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

Characterization of rational conformal QFTs and their boundary conditions1

Marcel Bischoff

http://www.theorie.physik.uni-goettingen.de/~bischoff

Research Training Group 1493 Mathematical Structures in Modern Quantum Physics University of G¨

• ttingen

32nd Workshop ”Foundations and Constructive Aspects of QFT” Wuppertal, 31 May 2013

1work in progress with Roberto Longo and Yasuyuki Kawahigashi Marcel Bischoff (Uni G¨

• ttingen)

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¨

• ttingen)

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¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Conformal Nets

H Hilbert space, I = family of proper intervals on S1 ∼ = R I ∋ I − → A(I) = A(I)′′ ⊂ B(H)

• A. Isotony. I1 ⊂ I2 =

⇒ A(I1) ⊂ A(I2)

• B. Locality. I1 ∩ I2 = =

⇒ [A(I1), A(I2)] = {0}

• C. M¨
• bius covariance. There is a unitary representation U of the M¨
• bius

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 L0 of the rotation subgroup (conformal Hamiltonian) has positive spectrum.

• E. Vacuum. ker L0 = CΩ 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¨

• ttingen)

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¨

• ttingen)

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

• perator ∆I and and conjugation JI of the pair (A(I), Ω) are

U(Λ(−2πt)) = ∆it, t ∈ R dilation U(rI) = JI reflection

(Gabbiani and Fr¨

• hlich, 1993), (Guido and Longo, 1995)

◮ Haag duality. A(I′) = A(I)′. ◮ Factoriality. A(I) is III1-factor (in (Connes, 1973) classification) ◮ Additivity. I ⊂ i Ii =

⇒ A(I) ⊂

i A(Ii) (Fredenhagen and J¨

• rß, 1996).

example complete rationality Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Representations

A representation of A is a family of representations π = {πI : A(I) → B(Hπ)} on a Hilbert space Hπ such that πJ ↾ A(I) = πI I ⊂ J. Fact: Let π be a non-degenerated represenatation on a seperable space Hπ and let I ∈ I then there is a unitary equivalent representation ρ on H, such that:

• 1. ρI′ = idA(I′), i.e. ρ is localized in I.
• 2. ρJ(A(J)) ⊂ A(J) for all J ⊃ I, i.e. ρJ is an endomorphism of A(J).

We call ρJ an DHR endomorphism. It is enough to look into representation localized in I. RepI(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. [π1] × [π2] := [ρ1 ◦ ρ2] ρi ∈ [πi] localized in I

Marcel Bischoff (Uni G¨

• ttingen)

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 : [ρ] ⊕ [σ] = [Adw1 ◦ ρ + Adw2 ◦ σ] w1w∗

1 + w2w∗ 2 = 1, w∗ i wj = δij

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¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Braiding

Fusion coefficients: [ρ] × [σ] =

• [τ]

Nτ

ρσ[τ]

with fusion coefficients Nτ

ρσ = dim Hom(ρσ, τ).

The fusion is commutative [π1] × [π2] = [π2] × [π1] and there is a natural choice of unitaries, the braiding: ε(ρ, σ) = : ρ ◦ σ → σ ◦ ρ ρ, σ ∈ RepI(A) the braiding. Fulfills naturality (braiding fusion equations) and Yang-Baxter identity: =

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Conjugates

Let us consider RepI

f (A), i.e. only representations with finite statistical

dimension dρ < ∞, where [M : N] denotes the minimal (Jones) index: (dρ)2 =

• ρJ′(A(J′))′ : ρJ(A(J))
• ≡ [A(I) : ρI(A(I))]

For [ρ] one can define a conjugate DHR sector [¯ ρ] by ¯ ρI′ = j ◦ ρI ◦ 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: ¯ R = ρ ¯ ρ id R = ¯ ρ ρ id ; ρ ρ = ρ ρ ; ¯ ρ ¯ ρ = ¯ ρ ¯ ρ Unitary ribbon category

Marcel Bischoff (Uni G¨

• ttingen)

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¨

• ttingen)

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¨

• ttingen)

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¨

• ttingen)

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¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Modularity

If the net A is completely rational then Repf(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: Sρσ ∼ ρ σ ; Tρρ ∼ ρ ρ = conformal spin SS∗ = TT ∗ = 1, (ST)3 = S2, S4 = 1 Unitary representation of the “modular group” SL(2, Z) ∼ = Z4 ∗Z2 Z6.

