This paper explores the relationship between extensions of completely unitary vertex operator algebras (VOAs) and Q-systems, demonstrating that unitary extensions preserve complete unitarity.
Contribution
It establishes a connection between VOA extensions and Q-systems and proves that unitary extensions of completely unitary VOAs are themselves completely unitary.
Findings
01
Extensions of completely unitary VOAs correspond to (commutative) Q-systems.
02
Any unitary extension of a completely unitary VOA remains completely unitary.
03
The work provides a structural understanding of VOA extensions in the context of operator algebras.
Abstract
We relate extensions of completely unitary VOAs and (commutative) Q-systems. As an application, we show that any unitary extension of a completely unitary VOA is completely unitary.
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Q-systems and extensions of completely unitary vertex operator algebras
Bin Gui
Abstract
Complete unitarity is a natural condition on a CFT-type regular VOA which ensures that its modular tensor category is unitary. In this paper we show that any CFT-type unitary (conformal) extension U of a completely unitary VOA V is completely unitary. Our method is to relate U with a Q-system AUβ in the Cβ-tensor category Repu(V) of unitary V-modules. We also update the main result of [KO02] to the unitary cases by showing that the tensor category Repu(U) of unitary U-modules is equivalent to the tensor category Repu(AUβ) of unitary AUβ-modules as unitary modular tensor categories.
As an application, we obtain infinitely many new (regular and) completely unitary VOAs including all CFT-type c<1 unitary VOAs. We also show that the latter are in one to one correspondence with the (irreducible) conformal nets of the same central charge c, the classification of which is given by [KL04].
This is the first part in a series of papers to study the relations between unitary vertex operator algebra (VOA) extensions and conformal net extensions. We will always focus on rational conformal field theories, so our VOAs are assumed to be CFT-type, self-dual, and regular, so that the categories of VOA modules are modular tensor categories (MTCs).
Although both unitary VOAs and conformal nets are mathematical formulations of unitary chiral CFTs, they are defined and studied in rather different ways, with the former being more algebraic and geometric, and the latter mainly functional analytic. A systematic study to relate these two approaches was initiated by Carpi-Kawahigashi-Longo-Weiner [CKLW18] and followed by [Ten19a, Ten18b, Gui20, Ten19b, CW, CWX], etc. In these works the methods of relating the two approaches are transcendental and have a lot of analytic subtleties. Due to these subtleties, certain models (such as unitary Virasoro VOAs, and unitary affine VOAs especially of type A) are easier to analyze than the others. On the other hand, when studying the extensions and conformal inclusions of chiral CFTs, the main tools in the two approaches are quite similar: both are (commutative) associative algebras in a tensor category C (called C-algebras); see [KO02, HKL15, CKM17] for VOA extensions, and [LR95, KL04, BKLR15] for conformal net extensions; see also [FRS02, FS03] for the general notion of algebra objects inside a tensor category. In this and the following papers, we will see that C-algebras are also powerful tools for relating unitary VOA extensions and conformal net extensions in the above mentioned systematic and transcendental settings.
There is, however, one important difference between the C-algebras used in the two approaches: for conformal net extensions the C-algebras are unitary. Unitarity is an essential property for conformal nets and operator algebras but not quite necessary for VOAs. However, it is impossible to relate VOAs and conformal nets without adding unitary structures on VOAs (and their representation categories). This point is already clear in [Gui20], where we have seen that to relate the tensor categories of VOAs and conformal nets, one has to first make the VOA tensor categories unitary.
In this paper our main goal is to relate the Cβ-tensor categories of unitary VOA extensions with those of unitary C-algebras (also called Cβ-Frobenius algebras or (under slightly stronger condition) Q-systems [Lon94]). As applications, we prove many important unitary properties of VOA extensions, the most important of which are the complete unitarity of VOAs as defined below.
Complete unitarity of unitary VOA extensions
A CFT-type regular VOA V is called completely unitary if the following conditions are satisfied.
β’
V is unitary [DL14], which means roughly that V is equipped with an inner product and an antiunitary antiautomorphism Ξ which relates the vertex operators of V to their adjoints.
β’
Any V-module admits a unitary structure. Since V-modules are semisimple, it suffices to assume the unitarizability of irreducible V-modules.
