Semigroup elements associated to conformal nets and boundary quantum field theory

Transcription

1 Semigroup elements associated to conformal nets and boundary quantum field theory Marcel Bischoff Dipartimento di Matematica Università degli Studi di Roma Tor Vergata Meeting of GDRE GREFI-GENCO Institut Henri Poincaré Paris, 1 June 2011

2 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) Boundary Quantum Field Theory (BQFT) on Minkowski half-plane: (Longo and Witten, 2010)

3 Outline Conformal Nets Nets on Minkowski half-plane Standard subspaces Conformal nets associated to lattices Semigroup elements

4 Conformal Nets H Hilbert space, I = family of proper intervals on S 1 = R I I A(I) = A(I) B(H) 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. Möbius covariance. There is a unitary representation U of the Möbius 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 = CΩ and Ω (vacuum vector) is a unit vector cyclic for the von Neumann algebra I I A(I). Consequences Complete Rationality

5 Outline Conformal Nets Nets on Minkowski half-plane Standard subspaces Conformal nets associated to lattices Semigroup elements

6 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( 2πt)) = it, t R U(r I ) = J I dilation reflection (Gabbiani and Fröhlich, 1993), (Guido and Longo, 1995) 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 and Jörß, 1996). example complete rationality

7 Complete rationality Completely rational conformal net (Kawahigashi, Longo, Müger 2001) Split property. For every relatively compact inclusion of intervalls intermediate type I factor M ( ) ( ) A M A Strong additivity. Additivity for touching intervals: ( ) ( ) ( ) A A = A Finite µ-index: finite Jones index of subfactor ( ) ( ) ( ( ) ( )) A A A A where the intervals are splitting the circle. Consequences Only finite sectors, each sector has finite statistical dimension Modularity: The category of DHR sectors is modular, i.e. non degenerated braiding.

8 Complete rationality Completely rational conformal net (Kawahigashi, Longo, Müger 2001) Split property. For every relatively compact inclusion of intervalls intermediate type I factor M ( ) ( ) A M A Strong additivity. Additivity for touching intervals: ( ) ( ) ( ) A A = A Finite µ-index: finite Jones index of subfactor ( ) ( ) ( ( ) ( )) A A A A where the intervals are splitting the circle. Consequences Only finite sectors, each sector has finite statistical dimension Modularity: The category of DHR sectors is modular, i.e. non degenerated braiding.

9 Complete rationality Completely rational conformal net (Kawahigashi, Longo, Müger 2001) Split property. For every relatively compact inclusion of intervalls intermediate type I factor M ( ) ( ) A M A Strong additivity. Additivity for touching intervals: ( ) ( ) ( ) A A = A Finite µ-index: finite Jones index of subfactor ( ) ( ) ( ( ) ( )) A A A A where the intervals are splitting the circle. Consequences Only finite sectors, each sector has finite statistical dimension Modularity: The category of DHR sectors is modular, i.e. non degenerated braiding.

10 Complete rationality Completely rational conformal net (Kawahigashi, Longo, Müger 2001) Split property. For every relatively compact inclusion of intervalls intermediate type I factor M ( ) ( ) A M A Strong additivity. Additivity for touching intervals: ( ) ( ) ( ) A A = A Finite µ-index: finite Jones index of subfactor ( ) ( ) ( ( ) ( )) A A A A where the intervals are splitting the circle. Consequences Only finite sectors, each sector has finite statistical dimension Modularity: The category of DHR sectors is modular, i.e. non degenerated braiding.

11 Loop group net Example G compact Lie group Loop group: LG = C (S 1, G) (point wise multiplication) Projective representations representations of a central extension 1 T LG LG 1 π 0,k projective positive-energy and vacuum representation (classified by the level k) I A G,k (I) = π 0,k (L I G) is a conformal net; L I G loops supported in I. Example G = SU(n) gives completely rational conformal net (Xu, 2000)

12 Loop group net Example G compact Lie group Loop group: LG = C (S 1, G) (point wise multiplication) Projective representations representations of a central extension 1 T LG LG 1 π 0,k projective positive-energy and vacuum representation (classified by the level k) I A G,k (I) = π 0,k (L I G) is a conformal net; L I G loops supported in I. Example G = SU(n) gives completely rational conformal net (Xu, 2000)

