Characterization of rational conformal QFTs and their boundary - PowerPoint PPT Presentation

Characterization of rational conformal QFTs and their boundary conditions 1 Marcel Bischoff http://www.theorie.physik.uni-goettingen.de/~bischoff Research Training Group 1493 Mathematical Structures in Modern Quantum Physics University of G¨ ottingen 32nd Workshop ”Foundations and Constructive Aspects of QFT” Wuppertal, 31 May 2013 1 work in progress with Roberto Longo and Yasuyuki Kawahigashi Marcel Bischoff (Uni G¨ ottingen) Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Introduction ◮ Algebraic quantum field theory: A family of algebras containing all local observables associated to space-time regions. ◮ Many structural results, recently also construction of interesting models ◮ Conformal field theory (CFT) in 1 and 2 dimension described by AQFT quite successful, e.g. partial classification results (e.g. c < 1 ) (Kawahigashi and Longo, 2004) ◮ Boundary Conformal Quantum Field Theory (BCFT) on Minkowski half-plane: (Longo and Rehren, 2004) Marcel Bischoff (Uni G¨ ottingen) Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Outline Conformal Nets Nets on Minkowski space Nets on Minkowski half-plane Boundary conditions Marcel Bischoff (Uni G¨ ottingen) Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Conformal Nets H Hilbert space, I = family of proper intervals on S 1 ∼ = R → A ( I ) = A ( I ) ′′ ⊂ B( H ) I ∋ I �− 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. There is a unitary representation U of the M¨ obius group ( ∼ = PSL(2 , R ) on H such that U ( g ) A ( I ) U ( g ) ∗ = A ( gI ) . D. Positivity of energy. U is a positive-energy representation, i.e. generator L 0 of the rotation subgroup (conformal Hamiltonian) has positive spectrum. E. Vacuum. ker L 0 = C Ω and Ω (vacuum vector) is a unit vector cyclic for the von Neumann algebra � I ∈I A ( I ) . Consequences Complete Rationality Marcel Bischoff (Uni G¨ ottingen) Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Outline Conformal Nets Nets on Minkowski space Nets on Minkowski half-plane Boundary conditions Marcel Bischoff (Uni G¨ ottingen) Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Some consequences ◮ Irreducibility. � I ∈I A ( I ) = B ( H ) ◮ Reeh-Schlieder theorem. Ω is cyclic and separating for each A ( I ) . ◮ Bisognano-Wichmann property. The Tomita-Takesaki modular operator ∆ I and and conjugation J I of the pair ( A ( I ) , Ω) are U (Λ( − 2 πt )) = ∆ i t , t ∈ R dilation U ( r I ) = J I reflection ohlich, 1993) , (Guido and Longo, 1995) (Gabbiani and Fr¨ ◮ Haag duality. A ( I ′ ) = A ( I ) ′ . ◮ Factoriality. A ( I ) is III 1 -factor (in (Connes, 1973) classification) ◮ Additivity. I ⊂ � ⇒ A ( I ) ⊂ � i I i = i A ( I i ) (Fredenhagen and J¨ orß, 1996) . example complete rationality Marcel Bischoff (Uni G¨ ottingen) Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Representations A representation of A is a family of representations π = { π I : A ( I ) → B( H π ) } on a Hilbert space H π such that π J ↾ A ( I ) = π I I ⊂ J. Fact: Let π be a non-degenerated represenatation on a seperable space H π and let I ∈ I then there is a unitary equivalent representation ρ on H , such that: 1. ρ I ′ = id A ( I ′ ) , i.e. ρ is localized in I . 2. ρ J ( A ( J )) ⊂ A ( J ) for all J ⊃ I , i.e. ρ J is an endomorphism of A ( J ) . We call ρ J an DHR endomorphism. It is enough to look into representation localized in I . Rep I ( A ) is a full subcategory of End( N ) with N = A ( I ) . A sector is a unitary equivalence class [ π ] . We can define the fusion by composition of DHR endomorphisms. [ π 1 ] × [ π 2 ] := [ ρ 1 ◦ ρ 2 ] ρ i ∈ [ π i ] localized in I Marcel Bischoff (Uni G¨ ottingen) Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

