Modular Tensor Categories

is defined analogously. A monoidal category is called rigid (or autonomous) if it has a right and a left duality.

A ribbon category (or tortile category) is a right rigid, balanced monoidal category, i.e. a monoidal category with a braiding, a twist and a right duality such that the twist and the right duality are compatible in a certain way. Ribbon categories have a number of well-known properties. For instance they come automatically equipped with a left duality by means of the definitions^∗A:=A^∗ and

˜b_A:= (θ_A^∗⊗idA)◦c_A,A^∗◦b_A and ˜d_A:=d_A◦c_A,A^∗(θ_A⊗idA^∗)

for A ∈ C. This means that ribbon categories are rigid. In fact, ribbon categories are examples of sovereign categories, where there is a natural isomorphism ^∗A → A^∗ for A ∈ C. Ribbon categories are also pivotal categories, a concept essentially equivalent to the aforementioned sovereign categories. A pivotal category is a right rigid monoidal category equipped with a natural isomorphism

ψ_A:A→A^∗∗

forA∈ C that is monoidal, i.e. compatible in a certain way with the monoidal structure.

Via the braiding it is possible to define another natural morphism uA:A→A^∗∗ as A ^{(idA⊗bA∗)◦r}

−1

−−−−−−−−−−→A A⊗(A^∗⊗A^∗∗) ^α

−1 A,A∗,A∗∗

−−−−−−→(A⊗A^∗)⊗A^∗∗

cA,A∗⊗id_A∗∗

−−−−−−−−→(A^∗⊗A)⊗A^∗∗−−−−−−−−−→^{la◦(dA⊗idA∗∗)} A^∗∗,

which is well known to be an isomorphism. In fact,ψAand uA are precisely related via the twist isomorphismθA, i.e.

ψ_A=u_Aθ_A.

In pivotal categories we define the left and right traces trL(f),trR(f)∈MorC(1_C,1_C) of morphisms f ∈MorC(A, A) via

trL(f) :1_C−−→^bA∗ A^∗⊗A^{∗∗ idA∗⊗ψ}

−1

−−−−−−−A→A^∗⊗A−−−−−→^idA∗⊗f A^∗⊗A−→^dA 1_C, trR(f) :1_C−→^bA A⊗A^∗−^f−−−−^⊗idA→^∗ A⊗A^∗−−−−−−→^ψA⊗idA∗ A^∗∗⊗A^∗ −−→^dA∗ 1_C. Alternatively, in a sovereign category we can express the traces as

trL(f) :1_C−→^˜bA A^∗⊗A−−−−−→^idA∗⊗f A^∗⊗A−→^dA 1_C, trR(f) :1_C−→^bA A⊗A^∗−^f−−−−^⊗idA→^∗ A⊗A^∗−→^d˜A 1_C. Via the traces we define the (categorical)dimensions

d_L(A) = trL(idA) and d_R(A) = trR(idA).

We will later, in particular in the setting of modular tensor categories, call thesequantum dimensions. The dimensions only depend on the isomorphism class of the object.

Ribbon categories are also spherical categories, pivotal categories in which left and right traces coincide, i.e.

trL(f) = trR(f) =: tr(f).

Then alsod_L(A) =d_R(A) =:d(A). For reasons of clarity we will also write trA(f) = tr(f) for a morphismf ∈MorC(A, A).

We will later need the following result from [DGNO10] (Proposition 2.32), which even holds in any ribbon category (see [CKL15], Corollary 2.5):

Proposition 3.3.1. Let C be a ribbon category. Then trA⊗A(c⁻¹_A,A) = trA(θ⁻¹_A ) for any object A∈ C.

Modular Tensor Categories

Given a monoidal category C, we can demand the following additional structure: let C be an abelian,C-linear, i.e. enriched over VectC, and semisimple monoidal category such that there are only finitely many isomorphism classes of simple objects and the tensor unit1_C is simple.

An abelian category is called semisimple if every object is semisimple, i.e. a direct sum of simple objects. The C-linearity means that HomC(A, B) := MorC(A, B) is aC-vector space and the composition of morphisms isC-bilinear.

Let us also assume that the spaces HomC(A, B) are all finite-dimensional. Then Schur’s lemma holds, i.e.

HomC(A, B)∼= (

C ifA∼=B, 0 ifAB asC-vector spaces for simple objectsA, B ∈ C.

Definition 3.3.2 (Modular Tensor Category). A ribbon category C with these addi-tional properties and whose ribbon structure is compatible with the C-linear structure satisfying a certain non-degeneracy condition for the braiding, called modularity, is a modular tensor category.

From now on let C be a modular tensor category. Since we assumed the unit object 1_C ∈ C to be simple, the trace tr(f)∈EndC(1_C) is simply a complex multiple of id1C and we will view tr(f)∈Cin the following. In the same way, we view the quantum dimen-sionsd(A)∈C. For a simple objectA∈ Cthe twist isomorphismθA∈EndC(A) =CidA

can also be viewed as a non-zero complex number that only depends on the isomorphism class ofA.

