Simple-Current Extensions

Corollary 2.2.13. Let V be as in Assumptions SNP of central charge c ∈Z. Let FV

be the fusion group of V, a finite quadratic space by Theorem 2.2.7. Then the functions FW(τ) :=TW(v, τ)η(τ)^c

for W ∈ Irr(V) form a vector-valued modular form of weight k = wt[v] +c/2 for the Weil representation ρFV of Mp2(Z), i.e.

(cτ+d)^−kF_W(M.τ) = ^X

X∈Irr(V)

ρ_FV(M^f)W,XF_X(v, τ) for M = ^{a b}_{c d}∈SL2(Z).

In anI-graded extensionVI,V^α·V^β ⊆V^α+βby definition. This means that the vertex operation on V_I restricts to intertwining operators of type _W^Wαα+β_W^β, implying that the spaceV_W^Wα^α+β,W^β is at least one-dimensional. For anI-graded simple-current extension this clearly implies N_W^Wα^γ,W^β =δα+β,γ and hence

W^αV0 W^β ∼=W^α+β.

We summarise some properties of simple-current extensions. The following is essen-tially Proposition 1 in [LY08]:

Proposition 2.3.5([ABD04], [DM04b], Section 5, [Yam04], Lemma 2.6, [Lam01], The-orem 4.5). Let V⁰ be a simple, rational vertex operator algebra. Let V_I =^Lα∈IV^α be anI-graded simple-current extension of V⁰. Then:

(1) VI is rational.

(2) If V⁰ is C₂-cofinite and of CFT-type, then also V_I is C₂-cofinite.

(3) (Uniqueness) IfVˆ_I =^Lα∈IVˆ^α is another simple-current extension ofV⁰ such that Vˆ^α∼=V^α for all α∈I, then V_I andVˆ_I are isomorphic vertex operator algebras.

Item (3) shows that the vertex operator algebra structure of the simple-current exten-sion only depends on the isomorphism classes of the simple-currentV⁰-modules.

Modules for Simple-Current Extensions

In the following we will review the representation theory of simple-current extensions, which is developed in [Lam01, Yam04].

Proposition 2.3.6([SY03], Lemma 3.6). LetV_Ibe anI-graded simple-current extension of the simple and rational vertex operator algebra V⁰. Let X be a V_I-module. LetW be an irreducible V⁰-submodule of X. Then all

V^α·W = span_C{v_nw|v∈V^α, w∈W, n∈Z}, α∈I, are also irreducible V⁰-submodules ofX.

LetXbe an irreducibleVI-module. SinceV⁰ is rational, there is always an irreducible V⁰-submodule W of X. SinceX is irreducible, we get

X =VI·W =^M

α∈I

V^α·W.

In view of the above proposition we define

I_W :={α∈I |V^α·W ∼=W},

which is a subgroup ofI since bothV^α·(V^β·W) andV^α+β·W are irreducibleV⁰-modules by the previous proposition and by associativity they are isomorphic.

Definition 2.3.7 (I-Stability). A VI-module X is said to be I-stable if IW = {0} for some irreducibleV⁰-submodule W of X.

Clearly, ifIW ={0}for some irreducibleV⁰-submoduleW ofX, then the same is true for all V⁰-submodules of X. We will see that I-stable V_I-modules behave very nicely.

Proposition 2.3.8 ([SY03], Proposition 3.8). Let VI be an I-graded simple-current extension of the simple, rational vertex operator algebraV⁰. Then the structure of every I-stable, irreducible V_I-module is completely determined by its V⁰-module structure.

The statement of the proposition also follows from the more general results Theorems 2.8 and 2.14 in [Yam04].

Proposition 2.3.9 ([Yam04], Lemma 2.16, [SY03], Lemma 3.12). Let V₀ be a simple, rational, C2-cofinite vertex operator algebra of CFT-type and let VI be an I-graded simple-current extension ofV⁰. LetX¹, X², X³be irreducibleI-stableV_I-modules and let W¹, W², W³ be V⁰-submodules of X¹, X², X³, respectively. Then there is the following isomorphism of spaces of intertwining operators

X³ X¹X²

!

