Subfactors and Topological Defects in Conformal Quantum Field Theory

Transcription

1 Subfactors and Topological Defects in Conformal Quantum Field Theory Marcel Bischoff Department of Mathematics Vanderbilt University Nashville, TN San Antonio, TX, January 12, 2015 based on work with R. Longo, Y. Kawahigashi and K.-H. Rehren arxiv: , arxiv: , (see also: arxiv: )

2 Introduction Algebraic quantum field theory: A family of von Neumann algebras containing all local observables/operations associated with space-time regions. Conformal Quantum Field Theory (CQFT) in 1 and 2 dimension described by AQFT quite successful, e.g. partial classification results (e.g. c < 1 (Kawahigashi, Longo 04)). Topological Field Theory (TFT) construction of CFT on Riemann surfaces with boundaries/defects (Fuchs, Runkel, Schweigert 02+).

3 Outline Conformal Nets Boundaries / Defects Classification (modular case)

4 Conformal Nets on S 1 Conformal net A 0 on S 1 = R (compactified light-ray): S 1 I A 0 (I) B(H), H fixed Hilbert space 1. Isotony: I J A 0 (I) A 0 (J) 2. Locality: [A 0 (I), A 0 (J)] = {0} if I J =. 3. Covariance: U is a unitary positive-energy representation of the Möbius group/diffeomorphism group, s.t. U(g)A 0 (I)U(g) = A 0 (gi). 4. Vacuum: Ω is a (up to a phase) unique vector invariant under the Möbius group.

5 Conformal Nets on S 1 Conformal net A 0 on S 1 = R (compactified light-ray): S 1 I A 0 (I) B(H), H fixed Hilbert space 1. Isotony: I J A 0 (I) A 0 (J) 2. Locality: [A 0 (I), A 0 (J)] = {0} if I J =. 3. Covariance: U is a unitary positive-energy representation of the Möbius group/diffeomorphism group, s.t. U(g)A 0 (I)U(g) = A 0 (gi). 4. Vacuum: Ω is a (up to a phase) unique vector invariant under the Möbius group.

6 Conformal Nets on S 1 Conformal net A 0 on S 1 = R (compactified light-ray): S 1 I A 0 (I) B(H), H fixed Hilbert space 1. Isotony: I J A 0 (I) A 0 (J) 2. Locality: [A 0 (I), A 0 (J)] = {0} if I J =. 3. Covariance: U is a unitary positive-energy representation of the Möbius group/diffeomorphism group, s.t. U(g)A 0 (I)U(g) = A 0 (gi). 4. Vacuum: Ω is a (up to a phase) unique vector invariant under the Möbius group.

7 Conformal Nets on S 1 Conformal net A 0 on S 1 = R (compactified light-ray): S 1 I A 0 (I) B(H), H fixed Hilbert space 1. Isotony: I J A 0 (I) A 0 (J) 2. Locality: [A 0 (I), A 0 (J)] = {0} if I J =. 3. Covariance: U is a unitary positive-energy representation of the Möbius group/diffeomorphism group, s.t. U(g)A 0 (I)U(g) = A 0 (gi). 4. Vacuum: Ω is a (up to a phase) unique vector invariant under the Möbius group.

8 Complete Rationality and Modular Tensor Categories Representation of A 0 = {A 0 (I)} IS 1 is a family: π = {π I : A 0 (I) B(H π )}, which is compatible, i.e. π J A 0 (I) = π I for I J. For all I 0 exists a ρ = π on H, s.t. ρ I = id A0 (I) for I I 0 =. ρ J is localized DHR endomorphism: ρ J (A 0 (J)) A 0 (J) for all J I 0. Tensor product: composition of localized endomorphisms. Rep I (A 0 ) End(A 0 (I)) (full and replete). natural braiding {c ρ,σ : ρ σ σ ρ} (Fredenhagen, Rehren, Schroer (1989)). Theorem ((Kawahigashi, Longo, Müger (2001))) Let A 0 be a completely rational conformal net. Complete Rationality Then Rep(A 0 ) is a modular C -tensor category = unitary modular tensor category (UMTC).

