Simple-Current Abelian Intertwining Algebras I

Let us return to the situation in Section 2.2 about simple-current vertex operator algeb-ras, i.e. letV be as in AssumptionSNa simple, rational,C₂-cofinite, self-contragredient vertex operator algebra of CFT-type such that all irreducible modules are simple cur-rents. Then we saw that there is the structure of a finite abelian group onF_V, the fusion group ofV, and a finite quadratic formQ_ρ, given by the conformal weights modulo 1.²

It follows from results by Huang:

Theorem 3.2.1. Let V be as in AssumptionSN. Then the direct sum of the irreducible V-modules

A:= ^M

γ∈F_V

W^γ

carries the structure of an abelian intertwining algebra associated with some normalised abelian 3-cocycle (F,Ω) onF_V.

The abelian intertwining algebra structure on A is the unique one up to a normalised abelian 3-coboundary extending the given vertex operator algebra and module structures.

Remark 3.2.2. The uniqueness statement in the above theorem is to be understood in the following sense: the vertex operationY(·, x) onAhas to be composed of intertwining operators of type _W^Wα^α+βW^β

,α, β ∈F_V. These are unique up to a scalar. Hence, Y(·, x) may be multiplied by a function f: FV ×FV → C^× with f(α,0) = f(0, α) = 1 for all α∈F_V, resulting in a change of the abelian 3-cocycle (F,Ω) by the normalised abelian 3-coboundary (F_f,Ωf). In particular the quadratic formQ_Ω is unique, as isB_Ω.

It is part of the statement of the theorem that the bilinear forms B_ρ and B_Ω asso-ciated with the conformal weights and the abelian 3-cocycle (F,Ω), respectively, are the negatives of each other, i.e. Bρ =−B_Ω (as explained in item (2) of Remark 3.1.5).

From the theory of finite quadratic forms alone this does not imply that the quadratic formsQ_ρ and −Q_Ω are identical because of the fact that to every bilinear form on F_V, there are |F_V/2FV| many possible quadratic forms with that associated bilinear form (see Remark A.1.3).

However, using the theory of modular tensor categories and in particular Huang’s con-struction of modular tensor categories associated with certain vertex operator algebras, it is possible to show that in the situation of Theorem 3.2.1 and under Assumption P the quadratic forms Q_ρ and −Q_Ω are indeed the same.³ This is the statement of the following main theorem of this chapter, which is Theorem 2.7 in [HS14] but was stated there with an incomplete proof, as was pointed out by Scott Carnahan (see introduction of [Car14]):

2We had to additionally assume thatV satisfies AssumptionPbut the result on the quadratic form does not depend on that (except for the non-degeneracy, cf. Theorem 2.2.7). Indeed, that the conformal weights form a quadratic form is part of the statement of Theorem 3.2.1 and will be shown in Lemma 3.2.5, which is needed for the proof of the theorem.

3Also, since we use AssumptionP in the theorem, we know that the quadratic form Qρ = −QΩ is non-degenerate, i.e. (FV, Qρ) forms a finite quadratic space (see Theorem 2.2.7).

Theorem 3.2.3. LetV be as in AssumptionsSNPwith fusion groupF_V = (F_V, Q_ρ).

Then the direct sum

A:= ^M

γ∈F_V

W^γ

can be given the structure of an abelian intertwining algebra, the unique one up to a normalised abelian 3-coboundary extending the given vertex operator algebra and module structures, with associated normalised abelian 3-cocycle (F,Ω) such that

Q_Ω(γ) =−Q_ρ(γ)

for all γ ∈ F_V, i.e. the quadratic forms associated with the abelian 3-cocycle and the conformal weights are the negatives of each other. In other words: the finite quadratic space (FV, QΩ) associated with the abelian intertwining algebra A equals F_V = (F_V,−Q_ρ).

The level of the abelian intertwining algebra A is exactly the level of the finite quadratic space FV (or FV).

In the following we will present proofs of Theorems 3.2.1 and 3.2.3. The first the-orem is essentially well known and a special case of more general results by Huang.

