Orbifold Construction

In the following we combine our knowledge from this chapter about the fusion algebra of V^G under Assumption O with the results in Chapter 3 about abelian intertwining algebras in the case of group-like fusion (under Assumptions SNP) to construct a new

holomorphic vertex operator algebra V^e from the irreducible V^G-modules. This process is called orbifolding ororbifold construction.

Having determined the fusion algebra of V^G, we can use Theorem 3.2.3 to endow the direct sum of all irreducible V^G-modules up to isomorphism with the structure of an abelian intertwining algebra. Under AssumptionOPthe assumptions of Theorem 3.2.3 are fulfilled because of Theorem 4.1.5.

Theorem 4.9.1. Let V and G=hσi be as in Assumption OP and σ of type n{r}. Then the direct sum

A:= ^M

i,j∈Zn

W^(i,j)= ^M

γ∈FV G

W^γ

of alln² irreducibleV^G-modules up to isomorphism can be given the structure of an abelian intertwining algebra, the unique one up to a normalised abelian 3-coboundary extending the vertex operator algebra structure ofV and that of its irreducible mod-ules, with associated finite quadratic space F_VG = (F_VG,−Q_ρ).

By Theorem 3.5.1 we get a vertex operator algebra structure if we restrict to the modules corresponding to an isotropic subgroup of the finite quadratic spaceF_V^G.

Theorem 4.9.2. Let V and G=hσi be as in Assumption OP and σ of type n{r}. Let I be an isotropic subgroup of the fusion groupF_V^G. Then the direct sum

V_I =^M

γ∈I

W^γ

admits the unique structure of a vertex operator algebra extending the vertex operator algebra structure of V^G and that of its irreducible modules. V_I satisfies Assump-tions SNP.

If I =I^⊥, then V^e :=V_I is holomorphic.

We call the holomorphic vertex operator algebra V^e theorbifold ofV.³

By Proposition A.1.9 any subgroupI ofF_V^GwithI =I^⊥fulfils|I|=nand is maximal isotropic by Remark A.1.11, item (2).

Maximal Isotropic Subgroups

In order to apply the above theorem to construct new holomorphic vertex operator algebras V^e we have to find isotropic subgroupsI of the fusion groupF_V^G with I =I^⊥.

3Other authors use the term “orbifold” to refer to the fixed-point vertex operator subalgebraV^G.

Clearly, the group I0 := {(0, j) |j ∈ Zn} is isotropic with I0 = I₀^⊥ and gives back the original holomorphic vertex operator algebra V, namely

M

γ∈I₀

W^γ= ^M

j∈Zn

W^(0,j) = ^M

j∈Zn

V^j =V.

To obtain a new holomorphic vertex operator algebra, the isotropic subgroup I has to be chosen such that I 6=I0. In fact, I should have trivial intersection withI0. Indeed, in the following we will show that wheneverI andI₀ have non-trivial intersection, then the new vertex operator algebraV^e can be obtained as an orbifold of some smaller order automorphism with trivial intersection of the corresponding subgroups.

Lemma 4.9.3. Let V and G= hσi be as in Assumption OP and let I be an isotropic subgroup of F_VG with I = I^⊥ and intersection H := I ∩I₀ 6= {0} with I₀. Let V^e = L

γ∈IW^γ be the orbifold of V associated with the subgroupI. Then:

(1) |H|=:m divides nand ρ:=σ^m is an automorphism ofV of order n⁰=n/m.

(2) The n⁰² =|H^⊥/H| irreducible V^hρi-modules admit the structure of an abelian in-tertwining algebra with associated quadratic spaceH^⊥/H.

(3) The vertex operator algebras V^e and V are isomorphic to the direct sums of the irreducibleV^hρi-modules corresponding to I/H and I0/H, respectively.

(4) I/H∩I₀/H ={0}.