β’
For any irreducible unitary V-modules Wiβ,Wjβ,Wkβ, the non-degenerate invariant sesquilinear formΞ introduced in [Gui19b] and defined on the dual vector space of type (iΒ jkβ) intertwining operators of V is positive.
The importance of complete unitarity lies in the following theorem.
If V is a CFT-type, regular, and completely unitary VOA, then the unitary V-modules form a unitary modular tensor category.
However, compared to unitarity, complete unitarity is much harder to prove since not only vertex operators but also intertwining operators need to be taken care of. In this paper, our main result as follows provides a powerful tool for proving the complete unitarity.
Theorem 0.2**.**
Suppose that V is a CFT-type, regular, and completely unitary VOA, and U is a CFT-type unitary VOA extension of V. Then U is also completely unitary.
Roughly speaking, if we know that a unitary VOA U is an extension of a completely unitary VOA V, then U is also completely unitary. As applications, since the complete unitarity has been established for unitary affine VOAs and c<1 unitary Virasoro VOAs (minimal models) as well as their tensor products (proposition 3.31), we know that all their unitary extensions are completely unitary. In particular, these extensions have unitary modular tensor categories. We also show that these tensor categories are equivalent to the unitary modular tensor categories associated to the corresponding Q-system. To be more precise, we prove
Theorem 0.3**.**
Let V be a CFT-type, regular, and completely unitary VOA, and let U be a CFT-type unitary extension of V whose Q-system is AUβ. If Repu(U) is the category of unitary U-modules, and Repu(AUβ) is the category of unitary AUβ-modules, then Repu(U) is natually equivalent to Repu(AUβ) as unitary modular tensor categories.
A non-unitary version of the above theorem has already been proved in [CKM17]: Repu(U) is known to be equivalent to Repu(AUβ) as modular tensor categories. The above theorem says that the unitary structures of the two modular tensor categories are also equivalent in a natural way. We remark that the unitary tensor structures compatible with the β-structure of Repu(U) are unique by [Reu19, CCP]. Thus [Reu19, CCP] provide a different method of proving the above theorem.
We would like to point out that our results on the relation between unitary VOA extensions and Q-systems also provide a new method of proving the unitarity of VOAs. For instance, the irreducible c<1 conformal nets are classified as (finite index) extensions of Virasoro nets by Kawahigashi-Longo in [KL04] (table 3), and their VOA counterparts are given by Dong-Lin in [DL15]. However, Dong-Lin were not able to prove the unitarity of two exceptional cases: the types (A10β,E6β) and (A28β,E8β) (see remark 4.16 of [DL15]). But since these two types are realized as commutative Q-systems in [KL04], we can now show that the corresponding VOAs are actually unitary. This proves that the c<1 CFT-type unitary VOAs are in one to one correspondence with the irreducibile conformal nets with the same central charge c.
We have left several important questions unanswered in this paper. We see that Q-systems can relate unitary VOA extensions and conformal net extensions. But [CKLW18] also provides a uniform way of relating unitary VOAs and conformal nets using smeared vertex operators. Are these two relations compatible? Moreover, do the VOA extensions and the corresponding conformal net extensions have the same tensor categories? Answers to these questions are out of scope of this paper, so we leave them to future works.
Outline of the paper
In chapter 1 we review the construction and basis properties of VOA tensor categories due to Huang-Lepowsky. We also review various methods of constructing new intertwining operators from old ones, and translate them into tensor categorical language. The translation of adjoint and conjugate intertwining operators is the most important result of this chapter. Unitary VOAs, unitary representations, and the unitary structure on VOA tensor categories are also reviewed.