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Loop group net

Example G compact Lie group Loop group: LG = C∞(S1, G) (point wise multiplication) Projective representations ← → representations of a central extension 1 − → T − → LG − → LG − → 1 π0,k projective positive-energy and vacuum representation (classified by the level k) I − → AG,k(I) = π0,k(LIG)′′ is a conformal net; LIG loops supported in I. Example G = SU(n) gives completely rational conformal net (Xu, 2000)

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Loop group net

Example G compact Lie group Loop group: LG = C∞(S1, G) (point wise multiplication) Projective representations ← → representations of a central extension 1 − → T − → LG − → LG − → 1 π0,k projective positive-energy and vacuum representation (classified by the level k) I − → AG,k(I) = π0,k(LIG)′′ is a conformal net; LIG loops supported in I. Example G = SU(n) gives completely rational conformal net (Xu, 2000)

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Loop group net

Example G compact Lie group Loop group: LG = C∞(S1, G) (point wise multiplication) Projective representations ← → representations of a central extension 1 − → T − → LG − → LG − → 1 π0,k projective positive-energy and vacuum representation (classified by the level k) I − → AG,k(I) = π0,k(LIG)′′ is a conformal net; LIG loops supported in I. Example G = SU(n) gives completely rational conformal net (Xu, 2000)

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Loop group net

Example G compact Lie group Loop group: LG = C∞(S1, G) (point wise multiplication) Projective representations ← → representations of a central extension 1 − → T − → LG − → LG − → 1 π0,k projective positive-energy and vacuum representation (classified by the level k) I − → AG,k(I) = π0,k(LIG)′′ is a conformal net; LIG loops supported in I. Example G = SU(n) gives completely rational conformal net (Xu, 2000)

Marcel Bischoff (Uni G¨

• ttingen)

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¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Minkowski space M

◮ Minkowski space ds2 = dt2 − dx2 ◮ Double cone O = I1 × I2 where I1, I2 disjoint intervals

x t I2 I1 O

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Conformal nets on Minkowski space

x t I2 I1 O Let us fix a completely rational conformal net A on S1 ∼ = R. One can define a chiral conformal net on Minkowski space by A2(O) = A(I1) ⊗ A(I2) ⊂ B(H ⊗ H) Non-chiral nets are given by irreducible local extensions B2(O) ⊃ A2(O) ≡ A(I1) ⊗ A(I2)

• n a bigger Hilbert space HB2.

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Conformal nets on Minkowski space

x t I2 I1 O Let us fix a completely rational conformal net A on S1 ∼ = R. One can define a chiral conformal net on Minkowski space by A2(O) = A(I1) ⊗ A(I2) ⊂ B(H ⊗ H) Non-chiral nets are given by irreducible local extensions B2(O) ⊃ A2(O) ≡ A(I1) ⊗ A(I2)

• n a bigger Hilbert space HB2.

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Classification Problem I Given completely rational conformal net A on S1 ∼ = R. Find (up to unitary equivalence) all local irreducible extensions B2(O) ⊃ A2(O) ≡ A(I1) ⊗ A(I2). Only finitely many extensions (Izumi,Popa,Longo,. . . ).

Marcel Bischoff (Uni G¨

• ttingen)

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¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Minkowski half-plane M+

◮ Minkowski half-plane x > 0, ds2 = dt2 − dx2 ◮ Double cone O = I1 × I2 where I1, I2 disjoint intervals

t x

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Minkowski half-plane M+

◮ Minkowski half-plane x > 0, ds2 = dt2 − dx2 ◮ Double cone O = I1 × I2 where I1, I2 disjoint intervals

I1 I2 t x O

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Minkowski half-plane

I1 I2 t x O Given completely rational conformal net A on S1 ∼ = R. Trivial (=chiral) boundary conformal net

(Longo and Rehren, 2004) is given by

A+(O) = A(I1) ∨ A(I2) ⊂ B(H). General boundary conformal nets given by (M¨

• bius covariant) irreducible local

extensions B+(O) ⊂ A+(O) ≡ A(I1) ∨ A(I2)

• n a (in general bigger) Hilbert space HB+.