SinceCis semisimple, given two simple objectsA, B∈ C, we can decompose the tensor product as

A⊗B ∼=^M

C

N_A,B^C C

where the direct sum runs over the finitely many isomorphism classes of simple objects inC and the numbers

N_A,B^C = dimC(HomC(C, A⊗B))∈Z≥0

are calledfusion rules and only depend on the isomorphism classes ofA,B and C. This is of course completely analogous to the definition of the fusion rules for vertex operator algebras (see Section 1.6).

Frobenius-Schur Indicator

The Frobenius-Schur indicator is defined in [FS03] for any sovereign category and special-ises to the Frobenius-Schur indicator in a modular tensor category defined in [FFFS02].

First note that for a simple objectA in a modular tensor category

1 = dim_C(HomC(A, A)) = dim_C(HomC(1_C, A⊗A^∗)) = dim_C(HomC(A⊗A^∗,1_C)) and analogously with A^∗ andA interchanged.

We call an objectA∈ C self-dual ifA∼=A^∗.

Definition 3.3.3 (Frobenius-Schur Indicator). Let C be a modular tensor category and A a simple, self-dual object with isomorphism φ_A: A → A^∗. Then the space HomC(1_C, A^∗⊗A) is one-dimensional and the factor of proportionalityν(A) in

˜bA=ν(A)(φA⊗φ⁻¹_A )◦bA∈HomC(1_C, A^∗⊗A) is called theFrobenius-Schur indicator ofA.

Easy consequences of the definition areν(A)²= 1, i.e. ν(A) =±1,ν(1_C) = 1 and the fact thatν(A) only depends on the isomorphism class ofA (see [FFFS02], Lemma 2.1).

We can equivalently define the Frobenius-Schur indicator as

d˜A=ν(A)dA◦(φA⊗φ⁻¹_A )∈HomC(A⊗A^∗,1_C) or, more symmetrically, as

d(A) =ν(A)dA◦(φA⊗φ⁻¹_A )◦bA,

involving the quantum dimensiond(A) ofAand where both sides are non-zero multiples of id1C ∈EndC(1_C) =Cid1C since the unit object 1_C is simple.

For a simple object inA∈ C which is not self-dual we setν(A) := 0.

There is a remarkable formula due to Bántay, expressing the Frobenius-Schur indicator in terms of the quantum dimensions, the fusion rules and the twist isomorphism.

Proposition 3.3.4 (Bántay’s Formula, [Bá97], formula (1)). Let C be a simple, not necessarily self-dual object in a modular tensor category C. Then the Frobenius-Schur indicator is given by

ν(C) = 1 D²

X

A,B

θ_A²

θ²_BN_A,C^B d(A)d(B)∈ {0,±1}

with

D²= ^X

A,B

θ_A

θBd(A)²d(B)²

where in each case the sum ranges over the finitely many isomorphism classes of simple objects in C.

A proof of this statement can be found in [NS07], Theorem 7.5 and [Wan10], The-orem 4.25.

Huang’s Modular Tensor Category

In the following we describe Huang’s construction of a modular tensor category associ-ated with a suitably regular vertex operator algebra.

Theorem 3.3.5 ([Hua08a], Theorem 4.6). Let V be a vertex operator algebra satisfy-ing Assumption N. Then the category of V-modules naturally admits the structure of a modular tensor category.

The proof of the theorem, in particular showing rigidity, makes use of the Verlinde formula for vertex operator algebras, also proved by Huang [Hua08b] (see Theorem 1.9.2).

In the following we describe some elements of Huang’s construction in more detail.

Given a vertex operator algebraV, letC be the abelian,C-linear category ofV-modules and module maps. Assume thatV satisfies AssumptionN. The simple objects in C are the irreducible V-modules and the rationality of V implies the semisimplicity ofC and that there are only finitely many isomorphism classes of simple objects.

Recall that for each z ∈ C^× there is the Huang-Lepowsky tensor product (or fusion product) P(z). The tensor-product bifunctor ⊗ on C is taken to be the P(1)-tensor product P(1), simply denoted by . The unit object 1_C for the tensor product is the vertex operator algebra V itself.

When passing from a formal power series inxwith non-integer exponents to a function in a complex variable z, it is important to have a consistent choice of the logarithm.

Recall that we chose the branch of log(z) such that 0 ≤ Im(log(z)) < 2π. Also recall from Section 1.1 that given an intertwining operatorY(·, x) we use the convention

Y(·, z) =Y(·, x)|_xn=e^nlog(z), n∈C

for z ∈ C^×. Note that Y(·, z) is a P(z)-intertwining map (see [HL95b], Section 12 for the correspondence between intertwining operators and P(z)-intertwining maps). We also use the (conflicting) shorthand notation

Y(·,e^ζ) =Y(·, x)|_xn=e^nζ, n∈C

forζ ∈C. In particular, if l(z) is some other branch of the logarithm, then, using this notation,Y(·, z)6=Y(·,e^l(z)) even thoughz= e^l(z).