In chapter 2 we relate unitary VOA extensions and Q-systems as well as their (unitary) representations. The first two sections serve as background materials. In section 2.1 we review the relation between VOA extensions and commutative C-algebras as in [HKL15]. Their results are adapted to our unitary setting. In section 2.2 we review various notions concerning dualizable objects in Cβ-tensor categories. Most importantly, we review the construction of standard evaluations and coevaluations in Cβ-tensor categories necessary for defining quantum traces and quantum dimensions. Standard reference for this topic is [LR97]. We also explain why the naturally defined evaluations and coevaluations in the tensor categories associated to completely unitary VOAs are standard. In section 2.3 we define a notion of unitary C-algebras which is a direct translation of unitary VOA extensions in categorical language. This notion is related to Cβ-Frobenius algebras and Q-systems in section 2.4. The equivalence of c<1 unitary VOAs and conformal nets is also proved in that section. A VOA U is called strongly unitary if it satisfies the first two of the three conditions defining complete unitarity. Therefore, strong unitarity means the unitarity of U and the unitarizability of all U-modules. In section 2.5, we give two proofs that any unitary extension U of a completely unitary VOA V is strongly unitary. The first proof uses induced representations, and the second one uses a result of standard representations of Q-systems in [BKLR15].
In chapter 3 we use the Cβ-tensor categories of the bimodules of Q-systems to prove the complete unitarity of unitary VOA extensions. We review the construction and basic properties of these Cβ-tensor categories in the first four sections. Although these results are known to experts (cf. [NY16] chapter 6 or [NY18b] section 4.1), we provide detailed and self-contained proofs of all the relevant facts, which we hope are helpful to the readers who are not familiar with tensor categories. We present the theory in such a way that it can be directly compared with the (Hermitian) tensor categories of unitary VOA modules. So in some sense our approach is closer in spirit to [KO02, CKM17]. In section 3.5 we prove the main results of this paper: theorems 0.2 and 0.3 (which are theorem 3.30 of that section). Finally, applications are given in section 3.6.
Acknowledgment
Iβm grateful to Robert McRae for a helpful discussion of the rationality of VOA extensions; Marcel Bischoff for informing me of certain literature on conformal inclusions; and Sergey Neshveyev and Makoto Yamashita for many discussions and helpful comments.
1 Intertwining operators and tensor categories of unitary VOAs
1.1 Braiding, fusion, and contragredient intertwining operators
We refer the reader to [BK01, EGNO, Tur] for the general theory of tensor categories, and [Hua08] for the construction of the tensor category Rep(V) of V-modules. A brief review of this construction can also be found in [Gui19a] section 2.4 or [Gui20] section 4.1. Here we outline some of the key properties of Rep(V) which will be used in the future.
Representations of V are written as Wiβ,Wjβ,Wkβ, etc. If a V-module Wiβ is given, its contragredient module (cf. [FHL93] section 5.2) is written as Wiβ . Wiβ, the contragredient module of Wiβ, can naturally be identified with Wiβ. So we write i=i. Note that the symbol i is now reserved for representations. So we write the imaginary unit β1β as i. Let W0β be the vacuum V-module, which is also the identity object in Rep(V). The product of V-modules is constructed in such a way that for any Wiβ,Wjβ,Wkβ there is a canonical isomorphism of linear spaces
