n ℓ 2 , Degrees of freedom = k n , S = Cn log k = C A ℓ 2 log k , d S = C log k Conclusion. Black hole entropy ↓ Two-dimensional conformal quantum field theory with a “fuzzy” point of view Legenda: Fuzzy = noncommutative geometrical

� Symmetries in Physics Spacetime symme- Internal symme- tries tries Lorentz, Gauge, . . . Poincar´ e,. . . � ����������������� � � � � � � � � � � � � � � � � SUSY Bose-Fermi SUSY: H = Q 2 , Q odd operator, [ · , Q ] graded super-derivation interchanging Boson and Fermions Among consequences: Cancellation of some Higgs boson divergence

Conformal and superconformal ◮ Low dimension, conformal → infinite dim. symmetry ◮ Low dimension, conformal + SUSY → Superconformal symmetry (very stringent)

� � � � About three approaches to CFT Vertex Algebras � Wightman fields Kac (algebraic) (analytic) � � ��������� � � � � � � � Carpi − Weiner Fredenhagen − Jorss � � � ������� � � � � � � Operator Algebras (algebraic & ana- lytic) partial relations known

von Neumann algebras H Hilbert space, B ( H ) ∗ algebra of all bounded linear operators on H . Def. A von Neumann algebra M is a weakly closed non-degenerate ∗ -subalgebra of B ( H ). • von Neumann density thm. A ⊂ B ( H ) non-degenerate ∗ -subalgebra A − = A ′′ where ′ denotes the commutant A ′ = { T ∈ B ( H ) : TA = AT ∀ A ∈ A } Double aspect, analytical and algebraic M is a factor if its center M ∩ M ′ = C .

The tensor category End( M ) M an infinite factor → End( M ) is a tensor C ∗ -category : ◮ Objects : End( M ) ◮ Arrows : Hom( ρ, ρ ′ ) ≡ { t ∈ M : t ρ ( x ) = ρ ′ ( x ) t ∀ x ∈ M } ◮ Tensor product of objects : ρ ⊗ ρ ′ = ρρ ′ ◮ Tensor product of arrows : σ, σ ′ ∈ End( M ), t ∈ Hom( ρ, ρ ′ ), s ∈ Hom( σ, σ ′ ), t ⊗ s ≡ t ρ ( s ) = ρ ′ ( s ) t ∈ Hom( ρ ⊗ σ, ρ ′ ⊗ σ ′ ) ◮ Conjugation : ∃ isometries v ∈ Hom( ι, ρ ¯ ρ ) and ¯ v ∈ Hom( ι, ¯ ρρ ) such that ρ ( v ) = 1 v ∗ ⊗ 1 ¯ v ∗ ¯ (¯ ρ ) · (1 ¯ ρ ⊗ v ) ≡ ¯ d v ) = 1 ( v ∗ ⊗ 1 ρ ) · (1 ρ ⊗ ¯ v ) ≡ v ∗ ρ (¯ d for some d > 0.

Dimension The minimal d is the dimension d ( ρ ) [ M : ρ ( M )] = d ( ρ ) 2 (tensor categorical definition of the Jones index) d ( ρ 1 ⊕ ρ 2 ) = d ( ρ 1 ) + d ( ρ 2 ) d ( ρ 1 ρ 2 ) = d ( ρ 1 ) d ( ρ 2 ) d (¯ ρ ) = d ( ρ ) End(M) is a “universal” tensor category (cf. Popa, Yamagami) (generalising the Doplicher-Haag-Roberts theory)

Local conformal nets obius covariant net A on S 1 is a map A local M¨ I ∈ I → A ( I ) ⊂ B ( H ) I ≡ family of proper intervals of S 1 , that satisfies: ◮ A. Isotony . I 1 ⊂ I 2 = ⇒ A ( I 1 ) ⊂ A ( I 2 ) ◮ B. Locality . I 1 ∩ I 2 = ∅ = ⇒ [ A ( I 1 ) , A ( I 2 )] = { 0 } ◮ C. M¨ obius covariance . ∃ unitary rep. U of the M¨ obius group M¨ ob on H such that U ( g ) A ( I ) U ( g ) ∗ = A ( gI ) , g ∈ M¨ ob , I ∈ I . ◮ D. Positivity of the energy . Generator L 0 of rotation subgroup of U (conformal Hamiltonian) is positive. ◮ E. Existence of the vacuum . ∃ ! U -invariant vector Ω ∈ H (vacuum vector), and Ω is cyclic for � I ∈I A ( I ).

First consequences ◮ Irreducibility : � I ∈I A ( I ) = B ( H ). ◮ Reeh-Schlieder theorem : Ω is cyclic and separating for each A ( I ). ◮ Bisognano-Wichmann property : Tomita-Takesaki modular operator ∆ I and conjugation J I of ( A ( I ) , Ω), are U (Λ I (2 π t )) = ∆ it I , t ∈ R , dilations U ( r I ) = J I reflection (Fr¨ olich-Gabbiani, Guido-L.) ◮ Haag duality : A ( I ) ′ = A ( I ′ ) ◮ Factoriality : A ( I ) is III 1 -factor (in Connes classification) ◮ Additivity : I ⊂ ∪ i I i = ⇒ A ( I ) ⊂ ∨ i A ( I i ) (Fredenhagen, Jorss).