Indeed, Theorem 3.7 (and Remark 3.8) in [Hua05] states that for a rational,C2-cofinite vertex operator algebra V of CFT-type the direct sum of all irreducible V-modules up to isomorphism admits the structure of an intertwining operator algebra. Intertwining operator algebras were first introduced in [Hua97] but Definition 5.1 in [Hua00] gives an equivalent characterisation of them as natural non-abelian generalisations of abelian intertwining algebras. From this definition on can read off that the intertwining oper-ator algebra from Theorem 3.7 in [Hua05] becomes an abelian intertwining algebra if we additionally assume thatV is simple, self-contragredient and all modules are simple currents, i.e. under AssumptionSN.

For reasons of comprehensibility it seems appropriate to include a direct proof of Theorem 3.2.1, which also heavily relies on results by Huang. This will be the rest of this section (Section 3.2).

For the proof of Theorem 3.2.3 we need some knowledge of modular tensor categories, in particular those defined by Huang associated with certain vertex operator algebras.

These concepts will be introduced in Section 3.3. Finally, Section 3.4 gives two inde-pendent proofs of Theorem 3.2.3.

Proof of Theorem 3.2.1

The following proof is largely based on notes by van Ekeren [Eke15] and uses results by Huang from [Hua96b, Hua00, Hua05, Hua08b]. Recall that we interpret fractional powers of complex variables as zⁿ = e^nlog(z) where log(z) = log(|z|) + i arg(z) and 0≤arg(z)<2π (see Section 1.1).

We start our considerations with the following result due to Huang:

Proposition 3.2.4 (Huang). LetV be a rational, C₂-cofinite vertex operator algebra of CFT-type. Let A, B, C, D, P be V-modules and Y₁ ∈ V_{A P}^D and Y₂ ∈ V_{B C}^P intertwining operators. Fix vectors a∈A, b∈B, c∈C and d⁰ ∈D⁰. Then there exists a V-module Q and intertwining operators Y₃∈ V_{B Q}^D and Y₄ ∈ V_{A C}^Q such that the series

d⁰,Y₁(a, z)Y₂(b, w)c and d⁰,Y₃(b, w)Y₄(a, z)c

converge in the domains0<|w|<|z|and0<|z|<|w|, respectively, and are restrictions to these domains of a multi-valued function F(z, w), analytic on the domain {(z, w) ∈ C² | z, w, z−w 6= 0}. Moreover, there is a V-module R and intertwining operators Y₅ ∈ V_{R C}^D and Y₆ ∈ V_{A B}^R such that

d⁰,Y₅(Y₆(a, z−w)b, w)c

converges in the domain 0 < |z−w| < |w| and is the restriction to that domain of F(z, w).

Proof. The proposition is a special case of [Hua00], Lemma 4.1. The result is also given without the precise statement on the domain ofF in [Hua96b], Theorems 1.8 and 3.1. In both cases the result is proved under additional hypotheses onV called “convergence and extension properties” (see [Hua96b], p. 210). In [Hua05], Remark 3.8, these hypotheses are shown to hold ifV is rational, C2-cofinite and of CFT-type.

The proof of the following lemma is similar to that of [Yam04], Lemma 3.1, where a special case of item (1) is proved.

Lemma 3.2.5 ([Eke15], Lemma 4.11). Let V be as in Assumption SN. In the situation of Proposition 3.2.4 letA=W^α,B =W^β, C=W^γ andD=W^α+β+γ forα, β, γ∈F_V. Then:

(1) Q_ρ(α) = ρ(W^α) +Z, α ∈ F_V, defines a quadratic form on the abelian group F_V with associated bilinear form Bρ(α, β) =Qρ(α+β)−Qρ(α)−Qρ(β), α, β ∈FV. (2) There is an N ∈Q, depending only ona∈A and b∈B, such that