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Minkowski half-plane

I1 I2 t x O Given completely rational conformal net A on S1 ∼ = R. Trivial (=chiral) boundary conformal net

(Longo and Rehren, 2004) is given by

A+(O) = A(I1) ∨ A(I2) ⊂ B(H). General boundary conformal nets given by (M¨

• bius covariant) irreducible local

extensions B+(O) ⊂ A+(O) ≡ A(I1) ∨ A(I2)

• n a (in general bigger) Hilbert space HB+.

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Classification Problem II Given completely rational conformal net A on S1 ∼ = R. Find (up to unitary equivalence) all boundary conformal nets B+ ⊃ A+, i.e. all local irreducible extensions B+(O) ⊃ A+(O) ≡ A(I1) ∨ A(I2).

Marcel Bischoff (Uni G¨

• ttingen)

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¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Boundary conditions

Two covariant nets R1, R2 are called locally isomorphic if there is an family of isomorphism ΦO : R1(O) → R2(O) such that: Φ ˜

O ↾ R1(O) = ΦO

O ⊂ ˜ O U2(g)ΦO(x)U2(g)∗ = ΦgO(U1(g)xU1(g)∗) x ∈ R1(O) . The chiral conformal net A2 on M and the chiral boundary conformal net A+ on M+ are locally equivalent (due to split property), i.e. A2 ↾ M+ ∼ = A+ Given B2 ⊃ A2 we say the net B+ ⊃ A+ is a (conformal) boundary condition (with chiral symmetry A) if the above local equivalence extends to: B2 ↾ M+ ∼ = B+.

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Classification Problem III Given completely rational conformal net A on S1 ∼ = R and a local irreducible extensions B2(O) ⊃ A2(O) ≡ A(I1) ⊗ A(I2). Find all boundary conditions (with chiral symmetry A), i.e. all B+ ⊃ A+ B+ ∼ = B2 ↾ M+ Only finitely many extensions (Izumi,Popa,Longo,. . . ). All these boundary conditions can be obtained by an operator algebraic construction (Carpi, Kawahigashi and Longo 2012). Main motivation of this talk How do this boundary conditions look like? Partial answer was given already in (Longo and Rehren, 2004).

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

We got an immediate answer by the following “curious identity” due to

(Evans, 2002) for non-degenerate braided subfactors.

• µν∈N ∆N

Zµν[µ ◦ ¯ ν] =

• a∈N ∆M

[a ◦ ¯ a] but unfortunately this is not the whole story.

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Let us first go back to: Classification Problem I Given completely rational conformal net A on S1 ∼ = R. Find (up to unitary equivalence) all local irreducible extensions B2(O) ⊃ A2(O) ≡ A(I1) ⊗ A(I2). With a given conformal net B2 on M we can associate a maximal net Bmax

2

(O) ⊃ B2(O) by B2(O) → B+(O) → Bd

+(O) = B+(O′)′ → Bmax 2

(O) Then B2 is characterized by the intermediate subfactor A2(O) ⊂ B2(O) ⊂ Bmax

2

(O) Properties of Bmax

2

:

• 1. B2 has no DHR superselection sectors (µ = 1).
• 2. B2 is modular invariant.

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Modular invariance

Bmax

2

is modular invariant: let ι : A2(O) ֒ → Bmax

2

(O) then θ2 = ¯ ι ◦ ι ∈ End(A2(O)) is the dual canonical endomorphism which decomposes as: [θ2] =

• µ,ν∈Sect(A)

Zµν[µ] ⊗ [¯ ν] with multiplicities Zµν ∈ N0 and the matrix Z commutes with the S and T matrices. Let A ⊂ B be a chiral (non-local) extension. There are two canonical tensor functors α± : RepI

f (A) → End(B(I))

(Longo and Rehren, 1995; B¨

• ckenhauer and Evans, 1998) called α-induction

defined using the braiding and opposite braiding, respectively. Using this (Rehren, 2000) constructed a local extension A2 ⊂ B2, where B2 is maximal and Zµν = dim Hom(α+

µ , α− ν ) a modular invariant

(B¨

• ckenhauer, Evans, Kawahigashi 1999).