[TABLE]
where V(iΒ jkβ)=V(WiβWjβWkββ) is the (finite-dimensional) vector space of intertwining operators of V. For any w(i)βWiβ, we write YΞ±β(w(i),z)=βsβCβYΞ±β(w(i))sβzβsβ1 where z is a complex variable defined in CΓ:=Cβ{0} and YΞ±β(w(i))sβ:WjββWkβ is the s-th mode of the intertwining operator. We say that Wiβ,Wjβ,Wkβ are respectively the charge space, the source space, and the target space of the intertwining operator YΞ±β. Tensor products of morphisms are defined such that the following condition is satisfied: If FβHom(Wiβ²β,Wiβ),GβHom(Wjβ²β,Wjβ),KβHom(Wkβ,Wkβ²β), then for any w(iβ²)βWiβ²β,
[TABLE]
One way to realize the above properties is as follows: Notice first of all that V has finitely many equivalence classes of irreducible unitary V-modules. Fix, for each equivalence class, a representing element, and let them form a finite set E. We assume that the vacuum unitary module V=W0β is in E. If WtββE, we will use the notation tβE to simplify formulas. We then define Wiββ Wjβ to be β¨tβEβV(iΒ jtβ)ββWtβ [HL95] (here V(iΒ jtβ)β is the dual vector space of V(iΒ jtβ)). Then for each tβE there is a natural identification between Hom(Wiββ Wjβ,Wtβ) and V(iΒ jtβ), which can be extended to the general case Hom(Wiββ Wjβ,Wkβ)βV(iΒ jkβ) via the canonical isomorphisms
[TABLE]
The tensor structure of Rep(V) is defined in such a way that it is related to the fusion relations of the intertwining operators of V as follows. Choose non-zero z,ΞΆ with the same arguments (notation: argz=argΞΆ) satisfying 0<β£zβΞΆβ£<β£ΞΆβ£<β£zβ£. In particular, we assume that z,ΞΆ are on a common ray stemming from the origin. We also choose arg(zβΞΆ)=argΞΆ=argz. Suppose that we have Wiβ,Wjβ,Wkβ,Wlβ,Wpβ,Wqβ in Rep(V), and intertwining operators YΞ±ββV(iΒ plβ),YΞ²ββV(jΒ kpβ),YΞ³ββV(iΒ jqβ),YΞ΄ββV(qΒ klβ), such that for any w(i)βWiβ,w(j)βWjβ, the following fusion relation holds when acting on Wkβ:
[TABLE]
Then, under the identification of Wiββ (Wjββ Wkβ) and (Wiββ Wjβ)β Wkβ (which we denote by Wiββ Wjββ Wkβ) via the associativity isomorphism, we have the identity Ξ±(1iββΞ²)=Ξ΄(Ξ³β1kβ), which can be expressed graphically as
[TABLE]
Here we take the convention that morphisms go from top to bottom.
Convention 1.1**.**
When we consider fusion relations in the form (1.5), we always assume 0<β£zβΞΆβ£<β£ΞΆβ£<β£zβ£ and arg(zβΞΆ)=argΞΆ=argz.
The braided and the contragredient intertwining operators are two major ways of constructing new intertwining operators from old ones [FHL93]. As we shall see, they can all be translated into operations on morphisms. We first discuss braiding. Given YΞ±ββV(iΒ jkβ), we can define braided intertwining operatorsB+βYΞ±β,BββYΞ±β of type V(jΒ ikβ) in the following way: Choose any w(i)βWiβ,w(j)βWjβ. Then
[TABLE]
Then the braid isomorphism \ss=\ssi,jβ:Wiββ WjββWjββ Wiβ is constructed in such a way that BΒ±βYΞ±β=YΞ±β\ssΒ±1β. Write B+βYΞ±β=YB+βΞ±β and BββYΞ±β=YBββΞ±β. Then BΒ±βΞ±=Ξ±β\ssΒ±1. Using
and
to denote \ss and \ssβ1 respectively, this formula can be pictured as
[TABLE]
Let Yiβ=Yiβ(v,z) be the vertex operator associated to the module Wiβ. Then Yiβ is also a type (0Β iiβ) intertwining operator. Itβs easy to verify that B+βYiβ=BββYiβ as type V(iΒ 0iβ) intertwining operators, which we denote by YΞΊ(i)β and call the creation operator of Wiβ. Then the canonical isomorphism of the left multiplication by identity W0ββ WiβββWiβ is defined to be the one corresponding to Yiβ. Similarly the right multiplication by identity Wiββ W0βββWiβ is chosen to be ΞΊ(i).