13 Loop group net Example G compact Lie group Loop group: LG = C (S 1, G) (point wise multiplication) Projective representations representations of a central extension 1 T LG LG 1 π 0,k projective positive-energy and vacuum representation (classified by the level k) I A G,k (I) = π 0,k (L I G) is a conformal net; L I G loops supported in I. Example G = SU(n) gives completely rational conformal net (Xu, 2000)

14 Loop group net Example G compact Lie group Loop group: LG = C (S 1, G) (point wise multiplication) Projective representations representations of a central extension 1 T LG LG 1 π 0,k projective positive-energy and vacuum representation (classified by the level k) I A G,k (I) = π 0,k (L I G) is a conformal net; L I G loops supported in I. Example G = SU(n) gives completely rational conformal net (Xu, 2000)

15 Outline Conformal Nets Nets on Minkowski half-plane Standard subspaces Conformal nets associated to lattices Semigroup elements

16 Nets on the real line Conformal net on the real line identifying S 1 \ { 1} = R Conformal net on S 1 restriction Conformal net on R I

17 Minkowski half-plane M + Minkowski half-plane x > 0, ds 2 = dt 2 dx 2 Double cone O = I 1 I 2 where I 1, I 2 disjoint intervals t x

18 Minkowski half-plane M + Minkowski half-plane x > 0, ds 2 = dt 2 dx 2 Double cone O = I 1 I 2 where I 1, I 2 disjoint intervals t I 2 O I 1 x

19 Minkowski half-plane Boundary conformal quantum field theory (Longo and Rehren, 2004) t A + (O) = A(I 1 ) A(I 2 ) I 2 O Boundary quantum field theory (Longo and Witten, 2010) A V (O) = A(I 1 ) V A(I 2 )V I 1 x V unitary on H [V, T (t)] = 0, i.e. commutes with translation T (t) V A(R + )V A(R + )

20 Minkowski half-plane Boundary conformal quantum field theory (Longo and Rehren, 2004) t A + (O) = A(I 1 ) A(I 2 ) I 2 O Boundary quantum field theory (Longo and Witten, 2010) A V (O) = A(I 1 ) V A(I 2 )V I 1 x V unitary on H [V, T (t)] = 0, i.e. commutes with translation T (t) V A(R + )V A(R + )

21 Local nets on Minkowski half-plane A local (time) translation covariant net on Minkowski half-plane on a Hilbert space H is a map K + O B(O) B(H) which fulfills: 1. Isotony. O 1 O 2 implies B(O 1 ) B(O 2 ). 2. Locality. If O 1, O 2 K + are mutually space-like separated then [B(O 1 ), B(O 2 )] = {0}.

22 Local nets on Minkowski half-plane A local (time) translation covariant net on Minkowski half-plane on a Hilbert space H is a map K + O B(O) B(H) which fulfills: 1. Isotony. O 1 O 2 implies B(O 1 ) B(O 2 ). 2. Locality. If O 1, O 2 K + are mutually space-like separated then [B(O 1 ), B(O 2 )] = {0}. O 1 O 2

23 Local nets on Minkowski half-plane A local (time) translation covariant net on Minkowski half-plane on a Hilbert space H is a map K + O B(O) B(H) which fulfills: 1. Isotony. O 1 O 2 implies B(O 1 ) B(O 2 ). 2. Locality. If O 1, O 2 K + are mutually space-like separated then [B(O 1 ), B(O 2 )] = {0}. O 1 O 2

24 Local nets on Minkowski half-plane A local (time) translation covariant net on Minkowski half-plane on a Hilbert space H is a map K + O B(O) B(H) which fulfills: 1. Isotony. O 1 O 2 implies B(O 1 ) B(O 2 ). 2. Locality. If O 1, O 2 K + are mutually space-like separated then [B(O 1 ), B(O 2 )] = {0}. 3. Time-translation covariance an unitary one-parameter group T (t) = e itp with positive generator P such that: T (t)b(o)t (t) = B(O t ), O K +, O t = O + (t, 0) O t O

25 Local nets on Minkowski half-plane A local (time) translation covariant net on Minkowski half-plane on a Hilbert space H is a map K + O B(O) B(H) which fulfills: 1. Isotony. O 1 O 2 implies B(O 1 ) B(O 2 ). 2. Locality. If O 1, O 2 K + are mutually space-like separated then [B(O 1 ), B(O 2 )] = {0}. 3. Time-translation covariance an unitary one-parameter group T (t) = e itp with positive generator P such that: T (t)b(o)t (t) = B(O t ), O K +, O t = O + (t, 0) 4. Vacuum. Ω H is a up to the multiple unique T invariant vector and cyclic and separating for every B(O) for O K +.