Proof. H is a subgroup of the cyclic groupI₀ ∼=Zn and theV^G-modules corresponding to the elements in I0 are the eigenspaces of σ in V. Hence, H is cyclic of some order m := |H|, i.e. H = (n/m)I0, and VH = ^Lγ∈HW^γ is the fixed-point vertex operator subalgebra ofV under the automorphismρ=σ^mofV of ordern⁰ =n/m, i.e.VH ∼=V^hρi. By Theorem 3.5.1 the n⁰² =|H^⊥/H| irreducibleV^hρi-modules are given by X^α+H = L

γ∈α+HW^γ for α+H ∈H^⊥/H, i.e. they are indexed by the elements of H^⊥/H. The direct sum of modules^Lµ+H∈H^⊥/HX^µ+H admits the structure of an abelian intertwin-ing algebra with associated quadratic space H^⊥/H.

The vertex operator algebra V^e can be obtained by an orbifold construction as V^e = L

µ+H∈I/HX^µ+H =^Lγ∈IW^γ, i.e. as a direct sum of the irreducible V^hρi-modules cor-responding to the isotropic subgroupI/H of H^⊥/H.

The above lemma shows that any holomorphic vertex operator algebra V^e obtained as orbifold from Theorem 4.9.2 can be obtained as an orbifold of possibly smaller order whereI and I₀ intersect trivially. On the other hand, in the next lemma we show that an isotropic subgroup I with I^⊥ = I and trivial intersection with I0 can only exist in the case ofr = 0, i.e. whenσ is of type n{0}.

Lemma 4.9.4. Let V and G= hσi be as in Assumption OP and σ of type n{r}. Let I be an isotropic subgroup of F_V^G with I^⊥ = I and I ∩I0 = {0}. Then r = 0 and if the trace functions, i.e. the representations φi, are rescaled as in Corollary 4.7.11, then F_VG=E_c0 =Zn×Zn and I =Zn× {0} while I₀ ={0} ×Zn.

Proof. There are two isotropic subgroupsI andI0 with trivial intersection. This implies F_VG={0}^⊥= (I₀∩I)^⊥=I₀^⊥+I^⊥=I₀+I, i.e.I andI₀generate the fusion groupF_VG. Both together means thatF_VG is isomorphic to a semidirect product ofI and I₀. Since I, I0 and F_V^G are abelian, this semidirect product is in fact direct and F_V^G ∼= I×I0. It is easy to see that the only r ∈ Zn for which the finite quadratic space F_VG ∼=Ec2r

is isomorphic to the direct product of the two isotropic groups I and I₀ of order n, of which I0 is isomorphic to Zn, is r = 0. In this case, if the representations φi are chosen as in Corollary 4.7.11,F_VG =Ec0 =Zn×Zn,I0 ={0} ×Zn and I can only be I =Zn× {0}.

Together both lemmata imply:

Proposition 4.9.5. To construct new holomorphic vertex operator algebras V^e using the orbifold construction in Theorem 4.9.2 it suffices to consider σ of type n{0} so that the fusion group is F_VG ∼= Zn×Zn and to choose I ∼= Zn× {0} under this isomorphism, with equality if the representations φi are chosen as in Corollary 4.7.11.

Special Case: Type n{0}

By the above proposition we only need to consider the typen{0}. The fusion algebra in this special case is described in Corollary 4.7.16. Let us assume that representationsφ_i are chosen as in Corollary 4.7.11. Then the direct sum of irreducibleV^G-modules

A= ^M

(i,j)∈Zn×Zn

W^(i,j)

admits the structure of an abelian intertwining algebra with associated quadratic space (Zn×Zn,−Q_ρ) and the direct sum of irreducible V^G-modules

Ve := ^M

i∈Zⁿ

W^(i,0) (4.6)

corresponding to the isotropic subgroup Zn× {0} of Zn×Zn admits the structure of a holomorphic vertex operator algebra satisfying Assumptions SNPand extending the vertex operator algebra V^G. V^e is aZn-graded simple-current extension ofV^G.

Inverse Orbifold

We continue in above setting, i.e. let V and G=hσi satisfy AssumptionOPwith σ of type n{0}. We will show that the orbifolding process V → Ve can be reverted, i.e. we can find an automorphism σ on V^e such that orbifolding with K = h σ i gives back the original vertex operator algebraV.

Theorem 4.9.6. Let V andG=hσi be as in Assumption OP with σ of type n{0} and the representations φi chosen as in Corollary 4.7.11 and let V^e = ^Li∈ZnW^(i,0) be the orbifold vertex operator algebra. Then:

(1) The operator σ defined on V^e by σ v =ξⁱ_nv for v ∈ W^(i,0) is an automorphism of the vertex operator algebra V^e of type n{0}.

