Generalized Orbifolds in Conformal Field Theory:
Compact Hypergroups
Simone Del Vecchio

based on joint work with Marcel Bischoff and Luca Giorgetti

Virtual LQP / LQP45 workshop
17 June 2020

Simone Del Vecchio (Univ. Leipzig)

Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

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Let A be a conformal net and B ⊂ A a conformal subnet.
Question: Is there an “algebraic object” K acting on A for which B = AK ?
Conversely: If K suitably acts on A, is B := AK a conformal subnet?
Can then have statements of the type:
If B ⊂ A with B = AK then all intermediate conformal subnets C,
B ⊂ C ⊂ A,
are given by fixed points under the action of subobjects of K on A.

Simone Del Vecchio (Univ. Leipzig)

Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

2 / 22Compact Hypergroups

Definition
We say a compact Hausdorff space K with distinguished element e ∈ K andcontinuous involution · : K → K, k 7→ k̄, with ē = e is a compact hypergroup, if• The Banach space M b (K) of complex Radon measures has a convolution

product ∗ which is associative, bilinear, weakly continuous.
• For x, y ∈ K the measure δx ∗ δy is a probability measure.
• The mapping (x, y) → δx ∗ δy is continuous.
• δe ∗ δx = δx ∗ δe = δx for all x ∈ K.
• δx ∗ δy = δȳ ∗ δx̄ , where µ(k) = µ(k).
• ∃ faithful probability measure µK ∈ M b (K) such that for all f, g ∈ C(K),ZZ
f (x ∗ y)g(x)dµK (x) =Kf (x)g(x ∗ ȳ)dµK (x)Kwith

Zf (x ∗ y) :=f (k)d(δx ∗ δy )(k).
KSimone Del Vecchio (Univ. Leipzig)Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

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Conformal NetsLet I be the set of nonempty, nondense, open intervals of S 1 . A conformal net on
S 1 is a family A = {A(I) : I ∈ I} of von Neumann algebras, acting on an
infinite-dimensional separable complex Hilbert space H, satisfying the following
properties:• Isotony: A(I1 ) ⊂ A(I2 ), if I1 ⊂ I2 , I1 , I2 ∈ I.
• Locality: A(I1 ) ⊂ A(I2 )0 , if I1 ∩ I2 = ∅, I1 , I2 ∈ I.
• Möbius covariance: There exists a strongly continuous unitary representation

U of PSL(2, R) in H such that U (g)A(I)U (g)∗ = A(gI), I ∈ I,g ∈PSL(2, R), where PSL(2, R) acts on S 1 by Möbius transformations.
• Positivity of the energy: The generator of rotations L0 has nonnegativespectrum.
• Existence and uniqueness of the vacuum: There exists a unique (up to phase)U -invariant unit vector Ω ∈ H.
• Cyclicity of the vacuum: Ω is cyclic for the algebra A(S 1 ) := ∪I∈I A(I).

Simone Del Vecchio (Univ. Leipzig)

Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

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The tensor category End(N )Let A be a conformal net on S 1 , B ⊂ A a conformal subnet.
Let N := B(I), M := A(I) for a fixed I ∈ I (type III factors).
End(N ) is the category of unital endomorphisms of N withHom(ρ, σ) := {T ∈ N : T ρ(n) = σ(n)T for all n ∈ N }for ρ, σ ∈End(N ), and tensor productρ ⊗ σ := ρ ◦ σ,T1 ⊗ T2 := T1 ρ1 (T2 ) = σ1 (T2 )T1 .
for T1 ∈ Hom(ρ1 , σ1 ), T2 ∈ Hom(ρ2 , σ2 ), ρ1 , ρ2 , σ1 , σ2 ∈End(N ).