There is a natural isomorphism of vector spaces from the space of module maps HomV(W¹ P(z)W², W³) to the space of intertwining operators of type _W^W1_W^{3 2} (cf.

Proposition 1.6.7). This isomorphism is based on the choice of the logarithm. Then there

is a canonical intertwining operator of type ^W1_W^P1^(z)W^2W2

corresponding to the identity module map on W¹ P(z)W². We denote this intertwining operator by Y_W^P(z)_1,W2(·, x).

Givenw₁ ∈W¹,w₂∈W² we define forz∈C^× theP(z)-tensor-product element w1P(z)w2:=Y_W^P(z)1,W²(w1, z)w2 ∈W¹P(z)W²

where for any V-module W = ^`n∈CWn = ^Ln∈CWn we denote by W = ^Qn∈CWn its algebraic completion. The homogeneous components of w₁ P(z)w₂ for all w₁ ∈ W¹, w2∈W² span the tensor-product moduleW¹P(z)W². However, the set of all tensor-product elements has in fact almost no intersection with the tensor-tensor-product module. A notable exception is1P(z)wforw∈W,W aV-module, which lies inVP(z)W. Note that1 denotes the vacuum vector inV and not the unit object 1C inC.

The left and right unit isomorphisms lW: V W → W and rW:W V → W are characterised by

l_W(1w) =w and r_W(w1) = e^L−1w

forw ∈W where r_W denotes the natural extension of r_W to the completion W V of W V.

In order to describe the braiding onCwe need the notion ofparallel transportproviding an isomorphism betweenP(z)-tensor products for differentz. GivenV-modulesW¹ and W²,z1, z2 ∈C^× and a path γ inC^× from z1 to z2, the parallel-transport isomorphism T_γ:W¹P(z1)W²→W¹P(z2)W² is defined via

T_γ(w₁P(z1)w₂) =Y_W^P₁^(z_,W²⁾₂(w₁,e^l(z1))w₂

wherel(z₁) is the value of the logarithm ofz₁ determined uniquely by log(z₂) satisfying 0≤Im(log(z))<2πand the pathγ. Here,T_γ denotes the natural extension ofT_γ to the completionW¹P(z1)W² ofW¹P(z1)W². Clearly, the parallel-transport isomorphism only depends on the homotopy class ofγ inC^×.

The braiding isomorphism c_W1,W²:W¹W² → W²W¹ can now be described as follows: let γ₁⁻ be a path from −1 to 1 in H\ {0}, the closed upper half-plane with 0 deleted, with the corresponding parallel-transport isomorphism T_γ−

1 :W²P(−1)W¹ → W²W¹. Then

c_W¹_,W²(w₁w₂) = e^L−1T_γ−

1 (w₂P(−1)w₁)

forw₁ ∈W¹,w₂ ∈ W². In terms of intertwining operators, or P(z)-intertwining maps to be precise, this equals

c_W¹_,W²Y_W1,W²(w₁,1)w₂= e^L−1T_γ− 1

Y_W^P(−1)_2,W1(w₂,−1)w₁

= e^L−1Y_W2,W¹(w₂,e^l(−1))w₁

= e^L−1Y_W2,W¹(w₂,e^πi)w₁, wherel(−1) =πi due to the choice of the path γ₁⁻ from −1 to 1.

For aV-module W the twist isomorphismθW:W →W is given by θW(w) = e^(2πi)L0w

forw∈W. In particular, ifW is irreducible, then θW = e^(2πi)ρ(W)idW. Rigidity

We describe the sovereign structure on C. In fact, the proof of left and right rigidity for C is the main result in [Hua08a]. The left and right dual of a V-module W is given by the contragredient module W⁰. Recall that the rigid structure consists of the right and left duality morphisms b_W ∈ HomV(V, W W⁰), d_W ∈ HomV(W⁰ W, V),

˜b_W ∈HomV(V, W⁰W) and ˜d_W ∈HomV(W W⁰, V).

Since the category Cis semisimple, it suffices to consider irreducible modules. Let the finitely many isomorphism classes of irreducibleV-modules be labelled by the setF with the isomorphism class of V corresponding to 0 ∈F. We fix a choice of representatives W^α,α∈F, withW⁰ =V. Forα∈F, letα⁰ ∈F denote the index of the contragredient module of W^α, i.e. (W^α)⁰ ∼= W^α0. If an irreducible module W^α is self-contragredient, i.e. α = α⁰, then there is a module isomorphism φ_W^α: W^α → (W^α)⁰, unique up to a complex scalar, which defines a non-degenerate, invariant bilinear form (·,·)W^α on W^α via (u, w)W^α := hφ_W^α(u), wi for u, w ∈ W^α where h·,·i denotes the natural pairing between (W^α)⁰, the graded dual space of W^α, and W^α (see Section 1.5). On V, this non-degenerate, invariant bilinear form is symmetric and for convenience we assume that it is normalised as (1,1)V = 1. This fixes the isomorphism φV. We also fix choices of theφ_W^α,α∈F.