26 Semigroup E(A) associated to a conformal net A Semigroup E(A) of unitaries on H (associated to A) [V, T (t)] = 0, i.e. commutes with translation T (t) V A(R + )V A(R + ) V A(a + R + )V A(a + R + ) Trivial examples of elements in E(A): V = T (t) t > 0 positive translations V inner symmetry, i.e V A(I)V = A(I) for all proper I Construction Conformal net A on R + semigroup element V E(A) local net A V on M +

27 Outline Conformal Nets Nets on Minkowski half-plane Standard subspaces Conformal nets associated to lattices Semigroup elements

28 Standard subspaces H complex Hilbert space, H H real subspace. Symplectic complement: H = {x H : Im(x, H) = 0} = ih Standard subspace: closed, real subspace H H with H + ih = H and H ih = {0}. Define antilinear unbounded closed involutive (S 2 1) operator S H : x + iy x iy for x, y H. Conversely S densely defined closed, antilinear involution on H, H S = {x H : Sx = x} is a standard subspace: standard subspaces H 1:1 densely defined, closed, antilinear involutions S Modular Theory: Polar decomposition S H = J H 1/2 H J H H = H it HH = H

29 Standard subspaces H complex Hilbert space, H H real subspace. Symplectic complement: H = {x H : Im(x, H) = 0} = ih Standard subspace: closed, real subspace H H with H + ih = H and H ih = {0}. Define antilinear unbounded closed involutive (S 2 1) operator S H : x + iy x iy for x, y H. Conversely S densely defined closed, antilinear involution on H, H S = {x H : Sx = x} is a standard subspace: standard subspaces H 1:1 densely defined, closed, antilinear involutions S Modular Theory: Polar decomposition S H = J H 1/2 H J H H = H it HH = H

30 Standard subspaces H complex Hilbert space, H H real subspace. Symplectic complement: H = {x H : Im(x, H) = 0} = ih Standard subspace: closed, real subspace H H with H + ih = H and H ih = {0}. Define antilinear unbounded closed involutive (S 2 1) operator S H : x + iy x iy for x, y H. Conversely S densely defined closed, antilinear involution on H, H S = {x H : Sx = x} is a standard subspace: standard subspaces H 1:1 densely defined, closed, antilinear involutions S Modular Theory: Polar decomposition S H = J H 1/2 H J H H = H it HH = H

31 Standard subspaces H complex Hilbert space, H H real subspace. Symplectic complement: H = {x H : Im(x, H) = 0} = ih Standard subspace: closed, real subspace H H with H + ih = H and H ih = {0}. Define antilinear unbounded closed involutive (S 2 1) operator S H : x + iy x iy for x, y H. Conversely S densely defined closed, antilinear involution on H, H S = {x H : Sx = x} is a standard subspace: standard subspaces H 1:1 densely defined, closed, antilinear involutions S Modular Theory: Polar decomposition S H = J H 1/2 H J H H = H it HH = H

32 Standard subspaces H complex Hilbert space, H H real subspace. Symplectic complement: H = {x H : Im(x, H) = 0} = ih Standard subspace: closed, real subspace H H with H + ih = H and H ih = {0}. Define antilinear unbounded closed involutive (S 2 1) operator S H : x + iy x iy for x, y H. Conversely S densely defined closed, antilinear involution on H, H S = {x H : Sx = x} is a standard subspace: standard subspaces H 1:1 densely defined, closed, antilinear involutions S Modular Theory: Polar decomposition S H = J H 1/2 H J H H = H it HH = H

33 Standard subspaces and inner functions Standard pair. (H, T ) H H standard subspace with T (t) = e itp one-param. group with positive generator P T (t)h H for t 0 Theorem (Borchers Theorem for standard subspaces) Let (H, T ) be a standard pair, then is HT (t) is H = T (e 2πs t) (s, t R) J H T (t)j H = T ( t) (t R)

34 Standard subspaces and inner functions Standard pair. (H, T ) H H standard subspace with T (t) = e itp one-param. group with positive generator P T (t)h H for t 0 Theorem (Borchers Theorem for standard subspaces) Let (H, T ) be a standard pair, then is HT (t) is H = T (e 2πs t) (s, t R) J H T (t)j H = T ( t) (t R)