End( N ) Let N be a type III factor and End( N ) the C ∗ -tensor category with: ◮ objects: endomorphisms ρ ∈ End( N ) ◮ arrows: intertwiner t : ρ → σ with t ∈ Hom( ρ, σ ) = { s ∈ N : sρ ( n ) = σ ( n ) s for all n ∈ N} ◮ ⊗ -product: ρ ⊗ σ = ρ ◦ σ (composition), s : σ → σ ′ and t : τ → τ ′ then s ⊗ t : σ : τ ◦ σ → τ ′ ◦ σ ′ given by s ⊗ t = sσ ( t ) = σ ′ ( t ) s . A sector [ ρ ] is the unitary equivalence class ( ρ ∼ ρ ′ ⇔ ρ ( · ) = Uρ ′ ( · ) for some U ∈ N unitary). Direct sums : w 1 w ∗ 1 + w 2 w ∗ 2 = 1 , w ∗ [ ρ ] ⊕ [ σ ] = [Ad w 1 ◦ ρ + Ad w 2 ◦ σ ] i w j = δ ij Proposition (Longo) Irreducible finite depth subfactors ι ( N ) ⊂ M ← → Q-systems ( θ, w, x ) in End( N ) , where θ = ¯ ι ◦ ι ∈ End( N ) is the dual canonical endomorphism , an algebra object with unit w ∗ : θ → id and counit x : θ → θ ◦ θ . Marcel Bischoff (Uni G¨ ottingen) Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Braiding Fusion coefficients: � N τ [ ρ ] × [ σ ] = ρσ [ τ ] [ τ ] with fusion coefficients N τ ρσ = dim Hom( ρσ, τ ) . The fusion is commutative [ π 1 ] × [ π 2 ] = [ π 2 ] × [ π 1 ] and there is a natural choice of unitaries, the braiding : ρ, σ ∈ Rep I ( A ) ε ( ρ, σ ) = : ρ ◦ σ → σ ◦ ρ the braiding. Fulfills naturality (braiding fusion equations) and Yang-Baxter identity: = Marcel Bischoff (Uni G¨ ottingen) Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Conjugates Let us consider Rep I f ( A ) , i.e. only representations with finite statistical dimension dρ < ∞ , where [ M : N ] denotes the minimal (Jones) index: � � ( dρ ) 2 = ρ J ′ ( A ( J ′ )) ′ : ρ J ( A ( J )) ≡ [ A ( I ) : ρ I ( A ( I ))] For [ ρ ] one can define a conjugate DHR sector [¯ ρ ] by ¯ ρ I ′ = j ◦ ρ I ◦ j where j is the anti-automorphism of A ( I ) given by Bisognano–Wichmann property. Then there exist ¯ R ∈ Hom(id , ρ ◦ ¯ ρ ) and R ∈ Hom(id , ¯ ρ ◦ ρ ) fulfilling the zig-zag identity: ρ ρ ρ ¯ ρ ¯ ρ ¯ ρ ρ ρ ¯ ¯ R = R = ; = ; = id id ρ ρ ρ ¯ ρ ¯ Unitary ribbon category Marcel Bischoff (Uni G¨ ottingen) Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Complete rationality Completely rational conformal net (Kawahigashi, Longo, M¨ uger (2001)) ◮ Split property. For every relatively compact inclusion of intervals ∃ intermediate type I factor M � � � � A ⊂ M ⊂ A ◮ Finite µ -index: finite Jones index of subfactor � � � � � � � � �� ′ A ∨ A ⊂ A ∨ A where the intervals are splitting the circle. Consequences ◮ Strong additivity. (Longo and Xu, 2004) Additivity for touching intervals: � � � � � � A ∨ A = A ◮ Only finite sectors, each sector has finite statistical dimension ◮ Modularity: The category of DHR sectors is modular, i.e. non degenerated braiding. Marcel Bischoff (Uni G¨ ottingen) Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Complete rationality Completely rational conformal net (Kawahigashi, Longo, M¨ uger (2001)) ◮ Split property. For every relatively compact inclusion of intervals ∃ intermediate type I factor M � � � � A ⊂ M ⊂ A ◮ Finite µ -index: finite Jones index of subfactor � � � � � � � � �� ′ A ∨ A ⊂ A ∨ A where the intervals are splitting the circle. Consequences ◮ Strong additivity. (Longo and Xu, 2004) Additivity for touching intervals: � � � � � � A ∨ A = A ◮ Only finite sectors, each sector has finite statistical dimension ◮ Modularity: The category of DHR sectors is modular, i.e. non degenerated braiding. Marcel Bischoff (Uni G¨ ottingen) Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Complete rationality Completely rational conformal net (Kawahigashi, Longo, M¨ uger (2001)) ◮ Split property. For every relatively compact inclusion of intervals ∃ intermediate type I factor M � � � � A ⊂ M ⊂ A ◮ Finite µ -index: finite Jones index of subfactor � � � � � � � � �� ′ A ∨ A ⊂ A ∨ A where the intervals are splitting the circle. Consequences ◮ Strong additivity. (Longo and Xu, 2004) Additivity for touching intervals: � � � � � � A ∨ A = A ◮ Only finite sectors, each sector has finite statistical dimension ◮ Modularity: The category of DHR sectors is modular, i.e. non degenerated braiding. Marcel Bischoff (Uni G¨ ottingen) Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Complete rationality Completely rational conformal net (Kawahigashi, Longo, M¨ uger (2001)) ◮ Split property. For every relatively compact inclusion of intervals ∃ intermediate type I factor M � � � � A ⊂ M ⊂ A ◮ Finite µ -index: finite Jones index of subfactor � � � � � � � � �� ′ A ∨ A ⊂ A ∨ A where the intervals are splitting the circle. Consequences ◮ Strong additivity. (Longo and Xu, 2004) Additivity for touching intervals: � � � � � � A ∨ A = A ◮ Only finite sectors, each sector has finite statistical dimension ◮ Modularity: The category of DHR sectors is modular, i.e. non degenerated braiding. Marcel Bischoff (Uni G¨ ottingen) Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013