Note that the spaces of intertwining operators of types W^α

V W^α

!

, W^α W^αV

!

, V⁰ W^α(W^α)⁰

!

, V

W^α,(W^α)⁰

!

, V

(W^α)⁰, W^α

!

are all one-dimensional since they are isomorphic to EndV(W^α) by Proposition 1.6.7 and theS3-symmetry and this space is one-dimensional by Schur’s lemma.

We recall thisS₃-symmetry of the intertwining operators (see Section 1.6). LetY(·, x) be an intertwining operator of type _W^WαW^{γ β}

. Then σ₁₂(Y) is defined to be the inter-twining operator of type _W^Wβ_W^γαgiven by

σ12(Y)(wα, x)wβ = eπi(ρ(γ)−ρ(α)−ρ(β))e^xL−1Y(wβ,e^−πix)wα

for wα ∈ W^α, wβ ∈ W^β where ρ(α) := ρ(W^α) denotes the conformal weight of W^α, α∈F. Again, to clarify the notation, recall from Section 1.1 that given an intertwining operatorY(·, x) we defineY(·,e^ζx) for ζ ∈C as

Y(·,e^ζx) =Y(·, y)|_yn=e^nζxⁿ, n∈C

wherexandyare formal variables. Furthermore,σ₂₃(Y) is defined to be the intertwining operator of type _W^(Wα(W^β)0γ)⁰

determined by

Dσ₂₃(Y)(w_α, x)w_γ⁰, w_β^E= e^πiρ(α)Dw_γ⁰,Y(e^xL1e^−πiL0x^−2L0w_α, x⁻¹)w_β^E

forwα ∈ W^α, wβ ∈W^β, w_γ⁰ ∈(W^γ)⁰. Upon identifying W^α with (W^α)⁰⁰ for all α ∈F one obtains

σ₁₂² = 1, σ₂₃² = 1 and σ₁₂σ₂₃σ₁₂=σ₂₃σ₁₂σ₂₃,

which shows thatσ₁₂ and σ₂₃ define a representation ofS₃ on the space of intertwining operators^Lα,β,γ∈FV_W^Wα^γW^β.

In the following we will define certain non-zero intertwining operators in the one-dimensional spaces of intertwining operators introduced above. Let

Y_V,W^Wαα :=Y_W^α

be the module vertex operation of V on W^α. This is an intertwining operator of type

W^α V W^α

. Furthermore, we define

Y_W^Wα^α,V :=σ12(Y_V,W^Wαα) =σ12(YW^α), which is an intertwining operator of type _W^Wα^αV

. We then define Y_W^V α,(W^α)⁰ :=φ⁻¹_V σ23(Y_W^Wα^α,V) =φ⁻¹_V σ23σ12(YW^α) of type _W^α_(W^{V α}₎⁰whereφV is the isomorphism V →V⁰. Finally,

Y_(W^V α)⁰,W^α :=σ₁₂(Y_W^Vα,(W^α)⁰) =φ⁻¹_V σ₁₂σ₂₃σ₁₂(Y_W^α) defines an intertwining operator of type _(Wα^V)⁰W^α

.

Then, the right evaluation morphismd_W^α: (W^α)⁰W^α→V is defined via dW^α(w⁰_α⊗wα) :=d(α)Y_(W^V α)⁰,W^α(w⁰_α,1)wα

and the left one ˜dW^α:W^α(W^α)⁰ →V via

d˜W^α(wα⊗w⁰_α) :=d(α)Y_W^V α,(W^α)⁰(wα,1)w_α⁰

for w_α ∈ W^α, w_α⁰ ∈ (W^α)⁰ where d(α) is the quantum dimension of W^α, which only depends on the isomorphism class ofW^α. Again,dW^α: (W^α)⁰W^α →V is the natural extension ofd_W^α: (W^α)⁰W^α→V and similarly for ˜d_W^α.

We omit the definition of the coevaluation morphisms b_W^α and ˜b_W^α but remark that if we choose two lowest-weight vectorsw∈W^α,w⁰ ∈W^α withhw⁰, wi= 1, then

b_W^α(1) =P₀(ww⁰)∈W^α(W^α)⁰

where P₀ is the projection onto the weight-zero space (see proof of Theorem 3.9 in [Hua08a]).