35 Standard subspaces and inner functions E(H) = unitaries V on H such that V H H and [V, T (t)] = 0. Analog of the Beurling-Lax theorem. Characterization of E(H). (Longo and Witten, 2010) (H, T ) irreducible standard pair, then are equivalent 1. V E(H), i.e. V H H with V unitary on H commuting with T. 2. V = ϕ(p ) with ϕ boundary value of a symmetric inner analytic L function ϕ : R + ir + C, where symmetric ϕ(p) = ϕ( p) for p 0 inner ϕ(p) = 1 for p R. R + ir + C

36 Standard subspaces and inner functions E(H) = unitaries V on H such that V H H and [V, T (t)] = 0. Analog of the Beurling-Lax theorem. Characterization of E(H). (Longo and Witten, 2010) (H, T ) irreducible standard pair, then are equivalent 1. V E(H), i.e. V H H with V unitary on H commuting with T. 2. V = ϕ(p ) with ϕ boundary value of a symmetric inner analytic L function ϕ : R + ir + C, where symmetric ϕ(p) = ϕ( p) for p 0 inner ϕ(p) = 1 for p R. R + ir + C

37 Standard subspaces and inner functions E(H) = unitaries V on H such that V H H and [V, T (t)] = 0. Analog of the Beurling-Lax theorem. Characterization of E(H). (Longo and Witten, 2010) (H, T ) irreducible standard pair, then are equivalent 1. V E(H), i.e. V H H with V unitary on H commuting with T. 2. V = ϕ(p ) with ϕ boundary value of a symmetric inner analytic L function ϕ : R + ir + C, where symmetric ϕ(p) = ϕ( p) for p 0 inner ϕ(p) = 1 for p R. R + ir + C

38 Outline Conformal Nets Nets on Minkowski half-plane Standard subspaces Conformal nets associated to lattices Semigroup elements

39 Net of free bosons. Net of standard subspaces (prequantised theory) LR = C (S 1, R) yields a Hilbert space H = LR using semi-norm. f = k>0 k ˆf k complex-structure. J : ˆfk i sign(k) ˆf k symplectic form. ω(f, g) = Im(f, g) = 1/(4π) gdf Local spaces: L I R = {f LR : suppf I} Conformal net of a free boson I H(I) = L I R H Second quantization. Conformal net on the symmetric Fock space e H by CCR functor (Weyl unitaries): I A(I) := CCR(H(I)) B(e H ) Weyl unitaries W (f)w (g) = e iω(f,g) W (f + g), Vacuum state φ(w (f)) = (Ω, W (f)ω) = e 1/2 f 2

40 Net of free bosons. Net of standard subspaces (prequantised theory) LR = C (S 1, R) yields a Hilbert space H = LR using semi-norm. f = k>0 k ˆf k complex-structure. J : ˆfk i sign(k) ˆf k symplectic form. ω(f, g) = Im(f, g) = 1/(4π) gdf Local spaces: L I R = {f LR : suppf I} Conformal net of a free boson I H(I) = L I R H Second quantization. Conformal net on the symmetric Fock space e H by CCR functor (Weyl unitaries): I A(I) := CCR(H(I)) B(e H ) Weyl unitaries W (f)w (g) = e iω(f,g) W (f + g), Vacuum state φ(w (f)) = (Ω, W (f)ω) = e 1/2 f 2

41 Localized automorphisms Conformal net of n free bosons A n (I) = A n 1 (I) = CCR(H(I) H(I)) Local endomorphisms (representations) of A n = A n l : S 1 R n smooth with compact support in I I gives localized automorphism ρ l (W (f)) = e i R l,f Rn 2π W (f) Charge: q l = 1 l R n 2π S 1 Statistics operator: ɛ(ρ l, ρ m ) = e ±iπ q l,q m R n ρ l = ρm q l = q m Local extension: If q l, q l 2Z then ɛ(ρ l, ρ l ) = 1 local extension (by cross product).

42 Localized automorphisms Conformal net of n free bosons A n (I) = A n 1 (I) = CCR(H(I) H(I)) Local endomorphisms (representations) of A n = A n l : S 1 R n smooth with compact support in I I gives localized automorphism ρ l (W (f)) = e i R l,f Rn 2π W (f) Charge: q l = 1 l R n 2π S 1 Statistics operator: ɛ(ρ l, ρ m ) = e ±iπ q l,q m R n ρ l = ρm q l = q m Local extension: If q l, q l 2Z then ɛ(ρ l, ρ l ) = 1 local extension (by cross product).