9 Complete Rationality and Modular Tensor Categories Representation of A 0 = {A 0 (I)} IS 1 is a family: π = {π I : A 0 (I) B(H π )}, which is compatible, i.e. π J A 0 (I) = π I for I J. For all I 0 exists a ρ = π on H, s.t. ρ I = id A0 (I) for I I 0 =. ρ J is localized DHR endomorphism: ρ J (A 0 (J)) A 0 (J) for all J I 0. Tensor product: composition of localized endomorphisms. Rep I (A 0 ) End(A 0 (I)) (full and replete). natural braiding {c ρ,σ : ρ σ σ ρ} (Fredenhagen, Rehren, Schroer (1989)). Theorem ((Kawahigashi, Longo, Müger (2001))) Let A 0 be a completely rational conformal net. Complete Rationality Then Rep(A 0 ) is a modular C -tensor category = unitary modular tensor category (UMTC).

10 Complete Rationality and Modular Tensor Categories Representation of A 0 = {A 0 (I)} IS 1 is a family: π = {π I : A 0 (I) B(H π )}, which is compatible, i.e. π J A 0 (I) = π I for I J. For all I 0 exists a ρ = π on H, s.t. ρ I = id A0 (I) for I I 0 =. ρ J is localized DHR endomorphism: ρ J (A 0 (J)) A 0 (J) for all J I 0. Tensor product: composition of localized endomorphisms. Rep I (A 0 ) End(A 0 (I)) (full and replete). natural braiding {c ρ,σ : ρ σ σ ρ} (Fredenhagen, Rehren, Schroer (1989)). Theorem ((Kawahigashi, Longo, Müger (2001))) Let A 0 be a completely rational conformal net. Complete Rationality Then Rep(A 0 ) is a modular C -tensor category = unitary modular tensor category (UMTC).

11 Complete Rationality and Modular Tensor Categories Representation of A 0 = {A 0 (I)} IS 1 is a family: π = {π I : A 0 (I) B(H π )}, which is compatible, i.e. π J A 0 (I) = π I for I J. For all I 0 exists a ρ = π on H, s.t. ρ I = id A0 (I) for I I 0 =. ρ J is localized DHR endomorphism: ρ J (A 0 (J)) A 0 (J) for all J I 0. Tensor product: composition of localized endomorphisms. Rep I (A 0 ) End(A 0 (I)) (full and replete). natural braiding {c ρ,σ : ρ σ σ ρ} (Fredenhagen, Rehren, Schroer (1989)). Theorem ((Kawahigashi, Longo, Müger (2001))) Let A 0 be a completely rational conformal net. Complete Rationality Then Rep(A 0 ) is a modular C -tensor category = unitary modular tensor category (UMTC).

12 Complete Rationality and Modular Tensor Categories Representation of A 0 = {A 0 (I)} IS 1 is a family: π = {π I : A 0 (I) B(H π )}, which is compatible, i.e. π J A 0 (I) = π I for I J. For all I 0 exists a ρ = π on H, s.t. ρ I = id A0 (I) for I I 0 =. ρ J is localized DHR endomorphism: ρ J (A 0 (J)) A 0 (J) for all J I 0. Tensor product: composition of localized endomorphisms. Rep I (A 0 ) End(A 0 (I)) (full and replete). natural braiding {c ρ,σ : ρ σ σ ρ} (Fredenhagen, Rehren, Schroer (1989)). Theorem ((Kawahigashi, Longo, Müger (2001))) Let A 0 be a completely rational conformal net. Complete Rationality Then Rep(A 0 ) is a modular C -tensor category = unitary modular tensor category (UMTC).

13 Complete Rationality and Modular Tensor Categories Representation of A 0 = {A 0 (I)} IS 1 is a family: π = {π I : A 0 (I) B(H π )}, which is compatible, i.e. π J A 0 (I) = π I for I J. For all I 0 exists a ρ = π on H, s.t. ρ I = id A0 (I) for I I 0 =. ρ J is localized DHR endomorphism: ρ J (A 0 (J)) A 0 (J) for all J I 0. Tensor product: composition of localized endomorphisms. Rep I (A 0 ) End(A 0 (I)) (full and replete). natural braiding {c ρ,σ : ρ σ σ ρ} (Fredenhagen, Rehren, Schroer (1989)). Theorem ((Kawahigashi, Longo, Müger (2001))) Let A 0 be a completely rational conformal net. Complete Rationality Then Rep(A 0 ) is a modular C -tensor category = unitary modular tensor category (UMTC).