Frobenius-Schur Indicator for Vertex Operator Algebras

Let V be a vertex operator algebra and W some irreducible, self-contragredient V -module. We saw in Section 1.5 that, given an isomorphism φ_W:W →W⁰,

(u, w)W :=hφ_W(u), wi,

u, w∈W, defines a non-degenerate, invariant bilinear form on W and this is the unique non-degenerate, invariant bilinear form on W up to a scalar. Moreover, this bilinear form is either symmetric or antisymmetric.

Definition 3.3.6(Frobenius-Schur Indicator). LetV be a vertex operator algebra. For an irreducible, self-contragredient module W we define the Frobenius-Schur indicator ν(W) := 1 and ν(W) :=−1 if the bilinear form (·,·)W onW is symmetric and antisym-metric, respectively. For an irreducible moduleW that is not self-contragredient we set ν(W) := 0.

If V is simple, then Proposition 1.5.9 shows that ν(V) = 1.

There are two notions of Frobenius-Schur indicator for a vertex operator algebraV: (1) the modular tensor categorical definition of the Frobenius-Schur indicator

(Defin-ition 3.3.3) applied to Huang’s construction of the modular tensor category asso-ciated withV (Theorem 3.3.5) if V satisfies AssumptionN,

(2) the definition via the symmetry of the invariant bilinear form (Definition 3.3.6).

The following proposition shows that these two notions coincide.

Proposition 3.3.7. Let V be a vertex operator algebra satisfying Assumption N. Then the two notions (1) and (2) of Frobenius-Schur indicator agree.

Proof. We compute the Frobenius-Schur indicator in the modular tensor category C obtained from Huang’s construction and show that it is +1 or −1 for an irreducible module W^α depending on whether the non-degenerate, invariant bilinear form on W^α is symmetric or antisymmetric.

Let W^α be an irreducible, self-contragredient V-module. We introduced the module isomorphism φW^α: W^α → (W^α)⁰ defining a non-degenerate, invariant bilinear form (·,·)W^α on W^α via (u, w)W^α :=hφ_W^α(u), wi foru, w∈W^α.

By definition, the Frobenius-Schur indicator ν(α) is given as the factor of proportion-ality between

d˜_W^α◦b_W^α and d_W^α◦(φ_W^αφ⁻¹_Wα)◦b_W^α, which lie in EndV(V) =CidV. For the left-hand side we get

d˜W^α(bW^α(1)) =d(α) idV

by the definition of the quantum dimension. The computation of the right-hand side is more involved. We evaluate the expression at the vacuum1. We stated above that

b_W^α(1) =P₀(ww⁰)∈W^α(W^α)⁰

for two lowest-weight vectorsw∈W^α and w⁰∈(W^α)⁰ withhw⁰, wi= 1. Then d_W^α((φ_W^αφ⁻¹_Wα)(b_W^α(1))) =d_W^α((φ_W^αφ⁻¹_Wα)(P₀(ww⁰)))

=d_W^α(P₀(φ_W^α(w)φ⁻¹_Wα(w⁰)))

=P0

dW^α(φW^α(w)φ⁻¹_Wα(w⁰))

=d(α)P0

Y_(W^V α)⁰,W^α(φW^α(w),1)φ⁻¹_W^α(w⁰)

=d(α) Resx

x^2ρ(α)−1Y_(W^V α)⁰,W^α(φ_W^α(w), x)φ⁻¹_Wα(w⁰) where Resx(·) is the coefficient ofx⁻¹. By definition ofY_(W^V α)⁰,W^α(·, x) the above is

d(α)e^{−(2πi)ρ(α)}Resx

x^2ρ(α)−1e^xL−1Y_W^Vα,(W^α)⁰(φ⁻¹_Wα(w⁰),e^−πix)φW^α(w).

Then we take the bilinear form of the above expression and 1∈V and by definition of Y_W^V α,(W^α)⁰(·, x) we obtain