VI

∼=^M

α∈I

V^αV⁰ W³ W¹W²

!

V⁰

.

Let VI be an I-graded simple-current extension of V0. We consider the group ˆI of characters χ: I → C^×. Then the elements χ of ˆI define automorphisms of V_I which leaveV⁰ pointwise invariant, i.e.V⁰ ⊆(V_I)^χfor all χ∈Iˆwhere (V_I)^χ denotes the fixed points of VI underχ. In fact the converse is also true:

Proposition 2.3.10 ([Yam04], Lemma 2.18). An automorphism χ ∈ Aut(V) satisfies V⁰ ⊆(VI)^χ if and only if χ∈Iˆ.

Let W be an irreducibleV⁰-module. We consider the function χW:I →C^×, χ_W(α) := e^{(2πi)(ρ(VαW)−ρ(W))}.

That this is a character, i.e. χ_W ∈ Iˆ, is shown under suitable assumptions in [Yam04], Lemma 3.1.

Theorem 2.3.11 ([Yam04], Theorems 3.2 and 3.3). Let V₀ be a simple, rational, C₂ -cofinite vertex operator algebra of CFT-type and let V_I be an I-graded simple-current extension of V⁰. Then every irreducible V⁰-module W is contained in an irreducible χ_W-twisted V_I-module whereχ_W is the character defined above.

Remark 2.3.12. Note that in [Yam04] the characterχW is defined with opposite sign.

It has to be chosen such that it is in agreement with the sign convention in the definition of twisted modules (see Remark 1.10.1).

Simple-Current Extensions for Simple-Current Vertex Operator Algebras

For a simple-current extension V_I = ^Lα∈IV^α the irreducible V₀-modules V^α, α ∈ I, are all simple currents. However, this does not mean thatall irreducibleV_I-modules are simple currents. Indeed, given a simple-current extension VI there is a subsetI of the isomorphism classes of irreducibleV⁰-modules such that all these modules areZ-graded, simple currents andI is closed under fusion.

In the following we want to study the case where all irreducibleV⁰-modules are simple currents. Let V⁰ be a vertex operator algebra satisfying Assumption SN. Then the conformal weights define a quadratic form Q_ρ on the fusion group F_V. Note that by Remark 2.2.8, AssumptionPis not needed for the existence of the quadratic form, only for its non-degeneracy.

Clearly, any I-graded extension V_I of V⁰ will have to exist on a direct sum V_I = L

α∈IW^α for some isotropic subgroup I ofFV and this extension will automatically be a simple-current extension.

In the following we study the representation theory of such an extensionV_I and give a classification of the irreducibleV_I-modules.

Proposition 2.3.13. Let V⁰ fulfil Assumption SN. Assume that for some isotropic subgroup I ≤ FV of the fusion group the direct sum VI = ^Lα∈IW^α is an I-graded simple-current extension of V⁰. Then any irreducible V_I-moduleX is of the form

X∼=^M

α∈I

W^α+γ=:X^γ+I for some γ ∈FV.

Proof. AsV⁰-module,Xis a direct sum of irreducibleV⁰-modules. SinceX is non-zero, let us assume that X containsW^γ for someγ ∈F_V. Then W^α·W^γ⊆W^α+γ has to be in X for any α ∈ I and hence any W^α+γ is contained in the decomposition of X into irreducibleV0-modules. This shows that

X ⊇^M

α∈I

W^α+γ

up to isomorphism. Both sides are irreducibleV_I-modules and hence we get equality up to isomorphism.

On the other hand, one can ask the question which objects of the form X^γ+I = L

α∈IW^α+γ for someγ ∈F_V are irreducible V_I-modules. The answer is given by The-orem 2.3.11.

Theorem 2.3.14. Let V⁰ fulfil Assumption SN and let V_I be as in Proposition 2.3.13.