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Two extensions are A ⊂ Ba, Bb are called Morita equivalent if the two module categories N XM• generated by sectors

A∆B• = {[a] ⊂ [ρ¯

ι•] : ρ ∈ Repf(A), a irred.},

• = a, b

respectively, are isomorphic as module categories. (ι• : A ֒ → B• is the inclusion and ¯ ι• its conjugate.) Give rise to NIMreps (non-negative integer representations) of the fusion rules in Repf(A). To a ∈ A∆B one can relate an extension A ⊂ Ba which is Morita equivalent to A ⊂ B. All Morita equivalent extensions are given this way

(Ostrik, 2003).

α-induction construction applied to Morita equivalent extensions A ⊂ B give the same two-dimensional extensions A2 ⊂ B2 (Longo and Rehren, 2004). Proposition (conjectured by (Kong and Runkel, 2010)) The α-induction construction coincides with the full centre construction in the categorical framework.

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

This enables us to use all the results about the full centre. The full centre maps of two ”extensions” is equivalent iff the extensions are Morita equivalent (Kong and Runkel, 2008). Classification Let A be a completely rational net. There is a one-to-one correspondence (up to unitary equivalence) between:

◮ Maximal two-dimensional extensions A2 ⊂ B2. ◮ Morita equivalence classes of extensions A ⊂ B. ◮ Morita equivalence classes of Q-systems in Repf(A).

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Given now a B2 ⊃ A2 maximal extension. What are its boundary conditions? For two extensions: A ⊂ B we define BXB the category defined by:

B∆B = {[β] ⊂ [ι ◦ ρ ◦ ¯

ι] : ρ ∈ Repf(A), β irred.},

• = a, b,

Proposition (Grossman and Snyder, 2012) Given A ⊂ B and a, b ∈ A∆B giving equivalent extensions A ⊂ Ba, Bb. Then there exists an invertible object β ∈ BXB, such that a = bβ. Proposition The simple elements in the Morita equivalence classe of A ⊂ B are in

• ne-to-one correspondence with

A∆B/B∆× B ≡ Skeleton

• AXB/Pic(BXB)
• ,

where B∆×

B = {β ∈ B∆B : β invertible} and

Pic(C) = {a ∈ C : ∃b ∈ C : a ⊗ b = id} is the Picard group of a tensor category C.

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Given B2 ⊃ A2 a maximal two-dimensional extension. Then it comes from the Morita equivalence class {A ⊂ Ba}a. Proposition Holography (Longo and Rehren, 2004) There is a one-to-one correspondence between Haag-dual local extensions A+ ⊂ B+ on Minkowski half-plane and non-local extensions A ⊂ B. Let us assume that B+ ⊃ A+ is a boundary condition for B2 and us the commutativity of the following diagram: B ⊃ A B+ ⊂ A+ B2 ⊃ A2 ≡ A ⊗ A α-induction ∼ removing bndry

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Proposition Let A be completely rational and B2 ⊃ A2 = A ⊗ A maximal local conformal net on Minkowski space. Then there exist a (up to Morita equivalents unique) A ⊂ B which gives B2. There is a one-to-one correspondence between

• 1. Boundary conditions Ba

+ of B2.

• 2. Extensions Ba ⊃ A Morita equivalent to B ⊃ A.
• 3. Elements in [a] ∈ A∆B/B∆×

B.

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Outlook

◮ Given chiral observables/symmetry in terms of a conformal net A we

can characterize all conformal theories on Minkowski space and all its boundary condition having this chiral symmetry

◮ Non-conformal boundary non-chiral boundary conditions (chiral

examples are given by (Longo, Witten (2011)), (B. (2012))).