d(α)e^{−(2πi)ρ(α)}Resx

x^2ρ(α)−1e^xL−1Y_W^Vα,(W^α)⁰(φ⁻¹_Wα(w⁰),e^−πix)φ^α_W(w),1

V

=d(α)e^{−(2πi)ρ(α)}Resx

x^2ρ(α)−1e^xL−1Y_W^Vα,(W^α)⁰(φ⁻¹_Wα(w⁰),e^−πix)φ_W^α(w),1

V

=d(α)e^{−(2πi)ρ(α)}Resx

x^2ρ(α)−1Y_W^Vα,(W^α)⁰(φ⁻¹_Wα(w⁰),e^−πix)φW^α(w),e^xL11

V

V

=d(α)e^{−(2πi)ρ(α)}Resx

x^2ρ(α)−1Dφ_V Y_W^Vα,(W^α)⁰(φ⁻¹_Wα(w⁰),e^−πix)φ_W^α(w),1^E

=d(α)e^{−(2πi)ρ(α)}Resx

x^2ρ(α)−1D(σ₂₃Y_W^Wα^α,V)(φ⁻¹_Wα(w⁰),e^−πix)φ_W^α(w),1^E

=d(α)e^{−(2πi)ρ(α)}e^πiρ(α)Resx

x^2ρ(α)−1 DφW^α(w),Y_W^Wα^α,V

e^e−πixL1e^−πiL0(e^−πix)^−2L0φ⁻¹_Wα(w⁰),(e^−πix)⁻¹1^E,

where we used L11 = 0. Now we use that w⁰ and hence φ⁻¹_Wα(w⁰) is a lowest-weight vector so that L1φ⁻¹_Wα(w⁰) = 0 andL0φ⁻¹_Wα(w⁰) =ρ(α)φ⁻¹_Wα(w⁰). Then the above equals d(α)e^{−(2πi)ρ(α)}Resx

x^2ρ(α)−1Dφ_W^α(w),Y_W^Wα^α,V

(e^−πix)^−2ρ(α)φ⁻¹_Wα(w⁰),(e^−πix)⁻¹1^E

=d(α)e^{−(2πi)ρ(α)}Resx

x^2ρ(α)−1Dφ_W^α(w),Y_W^Wα^α,V

e^(2πi)ρ(α)x^−2ρ(α)φ⁻¹_Wα(w⁰),e^πix⁻¹1^E

=d(α) Resx

x^−1Dφ_W^α(w),Y_W^Wα^α,V

φ⁻¹_Wα(w⁰),e^πix⁻¹1^E,

where we simplified the expressions (e^−πix)^−2ρ(α) and (e^−πix)⁻¹ consistent with our

notational conventions. Finally, we use the definition of Y_W^Wα^α,V =σ12(YW^α) to obtain d(α) Resx

x^−1Dφ_W^α(w),e^{eπix−1L−1}Y_W^α1,e^−πie^πix⁻¹φ⁻¹_Wα(w⁰)^E

=d(α) Resx

x^−1De^eπix−1L1φ_W^α(w), φ⁻¹_Wα(w⁰)^E

=d(α) Resx

x^−1DφW^α(w), φ⁻¹_Wα(w⁰)^E

=d(α)^DφW^α(w), φ⁻¹_W^α(w⁰)^E

=d(α)w, φ⁻¹_Wα(w⁰)

W^α. Hence, in total we have computed

(dW^α ◦(φW^α φ⁻¹_Wα)◦bW^α)(1),1

V =d(α)w, φ⁻¹_Wα(w⁰)

W^α, which we compare with

( ˜d_W^α◦b_W^α)(1),1

V =d(α)(1,1)V =d(α),

recalling the normalisation (1,1)V = 1. On the other hand, we chose hw⁰, wi = 1 and, setting u:=φ⁻¹_Wα(w⁰), this reads (u, w)W^α = 1 in terms of the bilinear form on W^α.

Hence, the Frobenius-Schur indicator ofW^α is given by ν(α) =w, φ⁻¹_Wα(w⁰)

W^α = (w, u)W^α = (w, u)W^α

(u, w)W^α

,

i.e. it coincides with the definition via the symmetry of the invariant bilinear form on W^α for special non-zero choices ofu, w ∈W^α and consequently for all u, w∈W^α.

The Frobenius-Schur indicator does not only appear in the symmetry relation for the invariant bilinear form on an irreducible, self-contragredient module but also in a relation similar to the skew-symmetry formula (1.2) for a vertex operator algebra V:

Y_V(a, x)b= e^xL−1Y_V(b,−x)a fora, b∈V. Indeed:

Proposition 3.3.8. Let V be a vertex operator algebra satisfying AssumptionN and let W^α be an irreducible, self-contragredient module. LetY(x,·) be an intertwining operator of type _Wα^VW^α

. Then

Y(w, x)u=ν(α)e^{−(2πi)ρ(α)}e^xL−1Y(u,e^−πix)w for u, w∈W^α.

Proof. Note that the space of intertwining operators of type _Wα(W^{V α})⁰

is, as remarked above, one-dimensional. This means that

Y_W^Vα,(W^α)⁰(·, x) is proportional to

Y_(W^V α)⁰,W^α(φW^α(·), x)φ⁻¹_Wα. We call the constant of proportionalityν(W^α) and obtain

Y_W^V α,(W^α)⁰(w, x)w⁰ =ν(W^α)Y_(W^V α)⁰,W^α(φW^α(w), x)φ⁻¹_W^α(w⁰)

=ν(W^α)e^{−(2πi)ρ(α)}e^xL−1Y_W^Vα,(W^α)⁰(φ⁻¹_Wα(w⁰),e^−πix)φW^α(w) forw∈W^α andw⁰∈(W^α)⁰. Replacing the formal variablexbyz= 1∈Cand recalling the definition of ˜d_W^α and d_W^α in terms of Y_W^Vα,(W^α)⁰(·,1) and Y_(W^V α)⁰,W^α(·,1) we see thatν(W^α) =ν(α) is exactly the Frobenius-Schur indicator.