14 Complete Rationality and Modular Tensor Categories Representation of A 0 = {A 0 (I)} IS 1 is a family: π = {π I : A 0 (I) B(H π )}, which is compatible, i.e. π J A 0 (I) = π I for I J. For all I 0 exists a ρ = π on H, s.t. ρ I = id A0 (I) for I I 0 =. ρ J is localized DHR endomorphism: ρ J (A 0 (J)) A 0 (J) for all J I 0. Tensor product: composition of localized endomorphisms. Rep I (A 0 ) End(A 0 (I)) (full and replete). natural braiding {c ρ,σ : ρ σ σ ρ} (Fredenhagen, Rehren, Schroer (1989)). Theorem ((Kawahigashi, Longo, Müger (2001))) Let A 0 be a completely rational conformal net. Complete Rationality Then Rep(A 0 ) is a modular C -tensor category = unitary modular tensor category (UMTC).

15 Complete Rationality and Modular Tensor Categories Representation of A 0 = {A 0 (I)} IS 1 is a family: π = {π I : A 0 (I) B(H π )}, which is compatible, i.e. π J A 0 (I) = π I for I J. For all I 0 exists a ρ = π on H, s.t. ρ I = id A0 (I) for I I 0 =. ρ J is localized DHR endomorphism: ρ J (A 0 (J)) A 0 (J) for all J I 0. Tensor product: composition of localized endomorphisms. Rep I (A 0 ) End(A 0 (I)) (full and replete). natural braiding {c ρ,σ : ρ σ σ ρ} (Fredenhagen, Rehren, Schroer (1989)). Theorem ((Kawahigashi, Longo, Müger (2001))) Let A 0 be a completely rational conformal net. Complete Rationality Then Rep(A 0 ) is a modular C -tensor category = unitary modular tensor category (UMTC).

16 Complete rationality Completely rational conformal net (Kawahigashi, Longo, Müger (2001)) Split property. For every relatively compact inclusion of intervals 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.

17 Loop group net of SU(N) at level k: (see (Wassermann 98)) A SU(N),k (I) = π(l I SU(N)) (completely rational (Xu 00)) with π level k vacuum PER of loop group LSU(N) = C (S 1, SU(N)). Example G = SU(2): Irreducible representations {0, 1 2, 1,..., k 2 }. Rep(A SU(2),k ) is generated by 1 2-representation ρ and Hom(id, ρρ) : = d = = with Hom(ρρ, id) and braiding defined by Kaufmann bracket := = q q 1 2 ( ) where q = e iπ k+2, d = q + q 1 = 2 cos π k+2.

18 Loop group net of SU(N) at level k: (see (Wassermann 98)) A SU(N),k (I) = π(l I SU(N)) (completely rational (Xu 00)) with π level k vacuum PER of loop group LSU(N) = C (S 1, SU(N)). Example G = SU(2): Irreducible representations {0, 1 2, 1,..., k 2 }. Rep(A SU(2),k ) is generated by 1 2-representation ρ and Hom(id, ρρ) : = d = = with Hom(ρρ, id) and braiding defined by Kaufmann bracket := = q q 1 2 ( ) where q = e iπ k+2, d = q + q 1 = 2 cos π k+2.

19 Theorem (well-known: Popa,..., Hayashi, Yamagami) Every finite rigid C tensor category C is realizable as C End 0 (N) with N a III 1 factor. The strict rigid C tensor category End 0 (N), with N a type III factor. Objects: Endomorphisms ρ: N N with finite index [N : ρ(n)] <. Morphisms: t: ρ σ is a t N, such that tρ(x) = σ(x)t for all x N. Tensor product: ρ σ := ρ σ (composition of endomorphisms) Direct sums: ρ σ = Ad t 1 ρ + Ad t 2 σ with t i N, s.t. 2 i=1 t it i = 1, t i t j = δ i,j (Cuntz algebra).... Also as C bim N N (category of dualizable N N-bimodules) with N type II 1 factor. For purpose of talk: type III is more natural.