(x−y)^N[Y₁(a, x)Y₂(b, y)− Y₃(b, y)Y₄(a, x)] = 0. (3) Fora∈A, b∈B, c∈C, d⁰ ∈D⁰ the series

d⁰,Y₁(a, x)Y₂(b, y)cι_x,y(x−y)^{−Bρ(α,β)}x^{−Bρ(α,γ)}y^{−Bρ(β,γ)}, d⁰,Y₃(b, y)Y₄(a, x)cιy,x(x−y)^{−Bρ(α,β)}x^{−Bρ(α,γ)}y^{−Bρ(β,γ)}, d⁰,Y₅(Y₆(a, x−y)b, y)cιy,x−y(x−y)^{−Bρ(α,β)}x^{−Bρ(α,γ)}y^{−Bρ(β,γ)}

are the images of a common element of C[x^±1, y^±1,(x−y)⁻¹]under ιx,y,ιy,x and ιy,x−y, respectively.

By a slight abuse of notation we interpret Bρ(α, β) in a term like x^Bρ(α,β) as an arbitrary representative inQof Bρ(α, β)∈Q/Z.

Proof. For the proof we consider some expressions in terms of the complex variablesz, w rather than the formal variablesx, y. Fixa, b, c, d⁰ as in Proposition 3.2.4. By definition, Y(W^α, x)W^β ⊆x^Bρ(α,β)W^α+β((x)) for an intertwining operatorY(·, x) of type _W^Wαα+β_W^β. The expressionhd⁰,Y₅(Y₆(a, z−w)b, w)ci defines the multi-valued functionF(z, w) as a series inz−w,w. By the boundedness-from-below property forY₆ (see Definition 1.6.1) there exists anN ∈Q, depending only onaand b, such that (z−w)^NF(z, w) is regular atz−w= 0. More precisely, N ∈ −B_ρ(α, β).

We consider the series expansion

(z−w)^NF(z, w) =ι_z,w(z−w)^Nd⁰,Y₁(a, z)Y₂(b, w)c

in the domain 0<|w|<|z|. Since (z−w)^NF(z, w) is regular at z−w= 0, the series on the right-hand side actually converges for all |z|,|w|>0. It is a series in fractional powers of z andw but

z^{Bρ(α,β)−Bρ(α,β+γ)}w^{−Bρ(β,γ)}ιz,w(z−w)^Nd⁰,Y₁(a, z)Y₂(b, w)c contains clearly only integral powers of z andw. Similarly,

z^{−Bρ(α,γ)}w^{Bρ(α,β)−Bρ(β,α+γ)}ιw,z(z−w)^Nd⁰,Y₃(b, w)Y₄(a, z)c

is a convergent series in integral powers ofzandw. Since the functions defined by these two series are single-valued, so is their ratio

z^{Bρ(α,β)−Bρ(α,β+γ)+Bρ(α,γ)}w^{Bρ(β,α+γ)−Bρ(α,β)−Bρ(β,γ)}. This implies

Bρ(α, β)−Bρ(α, β+γ) +Bρ(α, γ) = 0 +Z, B_ρ(β, α+γ)−B_ρ(α, β)−B_ρ(β, γ) = 0 +Z,

which means that B_ρ is bilinear. Since a module and its contragredient have the same weight grading, Qρ(−α) = Qρ(α) and Qρ(0) = ρ(V) +Z = 0 +Z so that by Proposi-tion A.1.5Q_ρ is a quadratic form. Above we obtained two convergent series in integral powers of z and w whose ratio is of the form (z/w)^k for some k ∈ Z. One series has finitely many negative powers ofwand finitely many positive powers ofz, the other vice versa. Hence both lie inC[z^±1, w^±1], which proves items (2) and (3).

Remark 3.2.6. Recall from (1.1) that for consistency with the definitions of [DL93] we chose to interpret

ιy,x(x−y)ⁿ= (e^πi)⁻ⁿιy,x(y−x)ⁿ

forn∈C. This expression appears in Lemma 3.2.5 but the convention does not matter for the statement of the lemma because any ambiguity can be absorbed into the choice of Y₃ and Y₄.