43 Localized automorphisms Conformal net of n free bosons A n (I) = A n 1 (I) = CCR(H(I) H(I)) Local endomorphisms (representations) of A n = A n l : S 1 R n smooth with compact support in I I gives localized automorphism ρ l (W (f)) = e i R l,f Rn 2π W (f) Charge: q l = 1 l R n 2π S 1 Statistics operator: ɛ(ρ l, ρ m ) = e ±iπ q l,q m R n ρ l = ρm q l = q m Local extension: If q l, q l 2Z then ɛ(ρ l, ρ l ) = 1 local extension (by cross product).

44 Localized automorphisms Conformal net of n free bosons A n (I) = A n 1 (I) = CCR(H(I) H(I)) Local endomorphisms (representations) of A n = A n l : S 1 R n smooth with compact support in I I gives localized automorphism ρ l (W (f)) = e i R l,f Rn 2π W (f) Charge: q l = 1 l R n 2π S 1 Statistics operator: ɛ(ρ l, ρ m ) = e ±iπ q l,q m R n ρ l = ρm q l = q m Local extension: If q l, q l 2Z then ɛ(ρ l, ρ l ) = 1 local extension (by cross product).

45 Localized automorphisms Conformal net of n free bosons A n (I) = A n 1 (I) = CCR(H(I) H(I)) Local endomorphisms (representations) of A n = A n l : S 1 R n smooth with compact support in I I gives localized automorphism ρ l (W (f)) = e i R l,f Rn 2π W (f) Charge: q l = 1 l R n 2π S 1 Statistics operator: ɛ(ρ l, ρ m ) = e ±iπ q l,q m R n ρ l = ρm q l = q m Local extension: If q l, q l 2Z then ɛ(ρ l, ρ l ) = 1 local extension (by cross product).

46 Even lattices Let Q be an (positive-definite) even lattice (eg. root lattice) of rank n α Q: α, α 2N = integral α, β Q: α, β Z. dual lattice (characters) Q = {α E Q : α, Q Z} E Q Q Z R. (eg. weight lattice in case of root lattices). A 2 SU(3) corresponding torus T Q = E Q /Q

47 Even lattices Let Q be an (positive-definite) even lattice (eg. root lattice) of rank n α Q: α, α 2N = integral α, β Q: α, β Z. dual lattice (characters) Q = {α E Q : α, Q Z} E Q Q Z R. (eg. weight lattice in case of root lattices). A 2 SU(3) corresponding torus T Q = E Q /Q

48 Conformal nets associated to lattices Local extension. For a lattice Q of rank n there is A Q A n containing of the net A n of n free bosons. Locally A Q (I) = (A(I)... A(I)) Q (Buchholz, Mack, Todorov 1988) (n = 1) (Staszkiewicz, 1995) (Dong and Xu, 2006) Construction even lattice Q of rank n Completely rational conformal net A Q Conformal nets corresponding to Lattice Vertex Operator Algebras. Some properties: Sectors finite group Q /Q, each sector statistical dimension 1. Completely rational net µ = Q /Q (Dong and Xu, 2006).

49 Conformal nets associated to lattices Local extension. For a lattice Q of rank n there is A Q A n containing of the net A n of n free bosons. Locally A Q (I) = (A(I)... A(I)) Q (Buchholz, Mack, Todorov 1988) (n = 1) (Staszkiewicz, 1995) (Dong and Xu, 2006) Construction even lattice Q of rank n Completely rational conformal net A Q Conformal nets corresponding to Lattice Vertex Operator Algebras. Some properties: Sectors finite group Q /Q, each sector statistical dimension 1. Completely rational net µ = Q /Q (Dong and Xu, 2006).

50 Conformal nets associated to lattices Local extension. For a lattice Q of rank n there is A Q A n containing of the net A n of n free bosons. Locally A Q (I) = (A(I)... A(I)) Q (Buchholz, Mack, Todorov 1988) (n = 1) (Staszkiewicz, 1995) (Dong and Xu, 2006) Construction even lattice Q of rank n Completely rational conformal net A Q Conformal nets corresponding to Lattice Vertex Operator Algebras. Some properties: Sectors finite group Q /Q, each sector statistical dimension 1. Completely rational net µ = Q /Q (Dong and Xu, 2006).