Simone Del Vecchio (Univ. Leipzig)

Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

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Longo’s Canonical Endomorphism

Let N := B(I), M := A(I).
Let ι : N → M be the embedding homomorphism of N in M, thenγ := ιῑ ∈ End(M) is the canonical endomorphism of N ⊂ M
θ := ῑι ∈ End(N ) is the dual canonical endomorphism of N ⊂ M• Any E ∈ E(M, N ) is a Stinespring dilation of γ: E(·) = w∗ γ(·)w withw ∈ Hom(id, θ) [Longo 90].
γ and θ may be defined by Tomita-Takesaki modular theory even if ῑ does notexist as in a rigid tensor category [Longo 90].
Let hθi denote the rigid category generated by finite-dimensional
subendomorphisms of θ.

Simone Del Vecchio (Univ. Leipzig)

Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

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DiscretenessDefinition (Discreteness)
If N ⊂ M is an irreducible subfactor (M ∩ N 0 = C) with normal, faithfulconditional expectation E, thenN ⊂ M discrete ⇔ θ ∼= ⊕i ρi with dim(ρi ) < ∞.

Simone Del Vecchio (Univ. Leipzig)

Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

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Discreteness

Hom(θ, θ) ∼=M

Mnρ (C)[ρ]Definition (Charged Intertwiners)
Let ρ ∈ Obj(hθi).
Hρ := Hom(ι, ι ◦ ρ) = {ψ ∈ M : ψι(n) = ι(ρ(n))ψ for all n ∈ N }.
Recall E(·) = w∗ γ(·)w with w ∈ Hom(id, θ).
Tρ : Hρ × Hρ → Hom(θ, θ) ⊂ N(ψ1 , ψ2 ) 7→ γ(ψ1∗ )ww∗ γ(ψ2 )Simone Del Vecchio (Univ. Leipzig)

Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

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Construction of Hypergroup - Algebra of Trigonometric Polynomials

Hom(θ, θ) ∼=MMnρ (C)[ρ]

Recall E(·) = w∗ γ(·)w with w ∈ Hom(id, θ).
Tρ : Hρ × Hρ → Hom(θ, θ) ⊂ N(ψ1 , ψ2 ) 7→ γ(ψ1∗ )ww∗ γ(ψ2 )Definition (Algebra of Trigonometric Polynomials)
Trig(N ⊂ M) := SpanC {Tρ (ψ1 , ψ2 ) : ρ ∈ Obj(hθi), ψ1 , ψ2 ∈ Hρ },Tρ1 (ψ1 , ψ2 ) ∗ Tρ2 (ψ3 , ψ4 ) := Tρ1 ◦ρ2 (ψ1 ψ3 , ψ2 ψ4 ),• : ψ ∈ Hρ → ψ • := ψ ∗ ι(r̄ρ ) ∈ Hρ̄ ,(Tρ (ψ1 , ψ2 ))• := Tρ̄ (ψ1• , ψ2• ).
Simone Del Vecchio (Univ. Leipzig)

Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

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Construction of Hypergroup - Algebra of Trigonometric Polynomials

Theorem (M. Bischoff, S.D.V, L. Giorgetti)(Trig(N ⊂ M), ∗,• ) is a unital, associative algebra with involution.
If N ⊂ M is a local subfactor then (Trig(N ⊂ M), ∗,• ) is commutative.

Definition
A discrete, irreducible subfactor N ⊂ M is local ifψρ ψσ = ε±ρ,σ ψσ ψρ
for all ψρ ∈ Hρ , ψσ ∈ Hσ , ρ, σ ∈Obj(hθi) with ε±ρ,σ the DHR braiding.

Simone Del Vecchio (Univ. Leipzig)

Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

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Adjointable UCP maps

Let Ω be a cyclic and separating vector for M.
Definition
Let UCPN (M, Ω) denote the set of normal linear maps φ : M → M with thefollowing properties
1

φ is unital and completely positive (UCP).

2φ preserves the state given by Ω, i.e. (Ω, φ( · )Ω) = (Ω, · Ω).

3

φ is N -bimodular, i.e. φ(n1 mn2 ) = n1 φ(m)n2 for every n1 , n2 ∈ N ,m ∈ M.