In order to pass to a generalised Jacobi identity for the intertwining operators under AssumptionSNwe first recall some standard formulæ for Laurent polynomials in several formal variables:

Lemma 3.2.7. Let f(x0, x1, x2)∈C[x^±1₀ , x^±1₁ , x^±1₂ ]. Then:

(1) [FHL93], Proposition 3.1.1:

ι_x1,x0x⁻¹₂ δ

x₁−x₀ x₂

f(x₀, x₁, x₁−x₀)

=ιx1,x2x⁻¹₀ δ

x1−x2

x₀

f(x₁−x₂, x₁, x₂)

−ιx2,x1x⁻¹₀ δ

x2−x1

−x₀

f(x1−x2, x1, x2). (2) [FHL93], Proposition 3.1.1:

ι_x2,x0x⁻¹₁ δ

x₂+x₀ x₁

f(x₀, x₀+x₂, x₂) =ι_x1,x0x⁻¹₂ δ

x₁−x₀ x₂

f(x₀, x₁, x₁−x₀). (3) [FLM88], Proposition 8.8.22:

ι_x1,x0x⁻¹₂

x₁−x₀ x₂

m

δ

x₁−x₀ x₂

=ι_x2,x0x⁻¹₁

x₂+x₀ x₁

−m

δ

x₂+x₀ x₁

for everym∈C.

We continue in the situation of Lemma 3.2.5. Let g(x, y) ∈ C[x^±1, y^±1,(x−y)⁻¹] be the common element in item (3) there and let the rational function f(x₀, x₁, x₂) ∈ C[x^±1₀ , x^±1₁ , x^±1₂ ] be such that g(x, y) = f(x−y, x, y). Applying items (1) and (2) of Lemma 3.2.7 tof yields

ι_x1,x2(x₁−x₂)^{−Bρ(α,β)}x^−B₁ ^ρ(α,γ)x^−B₂ ^ρ(β,γ)x⁻¹₀ δ

x₁−x₂ x₀

Y₁(a, x₁)Y₂(b, x₂)c

−ι_x2,x1(x₁−x₂)^{−Bρ(α,β)}x^−B₁ ^ρ(α,γ)x^−B₂ ^ρ(β,γ)x⁻¹₀ δ

x₂−x₁

−x₀

Y₃(b, x₂)Y₄(a, x₁)c

=ιx2,x0x^−B₀ ^ρ(α,β)(x0+x2)^{−Bρ(α,γ)}x^−B₂ ^ρ(β,γ)x⁻¹₁ δ

x2+x0

x₁

Y₅(Y₆(a, x0)b, x2)c.

(3.2)

Indeed, the identity holds when paired withd⁰ for alld⁰ ∈(W^α+β+γ)⁰ and hence it holds withd⁰ omitted. The first term of (3.2) equals

ι_x1,x2x^−B₀ ^ρ(α,β)x^−B₁ ^ρ(α,γ)x^−B₂ ^ρ(β,γ)x⁻¹₀

x₁−x₂ x₀

−B_ρ(α,β)

δ

x₁−x₂ x₀

Y₁(a, x₁)Y₂(b, x₂)c.

Using item (3) of Lemma 3.2.7, the third term of (3.2) equates to ι_x2,x0x^−B₀ ^ρ(α,β)x^−B₁ ^ρ(α,γ)x^−B₂ ^ρ(β,γ)x⁻¹₁

x₂+x₀ x₁

−B_ρ(α,γ)

δ

x₂+x₀ x₁

Y₅(Y₆(a, x₀)b, x₂)c

=ι_x1,x0x^−B₀ ^ρ(α,β)x^−B₁ ^ρ(α,γ)x^−B₂ ^ρ(β,γ)x⁻¹₂

x₁−x₀ x2

Bρ(α,γ)