Let us define the intertwining operator Y(x,·) of type _Wα^VW^α

by Y(·, x)w:=Y_W^Vα,(W^α)⁰(·, x)φ_W^α(w). Then, settingu=φ⁻¹_Wα(w⁰), we obtain

Y(w, x)u=ν(α)e^{−(2πi)ρ(α)}e^xL−1Y(u,e^−πix)w

for u, w ∈ W^α, which is the relation we want to show. Since the space of intertwining operators of type _W^αV_W^αis also one-dimensional, this formula holds for any intertwining operator of that type.

Remark 3.3.9.

(1) The factorν(α)e^{−(2πi)ρ(α)}in the above proposition describes the failure of the skew-symmetry relation Y(a, x)b = e^xL−1Y(b,e^−πix)a to hold, which would be true if Y(·, x) were simply the vertex operation onV.

(2) The above proposition is a generalisation of Proposition 5.6.1 in [FHL93] and Lemma 2.1 in [Yam13], which deal with the special cases of ρ(α) ∈Zand ρ(α)∈ (1/2)Z, respectively. We do not impose any restrictions on the value of ρ(α).⁴ (3) The proposition actually holds without Assumption N if we use the definition of

the Frobenius-Schur indicator not involving modular tensor categories. This is the approach in [FHL93, Yam13].

4Note however that in the case of group-like fusion discussed below it is easy to see thatρ(α)∈(1/4)Z for any irreducible, self-contragredient moduleW^α.

Quantum Dimensions for Vertex Operator Algebras

We introduced the notion of quantum dimensions (or categorical dimensions) for modular tensor categories. We can in particular study them for the modular tensor categories associated with certain vertex operator algebras from Huang’s construction. There is also another notion of quantum dimensions defined directly for vertex operator algebras and it will turn out that under suitable regularity assumptions onV, notably AssumptionP, both notions agree.

Given a vertex operator algebra V and a V-module W, assume that the characters chV(τ) and chW(τ) are well-defined functions on the upper half-plane. This is the case for example if V is rational and C2-cofinite and W is an irreducible V-module (see Theorem 1.8.1). The quantum dimension qdimV(W) ofW is defined as the limit

qdimV(W) := lim

y→0⁺

chW(iy) chV(iy). It is a priori not clear that this number exists. However:

Proposition 3.3.10 ([DJX13], Lemma 4.2). Let V be a simple, rational, C2-cofinite vertex operator algebra of CFT-type which satisfies Assumption P. Then for any irredu-cible module W ∈Irr(V) the quantum dimension exists and

0<qdim_V(W) = S_W,V S_V,V <∞.

Proposition 3.3.11([DLN15], Proposition 3.11). LetV satisfy AssumptionsNP. Con-sider the modular tensor categoryC associated withV from Huang’s construction. Then for any irreducible V-module W

d(W) = qdimV(W)

where d(W) is the quantum dimension of the simple object W in C.

Quantum dimensions play an important rôle since they characterise simple-current modules:

Proposition 3.3.12([DJX13], Proposition 4.17). LetV satisfy Assumptions NP. Then W ∈Irr(V) is a simple current if and only if qdimV(W) = 1.

Group-Like Fusion

Finally, let us assume that the vertex operator algebra V has group-like fusion, i.e.

thatV satisfies Assumption SN. Then there is an abelian group structure on the setF indexing the isomorphism classes of irreducible modules {W^α |α ∈ F} and the fusion group FV = F carries the quadratic form Qρ(α) = ρ(W^α) +Z. In the following, in addition toQ_ρ, we also consider the multiplicative quadratic formq_ρ:F_V →C^×,

q_ρ(α) = e^{(2πi)ρ(Wα)} forα∈F_V.

In addition to the modular tensor category structure on the V-modules, the direct sum of all irreducible V-modules A = ^Lα∈F_V W^α carries the structure of an abelian intertwining algebra by Theorem 3.2.1 with the vertex operation on A composed of the intertwining operators Y_W⁺_α,Wβ(·, x) for α, β ∈ FV from Section 3.2. Part of this structure is the quadratic form q_Ω:F_V →C^×,

qΩ(α) = Ω(α, α) forα∈F_V.

In the following we express the twist isomorphismθW^α, the Frobenius-Schur indicator ν(α) in two different ways and the braidingc_Wα,W^β in terms of the quadratic forms q_Ω and q_ρ. Recall that the twist is defined by Huang to be

θ_W^α =q_ρ(α) idW^α

for any irreducible module W^α,α∈F_V.

The following is an immediate consequence of Proposition 3.3.8 and the braiding convention in Definition 3.2.9.