◮ Defects (Longo,Rehren). ◮ Deform CQFTs using defects to obtain integrable QFTs (Runkel,. . . ) ◮ Operator algebraic constuction on riemann surfaces, extended CFTs,

• etc. (Henriques)

◮ Minkowski CFT ↔ euclidean CFT (Wick rotation) ◮ Relation to 3D TFTs, for example “duality” of

1D Positive energy reps of LG. 2D Wess Zumino Witten model 3D Chern-Simons theory.

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

Vielen Dank f¨ ur die Aufmerksamkeit! Characterization of rational conformal QFTs and their boundary conditions2

Marcel Bischoff

http://www.theorie.physik.uni-goettingen.de/~bischoff

Research Training Group 1493 Mathematical Structures in Modern Quantum Physics University of G¨

• ttingen

32nd Workshop ”Foundations and Constructive Aspects of QFT” Wuppertal, 31 May 2013

2work in progress with Roberto Longo and Yasuyuki Kawahigashi Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

References I

B¨

• ckenhauer, J. and Evans, D. E. (1998). Modular invariants, graphs and

α-induction for nets of subfactors. I. Comm. Math. Phys., 197(2):361–386. B¨

• ckenhauer, J., Evans, D. E., and Kawahigashi, Y. (1999). On

α-induction, chiral generators and modular invariants for subfactors.

• Comm. Math. Phys., 208(2):429–487.

Carpi, S., Kawahigashi, Y., and Longo, R. (2012). How to add a boundary

• condition. arXiv preprint arXiv:1205.3924.

Connes, A. (1973). Une classification des facteurs de type III. Ann. Sci. ´ Ecole Norm. Sup.(4), 6:133–252. Evans, D. E. (2002). Fusion rules of modular invariants. Rev. Math. Phys., 14(7-8):709–731. Dedicated to Professor Huzihiro Araki on the

• ccasion of his 70th birthday.

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

References II

Fredenhagen, K. and J¨

• rß, M. (1996). Conformal Haag-Kastler nets,

pointlike localized fields and the existence of operator product

• expansions. Comm. Math. Phys., 176(3):541–554.

Gabbiani, F. and Fr¨

• hlich, J. (1993). Operator algebras and conformal

field theory. Comm. Math. Phys., 155(3):569–640. Grossman, P. and Snyder, N. (2012). The Brauer-Picard group of the Asaeda-Haagerup fusion categories. Guido, D. and Longo, R. (1995). An algebraic spin and statistics theorem.

• Comm. Math. Phys., 172:517–533.

Kawahigashi, Y. and Longo, R. (2004). Classification of local conformal

• nets. Case c < 1. Ann. Math., 160(2):493–522.

Kawahigashi, Y., Longo, R., and M¨ uger, M. (2001). Multi-Interval Subfactors and Modularity of Representations in Conformal Field

• Theory. Comm. Math. Phys., 219:631–669.

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

References III

Kong, L. and Runkel, I. (2008). Morita classes of algebras in modular tensor categories. Adv. Math., 219(5):1548–1576. Kong, L. and Runkel, I. (2010). Algebraic structures in Euclidean and Minkowskian two-dimensional conformal field theory. In Noncommutative structures in mathematics and physics, pages 217–238. K. Vlaam. Acad. Belgie Wet. Kunsten (KVAB), Brussels. Longo, R. and Rehren, K.-H. (1995). Nets of Subfactors. Rev. Math. Phys., 7:567–597. Longo, R. and Rehren, K.-H. (2004). Local Fields in Boundary Conformal

• QFT. Rev. Math. Phys., 16:909–960.

Longo, R. and Xu, F. (2004). Topological sectors and a dichotomy in conformal field theory. Comm. Math. Phys., 251(2):321–364. Ostrik, V. (2003). Module categories, weak Hopf algebras and modular

• invariants. Transform. Groups, 8(2):177–206.

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013

References IV

Rehren, K.-H. (2000). Canonical tensor product subfactors. Comm. Math. Phys., 211(2):395–406. Xu, F. (2000). Jones-Wassermann subfactors for disconnected intervals.

• Commun. Contemp. Math., 2(3):307–347.

Marcel Bischoff (Uni G¨

• ttingen)

Characterization of rational CQFTs and their BCs Wuppertal, 31 May 2013