Then the irreducible (untwisted) VI-modules are up to isomorphism exactly given by X^γ+I =^M

α∈I

W^α+γ

for γ ∈I^⊥, i.e. the irreducible V_I-modules are indexed by I^⊥/I.

Proof. The above proposition gives all possible candidates for the irreducibleVI-modules.

Consider X^γ+I = ^Lα∈IW^α+γ for some γ ∈ F_V. Then X^γ+I contains the irreducible V⁰-module W^γ. Theorem 2.3.11 on the other hand states that the V₀-module W^γ is contained in an irreducible, possibly twisted VI-module, which can only be X^γ+I by the above proposition since no other of theX^β+I forβ+I ∈F_V/I contains W^γ. This V_I-module is twisted by the character ofI

χW^γ(α) = e^{(2πi)(ρ(WαWγ)−ρ(Wγ))}

= e^{(2πi)(Qρ(α+γ)−Qρ(γ))}

= e^{(2πi)(Qρ(α+γ)−Qρ(γ)−Qρ(α))}

= e^{(2πi)Bρ(α,γ)},

α∈I, and is untwisted if and only if Bρ(α, γ) = 0 +Z for allα∈I, i.e. γ ∈I^⊥.

We remark that the irreducible V_I-modules X^γ+I, γ +I ∈ I^⊥/I, have conformal weights in Qρ(α) for some α in γ +I. This makes sense since γ ∈ I^⊥ and hence Q_ρ(α) =Q_ρ(γ) for all α∈γ+I.

We can also calculate the fusion rules of the irreducible V_I-modules Irr(VI) =ⁿX^α+I α+I ∈I^⊥/I^o.

The fusion group will turn out to beI^⊥/I, as expected.

Lemma 2.3.15. Let V⁰ fulfil Assumption SN and let V_I be as in Proposition 2.3.13.

Then every irreducibleVI-module X^α+I,α+I ∈I^⊥/I, is I-stable.

Proof. We determine theβ∈I for whichW^β·W^α ∼=W^βV⁰W^α∼=W^β+αis isomorphic to W^α. This is clearly only the case for β = 0 and hence X^α+I is an I-stable VI -module.

Theorem 2.3.16. Let V⁰ fulfil Assumption SN and let V_I be as in Proposition 2.3.13.

Then

X^α+IVI X^β+I∼=X^α+β+I

for allα+I, β+I ∈I^⊥/I. In particular, V_I is a simple-current vertex operator algebra, i.e. all irreducible VI-modules are simple currents.

Proof. By Proposition 2.3.9, X^γ+I X^α+IX^β+I

!

VI

∼=^M

δ∈I

W^δV⁰ W^γ W^αW^β

!

V⁰

=^M

δ∈I

W^δ+γ W^αW^β

!

V⁰

. Then

N_α+I,β+I^γ+I :=N_X^Xα+I^γ+I,X^β+I =^X

δ∈I

δα+β,γ+δ=δα+β+I,γ+I.

Finally the following proposition holds:

Proposition 2.3.17. LetV⁰ fulfil AssumptionSNand letV_Ibe as in Proposition 2.3.13.

Then (X^γ+I)⁰ ∼=X^−γ+I

for γ+I ∈I^⊥/I. In particularV_I is self-contragredient.

Proof. Consider (X^γ+I)⁰ = ^M

α∈I

W^α+γ

!0

∼=^M

α∈I

(W^α+γ)⁰∼=^M

α∈I

W^−α−γ =^M

α∈I

W^α−γ=X^−γ+I. Only the second step is non-trivial. But this follows directly from the definition of the contragredient module.

In total, we have shown:

Corollary 2.3.18. Let V⁰ fulfil Assumption SN and letV_I be as in Proposition 2.3.13.

Then the fusion group of VI is FVI =I^⊥/I.

Im Dokument A Cyclic Orbifold Theory for Holomorphic Vertex Operator Algebras and Applications (Seite 63-69)