51 Simply laced groups and root lattices G simply-connected simple-laced Lie group, e.g. A SU(n + 1), n 1 A n : D Spin(2n), n 3 D n : E Exceptional Lie Groups E 6, E 7, E 8 : Q root lattice spanned by simple roots {α 1,..., α n } 2 i = j Cartan matrix (C ij ) α i, α j = C ij = 1 i j 0

52 Simply laced groups and root lattices G simply-connected simple-laced Lie group, e.g. A SU(n + 1), n 1 A n D Spin(2n), n 3 D n E Exceptional Lie Groups E 6, E 7, E 8 Q root lattice spanned by simple roots {α 1,..., α n } Maximal torus (Q Z R)/Q = T G A T,1 A Q (Conjectured) equivalence (proofed in case G = SU(n) (Xu, 2009)) loop group net for such G at level 1 = A G,1 A Q = conformal net associated at Q

53 Outline Conformal Nets Nets on Minkowski half-plane Standard subspaces Conformal nets associated to lattices Semigroup elements

54 Second quantization unitaries H one-particle space of a bosons (completion of LR) H(R + ) standard subspace localized in R + ϕ : R C inner function, then V 0 = ϕ(p 0 ) = V 0 H(R + ) H(R + ), [V 0, e itp 0 ] = 0 P 0 generator of translation. By second quantization A(I) = CCR(H(I)). V = Γ(V 0 ) = V E(A)

55 Second quantization unitaries II More general for n bosons A n (R + ) = A(R + ) n = CCR(H(R + ) H(R + )) Theorem (Prequantized semigroup reducible case (Longo and Witten, 2010)) V 0 E(H(R + ) H(R + )), then V 0 = ϕ kl (P 0 ) matrices of functions such that ϕ kl (p) unitary matrix for almost all p > 0, ϕ kl boundary value of a L function analytic on the upper half-plane which is symmetric ϕ kl (p) = ϕ kl ( p). Theorem V = Γ(V 0 ) E(A n ) for the second quantization of V 0 given above.

56 Second quantization unitaries II More general for n bosons A n (R + ) = A(R + ) n = CCR(H(R + ) H(R + )) Theorem (Prequantized semigroup reducible case (Longo and Witten, 2010)) V 0 E(H(R + ) H(R + )), then V 0 = ϕ kl (P 0 ) matrices of functions such that ϕ kl (p) unitary matrix for almost all p > 0, ϕ kl boundary value of a L function analytic on the upper half-plane which is symmetric ϕ kl (p) = ϕ kl ( p). Theorem V = Γ(V 0 ) E(A n ) for the second quantization of V 0 given above.

57 Question Which elements of the semigroup E(A n ) extend to the local extensions by lattices? A Q (I) = A n (I) Q where Q even lattice of rank n

58 Induction for local extension by free abelian groups Extension of the endomorphism η = AdV of A n (R + ) with V E(A n ) to A Q (R + ) = A n (R + ) βi Q β i localized in R + Assume η and β i commute up to some cocycle z i A n (R + ) z i Hom(ηβ i, β i η) z i β i (η(x)) = η(β i (x))z i for all x A n (R + ) and the compatibility condition then η extends to η = AdṼ. z i β i (z j ) = z j β j (z i ) V E(A n ) extends Ṽ E(A Q)

59 Induction for local extension by free abelian groups Extension of the endomorphism η = AdV of A n (R + ) with V E(A n ) to A Q (R + ) = A n (R + ) βi Q β i localized in R + Assume η and β i commute up to some cocycle z i A n (R + ) z i Hom(ηβ i, β i η) z i β i (η(x)) = η(β i (x))z i for all x A n (R + ) and the compatibility condition then η extends to η = AdṼ. z i β i (z j ) = z j β j (z i ) V E(A n ) extends Ṽ E(A Q)

60 Semigroup elements for lattice models Existence of z i Hom An(R + )(ηβ i, β i η) with the above properties in our model ensure V = Γ(ϕ ik (P 0 )) E(A n ) extends? Ṽ E(A Q) Restrictions. Such z i can be constructed if Algebraic obstruction. The inner function matrix has to be constant on every component of the lattice Analytical obstruction. The inner function need to be Hölder continuous at 0, i.e. 1 ϕ(p) 2 p locally integrable at p = 0

61 Semigroup elements for lattice models Existence of z i Hom An(R + )(ηβ i, β i η) with the above properties in our model ensure V = Γ(ϕ ik (P 0 )) E(A n ) extends? Ṽ E(A Q) Restrictions. Such z i can be constructed if Algebraic obstruction. The inner function matrix has to be constant on every component of the lattice Analytical obstruction. The inner function need to be Hölder continuous at 0, i.e. 1 ϕ(p) 2 p locally integrable at p = 0