(2) The unique irreducible σ ⁱ-twistedV^e-module is given up to isomorphism byV^e( σ ⁱ)∼= L

j∈ZnW^(j,i), i∈Zn.

(3) The orbifold construction forV^e andK =h σ i yields^Li∈ZnW^(0,i)∼=V. The situation is shown in the following table:

Ve V^e( σ ¹) · · · Ve( σ ⁿ⁻¹)

V W^(0,0) W^(0,1) · · · W^(0,n−1)

V(σ¹) W^(1,0) W^(1,1) · · · W^(1,n−1)

... ... ... ... ...

V(σⁿ⁻¹) W^(n−1,0) W^(n−1,1) · · · W^{(n−1,n−1)}

Proof. The automorphism σ of V of order n is by definition an automorphism of the vector space V fixing the vacuum and the Virasoro vector and fulfilling

σYV(v, x)σ⁻¹ =YV(σv, x) for all v∈V. We can decompose V as

V = ^M

j∈Zn

W^(0,j), whereσ acts onW^(0,j) by multiplication withξ_n^j.

In analogy, we define a vector-space automorphism σ on the orbifolded vertex operator algebra

Ve = ^M

i∈Zn

W^(i,0)

by setting σ v = ξⁱ_nv for v ∈ W^(i,0). By construction, this fixes the vacuum and the Virasoro vectors, which lie in W^(0,0). So, for σ to be an automorphism of the vertex operator algebraV^e, the vertex operator Y

Ve(·, x) on V^e has to fulfil σ Y

Ve(v, x) σ ⁻¹ =Y

Ve( σ v, x) (4.7)

for allv∈Ve. Indeed, sinceV^e is a Zn-graded extension ofV^G,Y

Ve(v, x)w∈W^(i+i0,0){x}

forv∈W^(i,0) and w∈W^(i0,0). Then the left-hand side of the above equation acting on w is given by

σ Y

Ve(v, x) σ ⁻¹w=ξ_n⁻ⁱ⁰ σ Y

Ve(v, x)w=ξ_n⁻ⁱ⁰ξ_nⁱ⁺ⁱ⁰Y

Ve(v, x)w=ξⁱ_nY

Ve(v, x)w.

On the other hand, for the right-hand side we obtain Y

Ve( σ v, x)w=Y

Ve(ξⁱ_nv, x)w=ξⁱ_nY

Ve(v, x)w

and hence (4.7) is fulfilled and we conclude that σ is a vertex operator algebra auto-morphism ofV^e of ordern.

We saw that V^e is holomorphic, i.e. has exactly one irreducible module up to iso-morphism, namely the adjoint module V^e. Then, by Theorem 1.10.6,V^e has exactly one irreducible σ ^j-twisted module V^e( σ ^j) up to isomorphism for eachj∈Zn.

Let X ∼=V^e( σ ^j0) be such an irreducible σ ^j0-twisted module ofV^e. X is an untwisted Ve^K = W^(0,0) = V^G-module and hence a direct sum of some of the modules W^(i,j), i, j ∈Zn. By the definition of twisted modules the exponents of the formal variable in the vertex operation ofV^e onX should lie inj₀k/n+Zif we restrict toW^(k,0) ⊆Ve. On the other hand, the intertwining operators of type _W^{W(k,0)(i,k+j)}_W^(i,j) have exponents of the formal variable in kj/n+Z. Hence X can only consist of the modules W^(i,j0), i∈Zn. Since X is non-empty, we can assume that X contains W^(i0,j0) for some fixed i₀. But then it containsV^e·W^(i0,j0)=^Li∈ZⁿW^(i,0)·W^(i0,j0)=^Li∈ZⁿW^(i+i0,j0)=^Li∈ZⁿW^(i,j0). Using the irreducibility ofX we conclude that X ∼=^Li∈ZⁿW^(i,j0).

The last item follows immediately, which completes the proof.

Im Dokument A Cyclic Orbifold Theory for Holomorphic Vertex Operator Algebras and Applications (Seite 130-135)