Local conformal nets Diff ( S 1 ) ≡ group of orientation-preserving smooth diffeomorphisms of S 1 Diff I ( S 1 ) ≡ { g ∈ Diff ( S 1 ) : g ( t ) = t ∀ t ∈ I ′ } . A local conformal net A is a M¨ obius covariant net s.t. F. Conformal covariance . ∃ a projective unitary representation U of Diff ( S 1 ) on H extending the unitary representation of M¨ ob s.t. U ( g ) A ( I ) U ( g ) ∗ = A ( gI ) , g ∈ Diff ( S 1 ) , U ( g ) xU ( g ) ∗ = x , x ∈ A ( I ) , g ∈ Diff I ′ ( S 1 ) , − → unitary representation of the Virasoro algebra [ L m , L n ] = ( m − n ) L m + n + c 12( m 3 − m ) δ m , − n [ L n , c ] = 0, L ∗ n = L − n .

Representations A representation π of A on a Hilbert space H is a map I ∈ I �→ π I , normal rep. of A ( I ) on B ( H ) I ⊂ ˜ π ˜ I ↾ A ( I ) = π I , I π is automatically diffeomorphism covariant : ∃ a projective, pos. energy, unitary rep. U π of Diff ( ∞ ) ( S 1 ) s.t. π gI ( U ( g ) xU ( g ) ∗ ) = U π ( g ) π I ( x ) U π ( g ) ∗ for all I ∈ I , x ∈ A ( I ), g ∈ Diff ( ∞ ) ( S 1 ) (Carpi & Weiner) DHR argument: given I , there is an endomorphism of A localized in I equivalent to π ; namely ρ is a representation of A on the vacuum Hilbert space H , unitarily equivalent to π , such that ρ I ′ = id ↾ A ( I ′ ) . • Rep( A ) is a braided tensor category (DHR, FRS, L.)

Index-statistics theorem � DHR dimension d ( ρ ) = Jones index Ind( ρ ) tensor category Rep I ( A ) full functor − − restriction tensor category End( A ( I )) − − − − →

Black hole incremental free energy Define the incremental free energy F ( ϕ σ | ϕ ρ ) between the thermal states ϕ σ and ϕ ρ in reps ρ , σ localized in I ( β − 1 = Hawking temperature) F ( ϕ σ | ϕ ρ ) = ϕ ρ ( H ρ ) − β − 1 S ( ϕ σ | ϕ ρ ) S ( ϕ σ | ϕ ρ ) = − (log ∆ ξ σ ,ξ ρ ξ ρ , ξ ρ ) is Araki relative entropy Then F ( ϕ σ | ϕ ρ ) =1 2 β − 1 � � d ( σ ) − d ( ρ ) =1 2 β − 1 (log m − log n ) If the charges ρ , σ come from a spacetime of dimension d ≥ 2 + 1 then n , m integers by DHR restriction on the values d ( ρ ), d ( σ ).

Complete rationality I 1 , I 2 intervals ¯ I 1 ∩ ¯ I 2 = ∅ , E ≡ I 1 ∪ I 2 . µ A ≡ [ A ( E ′ ) ′ : A ( E )] µ -index : (Jones index). A conformal: A completely rational def = A split & µ A < ∞ Thm. (Y. Kawahigashi, M. M¨ uger, R.L.) A completely rational: then � d ( ρ i ) 2 µ A = i sum over all irreducible sectors. (F. Xu in SU ( N ) models); • A ( E ) ⊂ A ( E ′ ) ′ ∼ LR inclusion (quantum double); • Representations form a modular tensor category (i.e. non-degenerate braiding).

Weyl’s theorem M compact oriented Riemann manifold, ∆ Laplace operator on L 2 ( M ). Theorem (Weyl) Heat kernel expansion as t → 0 + : 1 Tr( e − t ∆ ) ∼ (4 π t ) n / 2 ( a 0 + a 1 t + · · · ) The spectral invariants n and a 0 , a 1 , . . . encode geometric information and in particular a 1 = 1 � a 0 = vol ( M ) , κ ( m ) d vol ( m ) , 6 M κ scalar curvature. n = 2: a 1 is proportional to the Euler 1 � characteristic = M κ ( m ) d vol ( m ) by Gauss-Bonnet theorem. 2 π

Modularity With ρ rep. of A , set L 0 ,ρ conf. Hamiltonian of ρ , e 2 π i τ ( L 0 ,ρ − c / 24) � � χ ρ ( τ ) = Tr Im τ > 0 . specialized character, c the central charge. A is modular if µ A < ∞ and � χ ρ ( − 1 /τ ) = S ρ,ν χ ν ( τ ) , ν � χ ρ ( τ + 1) = T ρ,ν χ ν ( τ ) . ν with S , T the (algebraically defined) Kac-Peterson, Verlinde Rehren matrices generating a representation of SL (2 , Z ). One has: • Modularity = ⇒ complete rationality • Modularity holds in all computed rational case, e.g. SU ( N ) k -models • A modular, B ⊃ A irreducible extension = ⇒ B modular. • All conformal nets with central charge c < 1 are modular.

Asymptotics A modular. The following asymptotic formula holds as t → 0 + : log Tr( e − 2 π tL 0 ) ∼ π c 1 t − 1 2 log µ A − π c 12 t 12 In any representation ρ , as t → 0 + : 2 log d ( ρ ) 2 log Tr( e − 2 π tL 0 ,ρ ) ∼ π c 1 t + 1 − π c 12 t 12 µ A