20 Theorem (well-known: Popa,..., Hayashi, Yamagami) Every finite rigid C tensor category C is realizable as C End 0 (N) with N a III 1 factor. The strict rigid C tensor category End 0 (N), with N a type III factor. Objects: Endomorphisms ρ: N N with finite index [N : ρ(n)] <. Morphisms: t: ρ σ is a t N, such that tρ(x) = σ(x)t for all x N. Tensor product: ρ σ := ρ σ (composition of endomorphisms) Direct sums: ρ σ = Ad t 1 ρ + Ad t 2 σ with t i N, s.t. 2 i=1 t it i = 1, t i t j = δ i,j (Cuntz algebra).... Also as C bim N N (category of dualizable N N-bimodules) with N type II 1 factor. For purpose of talk: type III is more natural.

21 Braided subfactors Let C End 0 (N) (full and replete) be a braided fusion category. Let N M be a finite index subfactor. Then exists the dual morphism ῑ: M N of the canonical inclusion ι: N M. If the dual canonical endomorphism θ = ῑ ι: N N is in C, we call the pair (N M, C) a braided subfactor.. θ has canonically the structure of a Q-system (algebra object) in C. Actually, there is a one-to-one correspondence between (Longo 94): Q-sytems in C braided subfactors (N M, C) analogously, if N is type II 1, then we ask N L 2 M N C bim N N

22 Braided subfactors Let C End 0 (N) (full and replete) be a braided fusion category. Let N M be a finite index subfactor. Then exists the dual morphism ῑ: M N of the canonical inclusion ι: N M. If the dual canonical endomorphism θ = ῑ ι: N N is in C, we call the pair (N M, C) a braided subfactor.. θ has canonically the structure of a Q-system (algebra object) in C. Actually, there is a one-to-one correspondence between (Longo 94): Q-sytems in C braided subfactors (N M, C) analogously, if N is type II 1, then we ask N L 2 M N C bim N N

23 Braided subfactors Let C End 0 (N) (full and replete) be a braided fusion category. Let N M be a finite index subfactor. Then exists the dual morphism ῑ: M N of the canonical inclusion ι: N M. If the dual canonical endomorphism θ = ῑ ι: N N is in C, we call the pair (N M, C) a braided subfactor.. θ has canonically the structure of a Q-system (algebra object) in C. Actually, there is a one-to-one correspondence between (Longo 94): Q-sytems in C braided subfactors (N M, C) analogously, if N is type II 1, then we ask N L 2 M N C bim N N

24 Braided subfactors Let C End 0 (N) (full and replete) be a braided fusion category. Let N M be a finite index subfactor. Then exists the dual morphism ῑ: M N of the canonical inclusion ι: N M. If the dual canonical endomorphism θ = ῑ ι: N N is in C, we call the pair (N M, C) a braided subfactor.. θ has canonically the structure of a Q-system (algebra object) in C. Actually, there is a one-to-one correspondence between (Longo 94): Q-sytems in C braided subfactors (N M, C) analogously, if N is type II 1, then we ask N L 2 M N C bim N N

25 Braided subfactors II Using the braiding of C End 0 (N), we can define: (N M, C) is called local if the Q-system is commutative: θ θ θ θ =. θ θ (N M, C) then exist local subfactors (N M ±, C), s.t. N M + M M Given N M a, M b there is a von Neumann algebra with finite center M = M a ± N M b N: N M a M ± M b M M M a ± N M b

26 Braided subfactors II Using the braiding of C End 0 (N), we can define: (N M, C) is called local if the Q-system is commutative: θ θ θ θ =. θ θ (N M, C) then exist local subfactors (N M ±, C), s.t. N M + M M Given N M a, M b there is a von Neumann algebra with finite center M = M a ± N M b N: N M a M ± M b M M M a ± N M b

27 Braided subfactors II Using the braiding of C End 0 (N), we can define: (N M, C) is called local if the Q-system is commutative: θ θ θ θ =. θ θ (N M, C) then exist local subfactors (N M ±, C), s.t. N M + M M Given N M a, M b there is a von Neumann algebra with finite center M = M a ± N M b N: N M a M ± M b M M M a ± N M b

28 Braided subfactors and extensions of conformal net Given a (conformal) net I A 0 (I). There is a one-to-one correspondence (up to equivalence) between: Finite index (non-local) extensions B 0, i.e. a net B 0 (I) A 0 (I), which is relatively locality w.r.t. A, i.e. [A 0 (I 1 ), B 0 (I 2 )] = {0} for I 1, I 2 disjoint. Braided subfactors (N M, C), where N = A(I) and C = Rep I (A 0 ). B 0 is a local net (N M, C) is local