One can also construct contragredient intertwining operatorsC+βYΞ±ββ‘YC+βΞ±β,CββYΞ±ββ‘YCββΞ±β of YΞ±β, which are of type (iΒ kjββ), such that for any w(i)βWiβ,w(j)βWjβ,w(k)βWkβ,
[TABLE]
Here, and also throughout this paper, we follow the convention argzr=rargz (rβR) unless otherwise stated. To express contragredient intertwining operators graphically, we first introduce, for any V-module Wiβ (together with its contragredient module Wiβ), two important intertwining operators Yevi,iβββV(iΒ i0β) and Yevi,iβββV(iΒ i0β), called the annihilation operators of Wiβ and Wiβ respectively. Recall that V is self dual. Fix an isomorphism V=W0ββW0β and identify W0β and W0β through this isomorphism. We now define
[TABLE]
The type of Yevi,iββ shows that evi,iββHom(Wiββ Wiβ,V), which plays the role of the evaluation map of Wiβ. evi,iββHom(Wiββ Wiβ,V) can be defined in a similar way. We write \mathrm{ev}_{i,\overline{i}}=\vbox{\hbox{{
\begin{picture}(1.0,0.48349)\put(0.0,0.0){\includegraphics[width=24.95526pt,page=1]{ev.pdf}}\end{picture}}}} and \mathrm{ev}_{\overline{i},i}=\vbox{\hbox{{
\begin{picture}(1.0,0.48349)\put(0.0,0.0){\includegraphics[width=24.95526pt,page=1]{ev-2.pdf}}\end{picture}}}}=\vbox{\hbox{{
\begin{picture}(1.0,0.48539)\put(0.0,0.0){\includegraphics[width=24.85754pt,page=1]{ev-3.pdf}}\end{picture}}}},
following the convention that a vertical line with label i but upward-pointing arrow means (the identity morphism of) Wiβ. We can now give a categorical description of CββΞ± with the help of the following fusion relation (cf. [Gui19b] remark 5.4)
[TABLE]
which can be translated to
[TABLE]
By [Hua08] section 3, there exist coevaluation maps \mathrm{coev}_{i,\overline{i}}=\vbox{\hbox{{
\begin{picture}(1.0,0.4939)\put(0.0,0.0){\includegraphics[width=24.429pt,page=1]{coev.pdf}}\end{picture}}}}\in\mathrm{Hom}(V,W_{i}\boxtimes W_{\overline{i}}) and \mathrm{coev}_{\overline{i},i}=\vbox{\hbox{{
\begin{picture}(1.0,0.4939)\put(0.0,0.0){\includegraphics[width=24.429pt,page=1]{coev-2.pdf}}\end{picture}}}}=\vbox{\hbox{{
\begin{picture}(1.0,0.48687)\put(0.0,0.0){\includegraphics[width=24.78175pt,page=1]{coev-3.pdf}}\end{picture}}}}\in\mathrm{Hom}(V,W_{\overline{i}}\boxtimes W_{i}) satisfying
[TABLE]
Thus, let Wjββ tensor both sides of equation (1.51) from the left, and then apply coevjβ,jββ1iββ1kβ to the tops, we obtain
Proposition 1.2**.**
For any V-modules Wiβ,Wjβ,Wkβ, and any YΞ±ββV(iΒ jkβ),
[TABLE]
Finally, we remark that the ribbon structure on Rep(V) is defined by the twist Ο=Οiβ:=e2iΟL0ββEnd(Wiβ) for any V-module Wiβ.
1.2 Unitary VOAs and unitary representations
In this section, we only assume that V is of CFT-type, and discuss the unitary conditions on V and its tensor category. We do not assume, at the beginning, that V is self-dual. In particular, we do not identify W0β with W0β. As we shall see, the unitary structure on V is closely related to certain V-module isomorphism Ο΅:W0ββW0β.
Note that the first and the third equations are acting on V, while the second one on V.
Proof.
Ο΅ and Ξ are related by Ο΅=βΞ. Then Ο΅v=Ξv,Ο΅βv=Ξβ1v. It is easy to see that (1.67) and (1.68) are equivalent to (1.64) and (1.66) respectively. The uniqueness of Ο΅ follows from that of Ξ. When V is unitary, βΟ΅=Ξ=Ξβ1=Ο΅ββ.
β
A V-module is called unitarizable if it can be equipped with an inner product under which it becomes a unitary V-module. An intertwining operator YΞ±β of V is called unitary if it is among unitary V-modules.
This equation (applied to i=0), together with (1.68) and (1.69), shows for any vβV that Ο΅Y0β(v,z)Ο΅β=Y0β(v,z), where Y0β=Y is the vertex operator of the vacuum module V, and Y0β is that of its contragredient module. We conclude:
Proposition 1.4**.**
If V is a unitary CFT-type VOA, then V is self-dual. The vacuum module V=W0β is unitarily equivalent to its contragredient module V=W0β via the reflection operator Ο΅.
Convention 1.5**.**
Unless otherwise stated, if V is unitary and CFT-type, the isomorphism W0βββW0β is always chosen to be the reflection operator Ο΅, and the identification of V=W0β and V=W0β is always assumed to be through Ο΅.