δ

x₁−x₀ x2

Y₅(Y₆(a, x₀)b, x₂)c

and the second term becomes

−ιx2,x1x^−B₀ ^ρ(α,β)

x1−x2

x₀

−Bρ(α,β)

x^−B₁ ^ρ(α,γ)x^−B₂ ^ρ(β,γ)x⁻¹₀ δ

x2−x1

−x₀

Y₃(b, x2)Y₄(a, x1)c

=−ι_x2,x1x^−B₀ ^ρ(α,β) e^−πi(x2−x1) x0

!−B_ρ(α,β)

x^−B₁ ^ρ(α,γ)x^−B₂ ^ρ(β,γ)x⁻¹₀ δ

x2−x1

−x₀

Y₃(b, x₂)Y₄(a, x₁)c

=−ι_x2,x1x^−B₀ ^ρ(α,β)

x₂−x₁ e^πix0

−Bρ(α,β)

x^−B₁ ^ρ(α,γ)x^−B₂ ^ρ(β,γ)x⁻¹₀ δ

x₂−x₁

−x₀

Y₃(b, x2)Y₄(a, x1)c,

where, passing from the first to the second line, we have used the convention in Re-mark 3.2.6. Cancellingx^−B₀ ^ρ(α,β)x^−B₁ ^ρ(α,γ)x^−B₂ ^ρ(β,γ)from all three terms we finally obtain

ιx1,x2x⁻¹₀

x₁−x₂ x0

−Bρ(α,β)

δ

x₁−x₂ x0

Y₁(a, x1)Y₂(b, x2)c

−ιx2,x1x⁻¹₀

x2−x1

e^πix0

−Bρ(α,β)

δ

x2−x1

−x₀

Y₃(b, x2)Y₄(a, x1)c

=ιx1,x0x⁻¹₂

x1−x0

x2

+Bρ(α,γ)

δ

x1−x0

x2

Y₅(Y₆(a, x0)b, x2)c.

This already resembles the generalised Jacobi identity for abelian intertwining algebras (see Definition 3.1.4).

Recall that in the situation of group-like fusion (AssumptionSN) we choseV-module representativesW^α forα∈FV withW⁰ =V. Now we also choose representatives of the one-dimensional spacesV_W^W_α^α+β_Wβ of intertwining operators of type _W^Wαα+β_W^β,α, β∈F_V. Definition 3.2.8(System of Scalars). Assume thatV satisfies AssumptionSN. We fix a choice of intertwining operatorY_W⁺_α,Wβ ∈ V_W^W_α^α+β_Wβ for eachα, β ∈FV. Each intertwining operatorY_k,k= 1, . . . ,6 in the discussion above is a scalar multiple of one of the fixed ones and we immediately obtain the following derived generalised Jacobi identity (cf.

Definition 3.1.4):

ι_x1,x2x⁻¹₀

x₁−x₂ x₀

−Bρ(α,β)

δ

x₁−x₂ x₀

Y⁺

W^α,W^β+γ(a, x₁)Y⁺

W^β,W^γ(b, x₂)c

−B(α, β, γ)ι_x2,x1x⁻¹₀

x₂−x₁ e^πix₀

−Bρ(α,β)

δ

x₂−x₁

−x₀

Y⁺

W^β,W^α+γ(b, x₂)Y_W⁺α,W^γ(a, x₁)c

=F(α, β, γ)ι_x1,x0x⁻¹₂

x₁−x₀ x₂

Bρ(α,γ)

δ

x₁−x₀ x₂

Y⁺

W^α+β,W^γ(Y⁺

W^α,W^β(a, x₀)b, x₂)c for some system of non-zero scalar factorsF, B:F_V ×F_V ×F_V →C^×.

We can immediately also define the function Ω: FV ×FV →C^×: Definition 3.2.9 (Braiding Convention). For allα, β ∈F_V we define Y^e+

W^α,W^β by Ye_W⁺_α,Wβ(a, x)b= e^xL−1Y_W⁺_β,Wα(b,e^−πix)a.

Then by [HL95a], Section 7,Y^e_W⁺_α,Wβ ∈ V_W^Wα^α+βW^β and we define Ω: FV ×FV →C^× by Y_W⁺_α,Wβ(a, x)b=Ω(^e α, β)Y^e_W⁺_α,Wβ(a, x)b

whereΩ(^e α, β) = Ω(β, α)⁻¹ as in Remark 3.1.2.

In the following we show that F, B and Ω satisfy the properties required to endow A = ^Lγ∈F_V W^γ with the structure of an abelian intertwining algebra whose vertex operation consists of the intertwining operators chosen in Definition 3.2.8. We begin by showing that F and Ω form an abelian 3-cocycle:

Proposition 3.2.10(Huang, [Eke15], Proposition 4.20). LetV be as in AssumptionSN.