29 Braided subfactors and extensions of conformal net Given a (conformal) net I A 0 (I). There is a one-to-one correspondence (up to equivalence) between: Finite index (non-local) extensions B 0, i.e. a net B 0 (I) A 0 (I), which is relatively locality w.r.t. A, i.e. [A 0 (I 1 ), B 0 (I 2 )] = {0} for I 1, I 2 disjoint. Braided subfactors (N M, C), where N = A(I) and C = Rep I (A 0 ). B 0 is a local net (N M, C) is local

30 Conformal nets on Minkowski space Conformal net A on Minkowski space R 1,1 : R 1,1 O A(O) B(H A ), H A fixed Hilbert space 1. Isotony: O 0 O 1 A(O 0 ) A(O 1 ) 2. Locality: [A(O 1 ), A(O 2 )] = {0} if O 1 and O 1 spacelike separated: t x O 1 O 2 3. Covariance: U A is a unitary positive-energy representation of the 2D conformal group, s.t. U A (g)a(o)u A (g) = A(gO). 4. Vacuum: Ω is unique translation invariant vector.

31 Chiral nets t One can define a chiral conformal net on Minkowski space by A(O) = A + (I 1 ) A (I 2 ) I1 O I 2 x where A ± are conformal nets on R. Non-chiral nets are given by local extensions B(O) A(O) A + (I 1 ) A (I 2 ). physically, A prescribes (generalized) symmetries, e.g. if A = A SU(2),k then it prescribes SU(2)-gauge and conformal transformations.

32 Chiral nets t One can define a chiral conformal net on Minkowski space by A(O) = A + (I 1 ) A (I 2 ) I1 O I 2 x where A ± are conformal nets on R. Non-chiral nets are given by local extensions B(O) A(O) A + (I 1 ) A (I 2 ). physically, A prescribes (generalized) symmetries, e.g. if A = A SU(2),k then it prescribes SU(2)-gauge and conformal transformations.

33 Outline Conformal Nets Boundaries / Defects Classification (modular case)

34 Local boundaries Left observables B L A. Right observables B R A. Boundary invisible for A, D(O) := B L (O) B R (O). O L O R t x Locality: [B L (O L ), B R (O R )] = {0} for O L spacelike left of O R. B L (O L ) B R (O < L ) D(O < L ). B R (O R ) B R (O > R ) D(O > R ).

35 Local boundaries Left observables B L A. Right observables B R A. Boundary invisible for A, D(O) := B L (O) B R (O). O L O R t x Locality: [B L (O L ), B R (O R )] = {0} for O L spacelike left of O R. B L (O L ) B R (O < L ) D(O < L ). B R (O R ) B R (O > R ) D(O > R ).

36 Local boundaries Left observables B L A. Right observables B R A. Boundary invisible for A, D(O) := B L (O) B R (O). O L O < L t x Locality: [B L (O L ), B R (O R )] = {0} for O L spacelike left of O R. B L (O L ) B R (O < L ) D(O < L ). B R (O R ) B R (O > R ) D(O > R ).

37 Local boundaries Left observables B L A. Right observables B R A. Boundary invisible for A, D(O) := B L (O) B R (O). O > R O R t x Locality: [B L (O L ), B R (O R )] = {0} for O L spacelike left of O R. B L (O L ) B R (O < L ) D(O < L ). B R (O R ) B R (O > R ) D(O > R ).

38 Local boundaries Left observables B L A. Right observables B R A. Boundary invisible for A, D(O) := B L (O) B R (O). O L O R t x Locality: [B L (O L ), B R (O R )] = {0} for O L spacelike left of O R. B L (O L ) B R (O < L ) D(O < L ). B R (O R ) B R (O > R ) D(O > R ).