62 Semigroup elements for lattice models Existence of z i Hom An(R + )(ηβ i, β i η) with the above properties in our model ensure V = Γ(ϕ ik (P 0 )) E(A n ) extends? Ṽ E(A Q) Restrictions. Such z i can be constructed if Algebraic obstruction. The inner function matrix has to be constant on every component of the lattice Analytical obstruction. The inner function need to be Hölder continuous at 0, i.e. 1 ϕ(p) 2 p locally integrable at p = 0

63 Results Theorem Let A be conformal net of the family A Q associated to an even irreducible lattice Q A G,1 for G = SU(n) (G simple, simply connected, simple-laced) A and ϕ Hölder cont. V E(A) local net A V on Minkowski half-plane Further U inner symmetry V E(A) = V U E(A) V i E(A i ) = V 1... V n E(A 1 A n )

64 Results Theorem Let A be conformal net of the family A Q associated to an even irreducible lattice Q A G,1 for G = SU(n) (G simple, simply connected, simple-laced) A and ϕ Hölder cont. V E(A) local net A V on Minkowski half-plane Further U inner symmetry V E(A) = V U E(A) V i E(A i ) = V 1... V n E(A 1 A n )

65 Other constructions using E(A) Models in 2D Minkowski space If there is a one-parameter group V t with V t E(A) for t 0 with negative generator local Poincaré covariant net on 2D Minkowski space (Longo). wedge-local Poincaré covariant net on 2D Minkowski space with non-trivial scattering (Tanimoto). Example For A the net of free boson (U(1)-current) and the inner function ϕ t (p) = e it/p we have V t = Γ(ϕ t (P 0 )) like above and the construction yields the free massive scalar boson on 2D Minkowski space. Works for all free field construction But because of the mentioned Hölder continuity this does not work for extensions by lattices.

66 Other constructions using E(A) Models in 2D Minkowski space If there is a one-parameter group V t with V t E(A) for t 0 with negative generator local Poincaré covariant net on 2D Minkowski space (Longo). wedge-local Poincaré covariant net on 2D Minkowski space with non-trivial scattering (Tanimoto). Example For A the net of free boson (U(1)-current) and the inner function ϕ t (p) = e it/p we have V t = Γ(ϕ t (P 0 )) like above and the construction yields the free massive scalar boson on 2D Minkowski space. Works for all free field construction But because of the mentioned Hölder continuity this does not work for extensions by lattices.

67 Other constructions using E(A) Models in 2D Minkowski space If there is a one-parameter group V t with V t E(A) for t 0 with negative generator local Poincaré covariant net on 2D Minkowski space (Longo). wedge-local Poincaré covariant net on 2D Minkowski space with non-trivial scattering (Tanimoto). Example For A the net of free boson (U(1)-current) and the inner function ϕ t (p) = e it/p we have V t = Γ(ϕ t (P 0 )) like above and the construction yields the free massive scalar boson on 2D Minkowski space. Works for all free field construction But because of the mentioned Hölder continuity this does not work for extensions by lattices.

68 Boson-Fermion correspondence Extensions by the lattice Z n (not even!) yields Fermi (=twisted local) net F = Fer n C. Even part A := F Z 2 local conformal net, i.e. Fer n C = A Z n But Fer C can be realized on antisymmetric Fock space (CAR). Using second quantization we have two methods to construct elements in E(F) (and E(A)). E(F) CCR : constructed as extensions by the lattice E(F) CAR : second quantization unitaries in the CAR algebra (analogous like the CCR case). E(F) CAR E(F) CCR = trivial elements.

69 Boson-Fermion correspondence Extensions by the lattice Z n (not even!) yields Fermi (=twisted local) net F = Fer n C. Even part A := F Z 2 local conformal net, i.e. Fer n C = A Z n But Fer C can be realized on antisymmetric Fock space (CAR). Using second quantization we have two methods to construct elements in E(F) (and E(A)). E(F) CCR : constructed as extensions by the lattice E(F) CAR : second quantization unitaries in the CAR algebra (analogous like the CCR case). E(F) CAR E(F) CCR = trivial elements.