LetF be as in Definition 3.2.8 andΩas in Definition 3.2.9. Then(F,Ω)is an abelian 3-cocycle. If we chooseY_V,W⁺ α to be theV-module action ofV onW^αandY_W⁺α,V :=Y^e_V,W⁺ α, then(F,Ω) is normalised.

Proof. The relation F(0, β, γ) = 1 follows from the definition of the Jacobi identity for intertwining operators (see Definition 1.6.1), the relation Ω(0, β) = 1 from our choice of Y_W⁺α,V. These two relations imply that (F,Ω) is normalised once we establish that it is a cocycle.

A proof of the cocycle condition is given by Huang in [Hua08b], Section 1, using the Huang-Lepowsky tensor-product theory.

The above proposition also implies that Q_Ω associated with Ω as in equation (3.1) is a quadratic form.

Lemma 3.2.11([Eke15], Lemma 4.23). Let V be as in Assumption SN,Ω as in Defin-ition 3.2.9 and let B_Ω and B_ρ be the bilinear forms associated with the quadratic forms QΩ andQρ, respectively. Then

B_Ω =−B_ρ.

Proof. Proposition 3.2.10 states that (F,Ω) is a normalised abelian 3-cocycle. It is straightforward, though tedious, to derive

Ω(α+β, α+β) = Ω(α, α)Ω(α, β)Ω(β, α)Ω(β, β) from Definition 3.1.1. Hence

b_Ω(α, β) = e^{(2πi)(QΩ(α+β)−QΩ(α)−QΩ(β))}= Ω(α, β)Ω(β, α).

On the other hand, applying Definition 3.2.9 twice gives Y⁺

W^α,W^β(a, x)b= Ω(β, α)⁻¹e^xL−1Y⁺

W^β,W^α(b,e^−πix)a

= Ω(β, α)⁻¹Ω(α, β)⁻¹Y⁺

W^α,W^β(a,e^−2πix)b

= Ω(β, α)⁻¹Ω(α, β)⁻¹e^{−(2πi)Bρ(α,β)}Y_W⁺_α,Wβ(a, x)b, so thatBΩ(α, β) =−B_ρ(α, β) for all α, β∈FV.

As a last step we show that F and B are related via Ω in the desired way.

Lemma 3.2.12 ([Eke15], Lemma 4.26). Let V be as in Assumption SN and let F and B be as in Definition 3.2.8 and Ω as in Definition 3.2.9. Then

B(α, β, γ) =F(β, α, γ)⁻¹Ω(α, β)F(α, β, γ) for all α, β, γ∈FV

Proof. In the following manipulations, for reasons of readability, we omit F, B,Ω and most of the subscriptsW^α, W^β, W^γforY⁺(·, x) and only reinsert them at the very end.

We begin with the generalised Jacobi identity in Definition 3.2.8:

ι_x1,x2x⁻¹₀

x₁−x₂ x₀

−Bρ(α,β)

δ

x₁−x₂ x₀

Y⁺(a, x₁)Y⁺(b, x₂)c

−ι_x2,x1x⁻¹₀

x₂−x₁ e^πix₀

−Bρ(α,β)

δ

x₂−x₁

−x₀

Y⁺(b, x₂)Y⁺(a, x₁)c

=ι_x1,x0x⁻¹₂

x₁−x₀ x₂

Bρ(α,γ)

δ

x₁−x₀ x₂

Y⁺(Y⁺(a, x₀)b, x₂)c

=ι_x1,x0x⁻¹₂

x₁−x₀ x2

Bρ(α,γ)

δ

x₁−x₀ x2

Y⁺(e^x0L−1Y⁺

W^β,W^α(b,e^−πix₀)a, x₂)c

=ι_x1,x0x⁻¹₂

x₁−x₀ x2

Bρ(α,γ)

δ

x₁−x₀ x2

Y⁺(Y⁺

W^β,W^α(b,e^−πix₀)a, x₂+x₀)c, using the translation axiom and Taylor expansion in the last step. Now