39 A topological boundaries (Chiral) symmetries A = A + A with A ± local nets on R. Left observables: local extension B L A. Right observables: local extension B R A. Definition An A topological B L B R -defect D is a (non-local) extension D A such that: D + D D B L B R A C L (O)D(O) D(O < ) D(O) + C R (O)D(O) D(O > ) D(O)

40 A topological boundaries (Chiral) symmetries A = A + A with A ± local nets on R. Left observables: local extension B L A. Right observables: local extension B R A. Definition An A topological B L B R -defect D is a (non-local) extension D A such that: D + D D B L B R A C L (O)D(O) D(O < ) D(O) + C R (O)D(O) D(O > ) D(O)

41 Do such defects exist in general? Can we classify them?

42 Universal construction Theorem If B L A and B R A are local, irreducible, finite index extensions then there exists a unique A topological B L B R defect D univ with the above properties. Its central decomposition gives all irreducible defects. Universal A topological B L B R defect D univ is given by the extension A B L, B R D univ characterized by (braided product): A(O) B L (O) B R (O) B L (O) + A(O) B R(O) =: D univ (O) D univ (O) has in general finite center D univ (O) D univ (O). Let e D univ (O) D univ (O) be a minimal central projection, then is an irreducible defect. A(O) D e (O) : ea(o) ed univ (O)

43 Universal construction Theorem If B L A and B R A are local, irreducible, finite index extensions then there exists a unique A topological B L B R defect D univ with the above properties. Its central decomposition gives all irreducible defects. Universal A topological B L B R defect D univ is given by the extension A B L, B R D univ characterized by (braided product): A(O) B L (O) B R (O) B L (O) + A(O) B R(O) =: D univ (O) D univ (O) has in general finite center D univ (O) D univ (O). Let e D univ (O) D univ (O) be a minimal central projection, then is an irreducible defect. A(O) D e (O) : ea(o) ed univ (O)

44 Universal construction Theorem If B L A and B R A are local, irreducible, finite index extensions then there exists a unique A topological B L B R defect D univ with the above properties. Its central decomposition gives all irreducible defects. Universal A topological B L B R defect D univ is given by the extension A B L, B R D univ characterized by (braided product): A(O) B L (O) B R (O) B L (O) + A(O) B R(O) =: D univ (O) D univ (O) has in general finite center D univ (O) D univ (O). Let e D univ (O) D univ (O) be a minimal central projection, then is an irreducible defect. A(O) D e (O) : ea(o) ed univ (O)

45 Universal construction Theorem If B L A and B R A are local, irreducible, finite index extensions then there exists a unique A topological B L B R defect D univ with the above properties. Its central decomposition gives all irreducible defects. Universal A topological B L B R defect D univ is given by the extension A B L, B R D univ characterized by (braided product): A(O) B L (O) B R (O) B L (O) + A(O) B R(O) =: D univ (O) D univ (O) has in general finite center D univ (O) D univ (O). Let e D univ (O) D univ (O) be a minimal central projection, then is an irreducible defect. A(O) D e (O) : ea(o) ed univ (O)

46 Outline Conformal Nets Boundaries / Defects Classification (modular case)

47 Characterization maximal local extensions Now we choose A ± = A 0 (parity symmetry) and A 0 to be a completely rational conformal net. Further, we now choose the irreducible local extensions B L, B R A = A 0 A 0 to be maximal. We call B L, B R full CFT. Theorem ((Rehren, Müger, Kawahigashi Longo, Kong Runkel, B K L,... )) Let A A 0 A 0 with A 0 completely rational. There is a one-to-one correspondence: Full CFTs B A, and Morita equivalence classes of a braided subfactor (N M, C), with N = A 0 (I), C = Rep I (A 0 ). N M B is given by the full center.

48 Characterization maximal local extensions Now we choose A ± = A 0 (parity symmetry) and A 0 to be a completely rational conformal net. Further, we now choose the irreducible local extensions B L, B R A = A 0 A 0 to be maximal. We call B L, B R full CFT. Theorem ((Rehren, Müger, Kawahigashi Longo, Kong Runkel, B K L,... )) Let A A 0 A 0 with A 0 completely rational. There is a one-to-one correspondence: Full CFTs B A, and Morita equivalence classes of a braided subfactor (N M, C), with N = A 0 (I), C = Rep I (A 0 ). N M B is given by the full center.

49 Let N = A 0 (I) and C = Rep I (A 0 ). Then the full CFTs B A are given by Morita equivalence classes of braided subfactors (N M, C). Example (A, D, E classification for SU(2) k (Ocneanu)) Let A 0 = A SU(2),k, then the Morita equivalence classes of braided subfactors (N M, C) are in correspondence given by A, D, E Coxeter diagrams with Coxeter number k + 1. A n (trivial subfactor), D even and E 6,8 are realizable by local subfactors.