Proposition 3.3.13. LetV be a vertex operator algebra satisfying AssumptionSN. Let α∈FV with2α= 0, i.e. W^α is self-contragredient. Then

ν(α) = q_ρ(α) q_Ω(α). Proof. The braiding (see Definition 3.2.9) implies

Y_W⁺α,W^α(w, x)u= 1

q_Ω(α)e^xL−1Y_W⁺α,W^α(u,e^−πix)w

for u, w ∈ W^α where Y_W⁺α,W^α(·, x) is a certain choice of intertwining operator of type

V W^αW^α

. Then the statement follows from Proposition 3.3.8.

We can also compute the Frobenius-Schur indicator using Bántay’s formula (see Pro-position 3.3.4) and using that the quantum dimensions are all one in the case of group-like fusion and under Assumption P.

Proposition 3.3.14. Let V be a vertex operator algebra satisfying Assumptions SNP. Let α∈F_V with 2α= 0, i.e. W^α is self-contragredient. Then

ν(α) = 1 qρ(α)².

Proof. Propositions 3.3.11 and 3.3.12 show that the quantum dimensions d(α) = 1 for all α∈F_V. Then Bántay’s formula gives

D²= ^X

α,β∈F_V

e^{(2πi)(Qρ(α)−Qρ(β))}= ^X

α,β∈F_V

e^{(2πi)(Qρ(α+β)−Qρ(β))}

= ^X

α,β∈F_V

e^{(2πi)(Bρ(α,β)+Qρ(α))} = ^X

α∈F_V



 X

β∈F_V

e^{(2πi)Bρ(α,β)}



e^{(2πi)Qρ(α)}

= ^X

α∈F_V

|F_V|δ_α,0e^{(2πi)Qρ(α)}=|F_V|,

using that 2γ = 0 and hence 2B_ρ(α, γ) = 0 +Z for allα∈F_V. Then ν(γ) = 1

|F_V| X

α,β∈F_V

e^{(2πi)(2Qρ(α)−2Qρ(β))}δ_α+γ,β = 1

|F_V| X

α∈F_V

e^{(2πi)(2Qρ(α)−2Qρ(α+γ))}

= 1

|F_V| X

α∈F_V

e^{(2πi)(−2Bρ(α,γ)−2Qρ(γ))}= e^{(2πi)(−2Qρ(γ))}= 1 q_ρ(α)² sinceN_α,γ^β =δ_α+γ,β.

Note that since 2α = 0 for a self-contragredient module,Q_ρ(α)∈(1/4)Zsince Q_ρ is a quadratic form and hence 1/q_ρ(α)² indeed lies in {±1}.

Proposition 3.3.15. Let V be a vertex operator algebra satisfying Assumption SN. Then the braiding isomorphism in the modular tensor categoryC is given by

c_W^α_,W^α =q_Ω(α)q_ρ²(α) idW^αW^α

for all α∈F_V.

Proof. Recall the definition of the braiding isomorphism c_Wα,W^β

Y_Wα,W^β(w,1)u= e^L−1Y_Wβ,W^α(u,e^πi)w

forw∈W^α,u∈W^β where Y_Wα,Wβ(·, x), α, β∈F_V, are canonical intertwining operat-ors of type ^W_W^αα^WW^ββ

.

On the other hand, to construct the abelian intertwining algebra in the case of group-like fusion, we chose intertwining operators Y⁺

W^α,W^β(·, x) of type _W^Wα^α+βW^β

. This cor-responds to choosing module isomorphisms ψ_W^α_,W^β: W^α W^β → W^α+β such that Y_W⁺_α,Wβ(·, x) =ψ_Wα,W^βY_Wα,Wβ(·, x).

The skew-symmetry formula (see Definition 3.2.9) gives Y_W⁺_α,Wβ(w, x)u= 1

Ω(β, α)e^xL−1Y_W⁺_β,Wα(u,e^−πix)w,

which we reformulate as ψ_Wα,W^β

Y_Wα,W^β(w, x)u= 1

Ω(β, α)b_ρ(α, β)e^xL−1ψ_Wβ,W^α

Y_Wβ,W^α(u,e^πix)w using that the exponents of Y_Wβ,W^α(·, x) lie inB_ρ(α, β) +Z. Finally, settingα=β we obtain

Y_Wα,W^α(w, x)u= 1

q_Ω(α)qρ(α)²e^xL−1Y_Wα,W^α(u,e^πix)w.

Inserting z= 1 for xand comparing with the definition of c_W^α_,W^α yields q_Ω(α)qρ(α)²Y_Wα,W^α(w,1)u= e^L−1Y_Wα,W^α(u,e^πi)w=c_Wα,W^β

Y_Wα,W^α(w,1)u, which proves the statement.

Im Dokument A Cyclic Orbifold Theory for Holomorphic Vertex Operator Algebras and Applications (Seite 83-98)