ιx1,x0x⁻¹₂

x₁−x₀ x2

Bρ(α,γ)

δ

x₁−x₀ x2

(x2+x0)ⁿ

=ιx2,x0x⁻¹₁

x2+x0

x1

−Bρ(α,γ)

δ

x2+x0

x1

(x2+x0)ⁿ

=ιx2,x0x⁻¹₁

x2+x0

x₁

Bρ(β,γ)

δ

x2+x0

x₁

xⁿ₁

for anyn∈Bρ(α+β, γ). Here we used item (3) of Lemma 3.2.7, the bilinearity ofBρand the following property of the delta function: δ(x/y)p(x) = δ(x/y)p(y) for a polynomial p with integral exponents. Using this we rewrite the expression above as

ιx2,x0x⁻¹₁

x₂+x₀ x1

Bρ(β,γ)

δ

x₂+x₀ x1

Y⁺(Y_W⁺_β,Wα(b,e^−πix0)a, x1)c.

Then we apply the derived generalised Jacobi identity from Definition 3.2.8 again:

ι_x2,tx⁻¹₁

x₂−t x₁

Bρ(β,γ)

δ

x₂−t x₁

Y⁺(Y⁺

W^β,W^α(b, t)a, x₁)c _t=e−πix

0

= ι_x2,x1t⁻¹

x₂−x₁ t

−Bρ(α,β)

δ

x₂−x₁ t

Y⁺(b, x₂)Y⁺(a, x₁)c

_t=e−πix0

−ιx1,x2t⁻¹

x1−x2

e^πit

−Bρ(α,β)

δ

x1−x2

−t

Y⁺(a, x1)Y⁺(b, x2)c

_t=e−πix0

=−ι_x2,x1x⁻¹₀ e^{−(2πi)Bρ(α,β)}x2−x1

e^πix0

−Bρ(α,β)

δ

x2−x1

−x₀

Y⁺(b, x2)Y⁺(a, x1)c +ιx1,x2x⁻¹₀

x1−x2

x₀

−Bρ(α,β)

δ

x1−x2

x₀

Y⁺(a, x1)Y⁺(b, x2)c.

Reinserting all the omitted scalar factors and comparing the coefficients we obtain 1 =F(α, β, γ)Ω(α, β)F^e (β, α, γ)⁻¹B(β, α, γ),

B(α, β, γ) =F(α, β, γ)Ω(^e α, β)F(β, α, γ)⁻¹b_ρ(α, β)⁻¹, which can be reorganised to read

B(α, β, γ) =F(β, α, γ)⁻¹Ω(α, β)F(α, β, γ),

B(α, β, γ) =F(α, β, γ)Ω(β, α)⁻¹F(β, α, γ)⁻¹b_ρ(α, β)⁻¹.

Hence, B satisfies the relation we aimed to prove and we also obtain Ω(α, β)Ω(β, α) = 1/b_ρ(α, β), which already follows from Lemma 3.2.11.

Finally, collecting all results, we can prove that the direct sum A of the irreducible V-modules up to isomorphism admits the structure of an abelian intertwining algebra.

We choose the vertex operation Y(·, x) onA to be composed of the intertwining oper-atorsY⁺

W^α,W^β(·, x) forα, β ∈F_V with the normalisation condition described in Propos-ition 3.2.10 fulfilled.

Proof of Theorem 3.2.1. With our choice of intertwining operators, they satisfy the gen-eralised Jacobi identity for an abelian intertwining algebra (see Definition 3.1.4) for some system of scalarsF andB(see Definition 3.2.8). We saw in Proposition 3.2.10 that (F,Ω) defines a normalised abelian 3-cocycle. It is necessary thatB be compatible withF and Ω, which is Lemma 3.2.12. For the grading condition on Y(·, x) to hold the bilinear forms associated withQ_Ω and Qρ have to be negatives of each other, which is shown in Lemma 3.2.11. The remaining axioms are easy to verify.

Im Dokument A Cyclic Orbifold Theory for Holomorphic Vertex Operator Algebras and Applications (Seite 74-83)