70 Boson-Fermion correspondence Extensions by the lattice Z n (not even!) yields Fermi (=twisted local) net F = Fer n C. Even part A := F Z 2 local conformal net, i.e. Fer n C = A Z n But Fer C can be realized on antisymmetric Fock space (CAR). Using second quantization we have two methods to construct elements in E(F) (and E(A)). E(F) CCR : constructed as extensions by the lattice E(F) CAR : second quantization unitaries in the CAR algebra (analogous like the CCR case). E(F) CAR E(F) CCR = trivial elements.

71 Boson-Fermion correspondence Extensions by the lattice Z n (not even!) yields Fermi (=twisted local) net F = Fer n C. Even part A := F Z 2 local conformal net, i.e. Fer n C = A Z n But Fer C can be realized on antisymmetric Fock space (CAR). Using second quantization we have two methods to construct elements in E(F) (and E(A)). E(F) CCR : constructed as extensions by the lattice E(F) CAR : second quantization unitaries in the CAR algebra (analogous like the CCR case). E(F) CAR E(F) CCR = trivial elements.

72 Boson-Fermion correspondence Extensions by the lattice Z n (not even!) yields Fermi (=twisted local) net F = Fer n C. Even part A := F Z 2 local conformal net, i.e. Fer n C = A Z n But Fer C can be realized on antisymmetric Fock space (CAR). Using second quantization we have two methods to construct elements in E(F) (and E(A)). E(F) CCR : constructed as extensions by the lattice E(F) CAR : second quantization unitaries in the CAR algebra (analogous like the CCR case). E(F) CAR E(F) CCR = trivial elements.

73 Boson-Fermion correspondence Extensions by the lattice Z n (not even!) yields Fermi (=twisted local) net F = Fer n C. Even part A := F Z 2 local conformal net, i.e. Fer n C = A Z n But Fer C can be realized on antisymmetric Fock space (CAR). Using second quantization we have two methods to construct elements in E(F) (and E(A)). E(F) CCR : constructed as extensions by the lattice E(F) CAR : second quantization unitaries in the CAR algebra (analogous like the CCR case). E(F) CAR E(F) CCR = trivial elements.

74 Summary We have constructed Elements of the semigroup E(A) for a large class of rational conformal field theories is found : New models of boundary quantum field theory. Open questions Loop group nets at higher level (Coset construction/orbifold) Restriction of a net of free fermions (semigroup elements by second quantization) should give more examples. Construction of 1+1D massive models one-parameter semigroup. Until yet just examples from free field construction.

75 Merci beaucoup!! Semigroup elements associated to conformal nets and boundary quantum field theory Marcel Bischoff Dipartimento di Matematica Università degli Studi di Roma Tor Vergata Meeting of GDRE GREFI-GENCO Institut Henri Poincaré Paris, 1 June 2011

76 References I Buchholz, D., Mack, G., and Todorov, I. (1988). The current algebra on the circle as a germ of local field theories. Nuclear Physics B-Proceedings Supplements, 5(2): Dong, C. and Xu, F. (2006). Conformal nets associated with lattices and their orbifolds. Adv. Math., 206(1): Fredenhagen, K. and Jörß, M. (1996). conformal haag-kastler nets, pointlike localized fields and the existence of operator product expansions. Communications in mathematical physics, 176(3): Gabbiani, F. and Fröhlich, J. (1993). Operator algebras and conformal field theory. Communications in mathematical physics, 155(3): Guido, D. and Longo, R. (1995). An algebraic spin and statistics theorem. Communications in Mathematical Physics, 172: /BF Kawahigashi, Y. and Longo, R. (2004). Classification of local conformal nets. Case c 1. The Annals of Mathematics, 160(2):

77 References II Kawahigashi, Y., Longo, R., and Müger, M. (2001). Multi-Interval Subfactors and Modularityof Representations in Conformal Field Theory. Communications in Mathematical Physics, 219: Longo, R. and Rehren, K. (2004). Local Fields in Boundary Conformal QFT. Reviews in Mathematical Physics, 16: Longo, R. and Witten, E. (2010). An Algebraic Construction of Boundary Quantum Field Theory. Communications in Mathematical Physics, 1:179. Staszkiewicz, C. (1995). Die lokale Struktur abelscher Stromalgebren auf dem Kreis. PhD thesis, Freie Universität Berlin. Xu, F. (2000). Jones-wassermann subfactors for disconnected intervals. Communications in Contemporary Mathematics, 2(3): Xu, F. (2009). On affine orbifold nets associated with outer automorphisms. Communications in Mathematical Physics, 291: /s y.