Lemma
Under our hypothesis for N ⊂ M, every φ ∈UCPN (M, Ω) is Ω-adjointable, i.e.
∃φ# ∈UCPN (M, Ω) such that(φ# (m1 )Ω, m2 Ω) = (m1 Ω, φ(m2 )Ω)
for every m1 , m2 ∈ M.
Simone Del Vecchio (Univ. Leipzig)

Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

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Duality Pairing

There is a duality pairing between UCPN (M, Ω) and (Trig(N ⊂ M), ∗,• )h · , · i : UCPN (M, Ω) × Trig(N ⊂ M) → Chφ, Tρ (ψ1 , ψ2 )i := ψ1∗ φ(ψ2 ) ∈ Hom(ι, ι) = C id
Denote by ωφ := hφ, ·i. ωφ is a positive linear functional.

Simone Del Vecchio (Univ. Leipzig)

Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

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Duality Pairing

There is a duality pairing between UCPN (M, Ω) and (Trig(N ⊂ M), ∗,• )h · , · i : UCPN (M, Ω) × Trig(N ⊂ M) → Chφ, Tρ (ψ1 , ψ2 )i := ψ1∗ φ(ψ2 ) ∈ Hom(ι, ι) = C id
Denote by ωφ := hφ, ·i. ωφ is a positive linear functional.
Consider ωE , with E conditional expectation of N ⊂ M.
Let λE be the GNS representation of (Trig(N ⊂ M), ∗,• ) wrt ωE .
Lemma
λE is a faithful representation of (Trig(N ⊂ M), ∗,• ) by bounded operators.

Simone Del Vecchio (Univ. Leipzig)

Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

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Reduced C ∗ -algebra of N ⊂ M

Let λE be the GNS representation of (Trig(N ⊂ M), ∗,• ) wrt ωE .
∗Denote by CE(N ⊂ M) the closure of the image of λE .

Remark∗CE(N ⊂ M) is a commutative and separable C ∗ -algebra and thus∗
CE(N ⊂ M) ∼= C(K)for some compact, metrizable topological space K.

Simone Del Vecchio (Univ. Leipzig)

Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

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Reduced C ∗ -algebra of N ⊂ M

Let λE be the GNS representation of (Trig(N ⊂ M), ∗,• ) wrt ωE .
∗Denote by CE(N ⊂ M) the closure of the image of λE .

Remark∗CE(N ⊂ M) is a commutative and separable C ∗ -algebra and thus∗
CE(N ⊂ M) ∼= C(K)for some compact, metrizable topological space K.
Theorem
The duality map φ → ωφ lifts to a map between UCPN (M, Ω) and states on∗CE(N ⊂ M).

Simone Del Vecchio (Univ. Leipzig)

Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

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The Hypergroup K∗
CE(N ⊂ M) ∼= C(K)for some compact, metrizable topological space K.
Theorem (M. Bischoff. S.D.V., L. Giorgetti)
The duality map
φ 7→ ωφis a homeomorphism between• UCPN (M, Ω)∗• the set of states ω on CE(N ⊂ M).

In particular it restricts to a homeomorphism between• Extr(UCPN (M, Ω)) (Extreme points of UCPN (M, Ω))• K.

Simone Del Vecchio (Univ. Leipzig)

Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

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The Hypergroup K

Corollaryb
UCPN (M, Ω) ∼(Extr(UCPN (M, Ω)))
= M+bwhere M+(Extr(UCPN (M, Ω))) denotes the positive Radon measures onExtr(UCPN (M, Ω)).

Theorem (M. Bischoff. S.D.V., L. Giorgetti)∼K =Extr(UCPN (M, Ω)) is a compact hypergroup with• convolution: φ1 ∗ φ2 := φ1 ◦ φ2
• adjoint: φ∼ := φ# , where φ# is the Ω-adjoint of φ.
• E is the Haar measure of K.

