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From vertex operator algebras to conformal nets. (Slides)

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From Vertex Operator Algebras to Conformal Nets

Sebastiano Carpi Universit` a di Chieti-Pescara Based on a joint work in progress with

• Y. Kawahigashi, R. Longo and M. Weiner

Wien, September 15, 2008

Introduction Vertex operator algebras and conformal nets

• f von Neumann algebras give two different

axiomatizations of chiral conformal quantum field theories. The first is purely algebraic while the second is operator algebraic (analytic) but the two approaches have many structural sim- ilarities. In particular many interesting chi- ral CFT models can be considered form both point of view with similar outputs. The aim of this work is to analyze the con- nection of these two approaches to CFT in a systematic way. Our investigation is deeply re- lated to the old, but not yet completely under- stood, problem of the relation between Wight- man and Haag-Kastler axioms for Quantum Field Theory.

Conformal nets on S1 Let S1 = {z ∈ C : |z| = 1} be the unit circle and let I be the family of open, nonempty, nondense intervals of S1. A (local) conformal net A on S1 is a map I → A(I) ⊂ B(HA) from I into the set of von Neumann alge- bras acting on the (complex) Hilbert space HA which satisfies:

• A. Isotony. I1 ⊂ I2 ⇒ A(I1) ⊂ A(I2)
• B. Locality. I1∩I2 = ∅ ⇒ [A(I1), A(I2)] = {0}
• C. Conformal covariance. There exists a pro-

jective unitary rep. U of the group Diff+(S1)

• f orientation preserving (smooth) diffeomor-

phisms of S1 on HA such that U(γ)A(I)U(γ)∗ = A(γI)

and γ|I′ = id ⇒ U(γ) ∈ A(I), where I′ denotes the interior of S1/I.

• D. Positivity of the energy. U is a positive en-

ergy representation, i.e. the self-adjoint gen- erator L0 of the rotation subgroup of U (con- formal Hamiltonian) is positive.

• E. Vacuum. KerL0 = CΩ, where Ω (the vac-

uum vector) is a unit vector cyclic for the von Neumann algebra

I∈I A(I).

Some consequences Irreducibility.

I∈I A(I) = B(HA).

Reeh-Schlieder property. Ω is cyclic and sepa- rating for each A(I).

Haag duality. A(I)′ = A(I′)

• Factoriality. Each A(I) is a III1-factor (or C if

HA is one-dimensional).

• Additivity. I ⊂ ∪iIi ⇒ A(I) ⊂ ∨iA(Ii).

Vertex operator algebras Let V be a vector space. A formal series a(z) =

• n∈Z a(n)z−n−1 with coefficients a(n) ∈ End(V )

is called a field, if for every b ∈ V we have a(n)b = 0 for n sufficiently large. A vertex operator algebra (VOA) over C is a Z graded complex vector space V =

• n∈Z

Vn, such that dimVn < ∞ ∀n ∈ Z,

Vn = {0} for n sufficiently small, equipped with a linear map a → Y (a, z) =

• n∈Z

a(n)z−n−1, (the state fields correspondence) from V into the space of fields which satisfies

• A. Vacuum. There exists a vector Ω ∈ V such

that Y (Ω, z) = 1, Y (a, z)Ω|z=0 = a ∀a ∈ V.

• B. Locality. ∀a, b ∈ V ,

(z − w)N[Y (a, z), Y (b, w)] = 0 for N ≫ 0.

• C. Conformal covariance. There exists a vec-

tor ν ∈ V (the conformal vector) such that the coefficients of the field Y (ν, z) =

n∈Z Lnz−n−2

satisfy the Virasoro algebra relations [Ln, Lm] = (n − m)Ln+m + c 12(n3 − n)δ−n,m1

with central charge c ∈ C and also satisfy the following properties: [L−1, Y (a, z)] = d dzY (a, z) ∀a ∈ V ; L0a = na ∀a ∈ Vn. The fields Y (a, z) are called vertex operators. If a ∈ V is homogeneous, i.e. L0a = daa for some da ∈ Z (the conformal dimension of a), is useful to introduce the notation an ≡ a(n+da−1) so that Y (a, z) =

n∈Z anz−n−da.

If a ∈ V is arbitrary then an is defined by linearity. In the following we shall denote by Aut(V ) the group of VOA automorphisms of V . It has a natural group topology.

Unitary vertex operator algebras Let V be a VOA and let (·|·) be a scalar product

• n V such that (Ω|Ω) = 1 (normalization) and

(Lna|b) = (a|L−nb) ∀a, b ∈ V ∀n ∈ Z (unitary Virasoro symmetry). Then for every a ∈ V and every n ∈ Z there exists a+

(n) ∈ EndV

(the adjoint of a(n) on V ) such that (a(n)b|c) = (b|a+

(n)c) ∀b, c ∈ V.