1.3 Adjoint and conjugate intertwining operators
We have seen in section 1.1 two ways of producing new intertwining operators from old ones: the braided and the contragredient intertwining operators. In the unitary case there are two extra methods: the conjugate and the adjoint intertwining operators. In this section, our main goal is to derive tensor-categorical descriptions of these two constructions of intertwining operators.
First we review the definition of these two constructions; see [Gui19a] section 1.3 for more details and basic properties. Let V be unitary and CFT-type. Choose unitary V-modules Wiβ,Wjβ,Wkβ. For any YΞ±ββV(iΒ jkβ), its conjugate intertwining operatorβYΞ±ββ‘YβΞ±ββ‘YΞ±β is of type V(iΒ jβkβ) defined by
[TABLE]
for any w(i)βWiβ. Despite its simple form, one cannot directly translate this definition into tensor categorical language, again due to the anti-linearity of β. Therefore we need to first consider the adjoint intertwining operatorYΞ±β ββ‘YΞ±β ββV(iΒ kjβ), defined by
[TABLE]
which will be closely related to the β-structures of the Cβ-tensor categories. Then for any w(i)βWiβ,
[TABLE]
Note that β is an involution: Ξ±β β =Ξ±. Moreover, by unitarity, up to equivalence of the charge spaces Ο΅:VββV, Yiβ is equal to its adjoint intertwining operator, and obviously also equal to the conjugate intertwining operator of Yiβ. Another important relation is
[TABLE]
by [Gui19a] formula (1.44). Recall from section 1.1 that, up to the isomorphism W0ββW0β, evi,iβ is defined to be CββΞΊ(i). Due to convention 1.5, the more precise definition is evi,iβ=Ο΅β1βCββΞΊ(i). We will use (1.75) more often than this definition in our paper.
We shall now relate Ξ±β with the unitarity structure of the tensor category of unitary V-modules. Let V be a CFT-type VOA. Assume that V is strongly unitary [Ten18a], which means that V is unitary, and any V-module is unitarizable. Then Repu(V), the category of unitary V-modules, is a Cβ-category, whose β structure is defined as follows: if Wiβ,Wjβ are unitary and TβHom(Wiβ,Wjβ), then TββHom(Wjβ,Wiβ) is simply the adjoint of T, defined with respect to the inner products of Wiβ and Wjβ.
Assume also that V is regular. To make Repu(V) a Cβ-tensor category, we have to choose, for any unitary V-modules Wiβ,Wjβ, a suitable unitary structure on Wiββ Wjβ. Note that it is already known that Wiββ Wjβ is unitarizable by the strong unitary of V. But here the unitary structure has to be chosen such that the structural isomorphisms become unitary. So, for instance, if Wkβ is also unitary, the associativity isomorphism (Wiββ Wjβ)β WkβββWiββ (Wjββ Wkβ) and the braid isomorphism \ss:Wiββ WjβββWjββ Wiβ have to be unitary. Then we can identify (Wiββ Wjβ)β Wkβ and Wiββ (Wjββ Wkβ) as the same unitary V-module called Wiββ Wjββ Wkβ.
To fulfill these purposes, recall that Wiββ Wjβ is defined to be β¨tβEβV(iΒ jtβ)ββWtβ. Since every Wtβ already has a unitary structure, it suffices to assume that the direct sum is orthogonal, and define a suitable inner product Ξ (called invariant inner product) on each V(iΒ jtβ)β.
We warn the reader the difference of the two notations Ξ±β and Ξ±β. If YΞ±ββV(iΒ jtβ) then Ξ±β βHom(Wiββ Wtβ,Wjβ), while Ξ±β, defined using the β-structure of Repu(V), is in Hom(Wtβ,Wiββ Wjβ). Thus, YΞ±β β is the adjoint intertwining operator while YΞ±ββ makes no sense. This is different from the notations used in [Gui19a, Gui19b], where Ξ±β is not defined but YΞ±ββ (also written as YΞ±β β) denotes the adjoint intertwining operator.
Despite such difference, Ξ±β and Ξ±β, the VOA adjoint and the categorical adjoint, should be related in a natural way. And it is time to review the construction of invariant Ξ introduced in [Gui19b].