Simone Del Vecchio (Univ. Leipzig)

Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

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Generalized Orbifolds

Corollary (Choquet-type decomposition of E)
Let m ∈ M, we haveZE(m) =φ(m)dωE (φ)(1)

K

where the integral is understood in the weak sense.
Theorem (M. Bischoff. S.D.V., L. Giorgetti)
Let N ⊂ M be an irreducible local discrete subfactor of type III. Let Ω ∈ H be astandard vector for M ⊂ B(H) such that the associated state is invariant wrt theunique normal faithful conditional expectation E : M → N .
Then there is a compact hypergroup K which acts on M by Ω-Markov maps and
N = MK := {m ∈ M : φ(m) = m for all φ ∈ K}.

Simone Del Vecchio (Univ. Leipzig)

Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

16 / 22Representations

Proposition
There is a bijective correspondence between{π : Irreducible Representations of K } ←→ {ρπ : Irreducible subsectors of θ }Simone Del Vecchio (Univ. Leipzig)

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Representations

Proposition
There is a bijective correspondence between{π : Irreducible Representations of K } ←→ {ρπ : Irreducible subsectors of θ }Theorem (R. Vrem 79, Y. Chapovski, L. Vainerman 99 - Schur’s Orthogonality
Relations)
Let π be an irreducible representation of K acting on the complex Hilbert space
Hπ . There is dπ ∈ R with dπ ≥dimHπ such that for any v ∈ Hπ with kvk = 1Z
1.
|(v, π(k)v)|2 dµK (k) =dπK
dπ is called the hyperdimension of π.

Simone Del Vecchio (Univ. Leipzig)

Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

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Representations

Proposition
There is a bijective correspondence between{π : Irreducible Representations of K } ←→ {ρπ : Irreducible subsectors of θ }Theorem (R. Vrem 79, Y. Chapovski, L. Vainerman 99 - Schur’s OrthogonalityRelations)
Let π be an irreducible representation of K acting on the complex Hilbert space
Hπ . There is dπ ∈ R with dπ ≥dimHπ such that for any v ∈ Hπ with kvk = 1Z
1.
|(v, π(k)v)|2 dµK (k) =dπ
Kdπ is called the hyperdimension of π.
Proposition
The hyperdimension dπ is equal to the statistical dimension dρπ .
Simone Del Vecchio (Univ. Leipzig)

Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

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The Case of Depth 2CorollaryHom(γ, γ) ∼= L∞ (K, dωE ).
together withTheorem (M. Enock, R. Nest 96)
Suppose N ⊂ M has depth 2, is semidiscrete, and Hom(γ, γ) is commutative.
Then ∃G compact group acting on M with N = MG .
we getCorollarySuppose N ⊂ M is a discrete local subfactor with depth 2. Then ∃G compact
group acting on M with N = MG .

Simone Del Vecchio (Univ. Leipzig)

Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

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Examples - Double Cosets

Proposition
Let N = MG ⊂ M for some compact group G.
Let H ⊂ G a closed subgroup. Then
N = (MH )G//H ⊂ MH
whereG//H = {HxH : x ∈ G},Z
δHxH ∗ δHyH :=δHxtyH dωH (t),H(δHxH )∼ := δHx−1 H .

Simone Del Vecchio (Univ. Leipzig)

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Examples

ExampleSO(3)Vir1 = SU (2)1

⊂ SU (2)1 .

Every discrete extension of Vir1 is of the form
Vir1 ⊂ SU (2)H
1for some closed subgroup H of SO(3).
Thus for any discrete extension B of Vir1 ,Vir1 = B K
where K = SO(3)//H, with H a closed subgroup of SO(3).

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Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

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Examples

ExampleSU (2)10 ⊂ Spin(5)1γ = id ⊕γ1 ,d γ1 = 2 +√3

K = {e, φγ1 }φγ1 ∗ φγ1 =√11+ 3√ e+√ φ γ12+ 32+ 3

φ#γ1 = φ γ1 .

Simone Del Vecchio (Univ. Leipzig)

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Open Problems and Further Directions• Inverse Problem: From action of compact hypergroup to discrete subnet?
• Galois type correspondence for Intermediate subnets?
• Non Local case ←→ Compact Quantum Hypergroups?
• General case (semidiscreteness)?

Simone Del Vecchio (Univ. Leipzig)

Generalized Orbifolds in Conformal Field Theory: Compact Hypergroups

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