50 Classification of irreducible boundaries by chiral data Let N = A 0 (I) and C = Rep I (A 0 ) End(N) Given (N M L, C) and (N M R, C) braided subfactor. They give full CFTs B L, B R A A 0 A 0 on Minkowski space. ι : N M inclusion map, ῑ : M N conjugate Theorem There is a one-to-one correspondence between: minimal central projections e D univ (O) D univ (O) and irreducible sectors [β] with β ι L ρ ῑ R : M L M R, ρ C Classification by braided subfactor data: M L N M R, C Rep I (A 0 ). # boundaries = tr(z L Z t R ) Z = modular invariant matrix

51 Classification of irreducible boundaries by chiral data Let N = A 0 (I) and C = Rep I (A 0 ) End(N) Given (N M L, C) and (N M R, C) braided subfactor. They give full CFTs B L, B R A A 0 A 0 on Minkowski space. ι : N M inclusion map, ῑ : M N conjugate Theorem There is a one-to-one correspondence between: minimal central projections e D univ (O) D univ (O) and irreducible sectors [β] with β ι L ρ ῑ R : M L M R, ρ C Classification by braided subfactor data: M L N M R, C Rep I (A 0 ). # boundaries = tr(z L Z t R ) Z = modular invariant matrix

52 Example: SU(2) at level 10 with E 6 modular invariant M L = M R = M with N = A SU(2),10 (I) A Spin(5),1 (I) = M. Then there are 12 irreducible defects corresponding to the sectors: { } { } id M, ψ, σ, α ± 1 α 1 ±, α± 9, α + 1 α 1, α + 1 α1 = α+ 1 α 1, α + 9 α1 = α+ 1 α }{{} Rep I (A Spin(5),1 ) σ id M α ± 1 2 α ± 1 α ± 9 2 ψ

53 Outlook Summary Boundaries: locality and inivisiblity for subnet A (conservation). Existence of universal defect for finite index case. Classification of irreducible boundaries by chiral data in the rational maximal case. Open problems Fusion of phase boundaries and defects. Phase boundaries without assuming conformal symmetry. Deformations of defects and relations to integrable QFT.

54 References I K. Fredenhagen, K.-H. Rehren, and B. Schroer. Superselection sectors with braid group statistics and exchange algebras. I. General theory. Comm. Math. Phys., 125(2): , Jürgen Fuchs, Ingo Runkel, and Christoph Schweigert. TFT construction of RCFT correlators. I. Partition functions. Nuclear Phys. B, 646(3): , Y. Kawahigashi and Roberto Longo. Classification of local conformal nets. Case c < 1. Ann. Math., 160(2): , Y. Kawahigashi, Roberto Longo, and Michael Müger. Multi-Interval Subfactors and Modularity of Representations in Conformal Field Theory. Comm. Math. Phys., 219: , 2001.

55 References II Roberto Longo. A duality for Hopf algebras and for subfactors. I. Comm. Math. Phys., 159(1): , Antony Wassermann. Operator algebras and conformal field theory III. Fusion of positive energy representations of LSU(N) using bounded operators. Invent. Math., 133(3): , Feng Xu. Jones-Wassermann subfactors for disconnected intervals. Commun. Contemp. Math., 2(3): , 2000.

56 Complete rationality Completely rational conformal net (Kawahigashi, Longo, Müger (2001)) Split property. For every relatively compact inclusion of intervals 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. return

57 Modularity If the net A is completely rational then Rep f (A) is a modular C -tensor category (unitary MTC): 1. Finite # of sectors. 2. The braiding is non-degenerated, i.e. ε(ρ, σ)ε(σ, ρ) = = 1 for all ρ = [σ] = N[id] identity is the only transparent object, with respect to the braiding or equivalently S-matrix (Rehren) is unitary: S ρσ ρ σ ; T ρρ ρ ρ = conformal spin SS = T T = 1, (ST ) 3 = S 2, S 4 = 1 Unitary representation of the modular group SL(2, Z) = Z 4 Z2 Z 6.

58 A n : D n : E {6,7,8} :