Moreover, the formal series Y (a, z)+ ≡

• n∈Z

a+

(n)zn+1 =

• n∈Z

a+

(−n−2)z−n−1

is a field on V and we say that Y (a, z) has a local adjoint if ∀b ∈ V , (z − w)N[Y (a, z)+, Y (b, w)] = 0 for N ≫ 0. We say that the VOA V is a unitary VOA if Y (a, z) has a local adjoint ∀a ∈ V .

Equivalently a VOA V with a normalized scalar product (·|·) is unitary if there exists an anti- linear VOA automorphism θ of V (the PCT

• perator) such that (θ · |·) is an invariant bilin-

ear form on V . In this case we say that (·|·) is invariant. A unitary VOA is simple ⇔ V0 = CΩ In general a unitary VOA can have many dif- ferent unitary structures which however are al- ways related by VOA automorphisms. If V is a unitary VOA we denote by Aut(·|·)(V ) the subgroup of unitary elements (i.e preserv- ing (·|·)) of Aut(V ). It is always a compact subgroup. If V is simple we have the follow- ing: Aut(V ) = Aut(·|·)(V ) ⇔ Aut(V ) is compact ⇔ Aut(·|·)(V ) is totally disconnected ⇔ V has a unique unitary structure.

As a consequence Aut(V ) is finite ⇔ Aut(·|·)(V ) is finite and in this case Aut(V ) = Aut(·|·)(V ). Strongly local VOAs From now on V is a simple unitary VOA. We also assume that V is energy bounded i.e. that for every a ∈ V there exist positive integers s, k and a constant M > 0 such that anb ≤ M(|n| + 1)s(L0 + 1)kb ∀n ∈ Z, ∀b ∈ V. Now, let HV be the Hilbert space completion of V and let f ∈ C∞(S1) with Fourier coefficients ˆ fn. For every a ∈ V we define the operator Y0(a, f) on HV with domain V by Y0(a, f)b =

• n∈Z

an ˆ fnb for b ∈ V. It is a closable operator and we denote its clo- sure by Y (a, f) (smeared vertex operator). If

a is homogeneous then we can use the formal notation Y (a, f) =

• S1 Y (a, z)f(z)zda dz

2πiz. We now define a map AV from the family of intervals I into the family of von Neumann al- gebras on HV by AV (I) = W ∗({Y (a, f) : a ∈ V, suppf ⊂ I}). It is clear that the map I → AV (I) satisfies isotony. We say that V is strongly local if AV satisfies locality. For a strongly local V we have the following results:

• 1. AV is a conformal net on S1.

2. Different unitary structure on V give rise to isomorphic (unitary equivalent) conformal nets.

• 3. The automorphism group Aut(AV ) of AV

coincides with Aut(·|·)(V ). In particular if the latter is finite we have Aut(AV ) = Aut(V ).

• 4. Every unitary subalgebra W ⊂ V is a strongly

local unitary simple VOA and the map W → AW gives a one-to-one correspondence between the family of unitary subalgebras of V and the family of covariant subnets of AV . Orbifold and coset subalgebras are mapped in to orb- ifold and cosets subnets respectively. Examples of strongly local VOAs Let V be an energy-bounded simple unitary

• VOA. An homogeneous element a ∈ V

and the corresponding vertex operator Y (a, z) are called quasi-primary if L1a = 0. A quasi-primary a ∈ V and Y (a, z) are called hermitian if a+

n =

a−n for all n ∈ Z. Let F ⊂ V be a family of her- mitian quasi-primary elements of V and define AF(I) = W ∗({Y (a, f) : a ∈ F, suppf ⊂ I}), I ∈ I.

Clearly the map I → AF(I) satisfies isotony and AF(I) ⊂ AV (I). We have the following result (inspired by a work of Driessler, Sum- mers and Wichmann): If I → AF(I) satisfies locality and F generates V then AF(I) = AV (I) for all I ∈ I. In partic- ular V is strongly local. Using standard estimates (Goodman and Wal- lach; Buchholz and Schulz-Mirbach) we obtain the following: If V is generated by hermitian currents and Virasoro fields then it is strongly local. Using this fact one can find many interesting examples of strongly local unitary VOAs. For example: VOAs generated affine lie algebras, their cosets and orbifold subalgebras; Virasoro

VOAs; the moonshine VOA V ♮ and its unitary subalgebras. The automorphism group V ♮ is known to be the monster group, the larger among the 26 sporadic finite simple groups. Recently Longo and Kawahigashi have con- structed a net A♮ such that Aut(A♮) = Aut(V ♮). Since it can be shown that A♮ = AV ♮ this is a special case of the equality Aut(AV ) = Aut(V ) which holds for every strongly local unitary VOA when Aut(V ) is finite. It is known that there is a unitary subalgebra ˜ V ⊂ V ♮ such that Aut(˜ V ) is the baby monster sporadic simple group. ˜ V , being a unitary subalgebra of V ♮, is strongly local and hence A˜

V is a conformal net such

that Aut(A˜

V ) is the baby monster group.

Outlook Question: Are there unitary VOAs that are not strongly local? Problem: Find a direct relation between the (unitary) representation theory of (sufficiently regular) strongly local VOAs and the represen- tation theory of the corresponding conformal nets.

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