Definition 1.7**.**
Let Wiβ,Wjβ be unitary V-modules. Then the invariant sesquilinear formΞ on V(iΒ jtβ)β for any tβE is defined such that the fusion relation holds for any w1(i)β,w2(i)ββWiβ:
It is not too hard to show that Ξ is Hermitian (i.e. Ξ(YβΞ±β£YβΞ²)=Ξ(YβΞ²β£YβΞ±)β). Indeed, one can prove this by applying [Gui19b] formula (5.34) to the adjoint of (1.77). (The intertwining operator YΟ~β in that formula could be determined from the proofs of [Gui19b] corollary 5.7 and theorem 5.5.) Moreover, from the rigidity of Rep(V) one can deduce that Ξ is non-degenerate; see [HK07] theorem 3.4,111The relation between the bilinear form in [HK07] theorem 3.4 and the sesquilinear form Ξ is explained in [Gui19b] section 8.3. or step 3 of the proof of [Gui19b] theorem 6.7. However, to make Ξ an inner product one has to prove that Ξ is positive. In [Gui19b, Gui19c] we have proved the positivity of Ξ for many examples of VOAs. As mentioned in the introduction, one of our main goal of this paper is to prove that all unitary extensions of these examples have positive Ξ. We first introduce the following definition.
Definition 1.8**.**
Let V be a regular and CFT-type VOA. We say that V is completely unitary, if V is strongly unitary (i.e., V is unitary and any V-module is unitarizable), and if for any unitary V-modules Wiβ,Wjβ and any tβE, the invariant sesquilinear form Ξ defined on V(iΒ jtβ)β is positive. In this case we call Ξ the invariant inner product of V.
As we have said, since Ξ is non-degenerate, when V is completely unitary Ξ becomes an inner product on V(iΒ jtβ)β (and hence also on V(iΒ jtβ)), which can be extended to an inner product on Wiββ Wjβ, also denoted by Ξ. Then Wiββ Wjβ becomes a unitary (but not just unitarizable) V-module. In other words it is an object not just in Rep(V) but also in Repu(V). Moreover, as shown in [Gui19b], Ξ is the right inner product which makes all structural isomorphisms unitary. More precisely:
If V is regular, CFT-type, and completely unitary, then Repu(V) is a unitary modular tensor category.
In the remaining part of this paper, we assume, unless otherwise stated, that V is regular, CFT-type, and completely unitary. The following relation is worth noting; see [Gui19b] section 7.3.
Proposition 1.10**.**
If Wiβ is unitary then coevi,iβ=evi,iββ.
We are now ready to state the main results of this section.
Theorem 1.11**.**
For any unitary V-modules Wiβ,Wjβ,Wkβ and any YΞ±ββV(iΒ jkβ), we have Ξ±β =(evi,iββ1jβ)(1iββΞ±β). In other words,
[TABLE]
Proof.
It suffices to assume Wkβ to be irreducible. So let us prove (1.98) for all k=tβE and any basis vector Ξ±βΞi,jtβ. Here we assume Ξi,jtβ to be orthonormal under Ξ. Choose Ξ±βHom(Wiββ Wjβ,Wtβ) such that Ξ±β equals (evi,iββ1jβ)(1iββΞ±β). We want to show Ξ±=Ξ±.
By proposition 1.6 we have Ξ±Ξ²β=δα,Ξ²β1tβ for any Ξ±,Ξ²βΞi,jtβ. This implies
[TABLE]
Tensor 1iβ from the left and apply evi,iββ1jβ to the bottom, we obtain
For any unitary V-modules Wiβ,Wjβ,Wkβ and any YΞ±ββV(iΒ jkβ),
[TABLE]
Proof.
We have Ξ±=C+βCββΞ±β=(CββΞ±)β . By propositions 1.2, 1.10, and the unitarity of \ss, (CββΞ±)β equals
[TABLE]
By theorem 1.11, one obtains (CββΞ±)β from (CββΞ±)β by bending the leg i to the top. Thus (1.133) becomes the right hand side of (1.128).
β
2 Unitary VOA extensions, Cβ-Frobenius algebras, and their representations
2.1 Preunitary VOA extensions and commutative C-algebras
The above discussion can be summarized as follows: The preunitary VOA extension U is a unitary V module (Waβ,Yaβ); its vertex operator YΞΌβ is in V(aΒ aaβ). Thus \mu\equiv\vbox{\hbox{{
\begin{picture}(1.0,1.41536)\put(0.0,0.0){\includegraphics[width=21.75864pt,page=1]{C-algebra.pdf}}\end{picture}}}}\in\mathrm{Hom}(W_{a}\boxtimes W_{a},W_{a}) by our notation in the last chapter. Let ΞΉ:W0ββWaβ denote the embedding of V into U. Then clearly ΞΉβHom(W0β,Waβ).
We write \iota=\vbox{\hbox{{
\begin{picture}(1.0,1.94741)\put(0.0,0.0){\includegraphics[width=8.81326pt,page=1]{C-algebra-2.pdf}}\end{picture}}}}~{}~{}. Set AUβ=(Waβ,ΞΌ,ΞΉ).
Then AUβ is a commutative associative algebra in Repu(V) (or commutative Repu(V)-algebra for short) [Par95, KO02], which means:
β’
(Associativity) ΞΌ(ΞΌβ1aβ)=ΞΌ(1aββΞΌ).
β’
(Commutativity) ΞΌβ\ss=ΞΌ.
β’
(Unit) ΞΌ(ΞΉβ1aβ)=1aβ.
Note that the associators and the unitors of Repu(V) have been suppressed to simplify discussions. We will also do so in the remaining part of this article.
Recall that \ss is the braid isomorphism \ssa,aβ:Waββ WaββWaββ Waβ. These three conditions can respectively be pictured as
[TABLE]
Indeed, associativity and commutativity are equivalent to the Jacobi identity for YΞΌβ. The unit property follows from the fact that YΞΌβ restricts to Yaβ. See [HKL15] for more details. Note that we also have
[TABLE]
for any nβZ. The first equation follows from induction and that ΞΌ=ΞΌ\ss\ssβ1=ΞΌ\ssβ1. The second equation holds because ΞΌ(1aββΞΉ)=ΞΌ\ssa,aβ(1aββΞΉ)=ΞΌ(ΞΉβ1aβ)\ssa,0β=1aβ\ssa,0β=1aβ.
A unitary VOA extension is clearly preunitary. Another useful fact is the following:
Proposition 2.3**.**
If U is a CFT-type unitary VOA extension of V, then the PCT operator ΞUβ of U restricts to the one ΞVβ of V. In particular, V is a ΞUβ-invariant subspace of U.
Thus we can let Ξ denote unambiguously both the PCT operators of U and of V.
Proof.
By relation (1.64) and the fact that ΞU2β=1Uβ,ΞV2β=1Vβ, for any vβVβU we have
[TABLE]
when evaluating between vectors in V. It should be clear to the reader how the condition that the normalized inner product of U restricts to that of V is used in the above equations. Thus ΞUββ£Vβ=ΞVβ.
β
2.2 Duals and standard evaluations in Cβ-tensor categories
Dualizable objects
Let C be a Cβ-tensor category (cf. [Yam04]) whose identity object W0β is simple. We assume tacitly that C is closed under finite orthogonal direct sums and orthogonal subojects, which means that for a finite collection {Wsβ:sβS} of objects in C there exists an object Wiβ and partial isometries {usββHom(Wiβ,Wsβ):sβS} satisfying utβusββ=Ξ΄s,tβ1sβ (βs,tβS) and βsβSβusββusβ=1iβ, and that for any object Wiβ and a projection pβEnd(Wiβ) there exists an object Wjβ and a partial isometry uβHom(Wiβ,Wjβ) such that uuβ=1jβ,uβu=p.222A morphism uβHom(Wiβ,Wjβ) is called a partial isometry if uβu and uuβ are projections. A morphism eβEnd(Wiβ) is called a projection if e2=e=eβ.
Assume that an object Wiβ in C has a right dual Wiβ, which means that there exist evaluation eviββHom(Wiββ Wiβ,W0β) and coevaluation coeviββHom(W0β,Wiββ Wiβ) satisfying (1iββeviβ)(coeviββ1iβ)=1iβ and (eviββ1iβ)(1iββcoeviβ)=1iβ. Set evi,iβ=eviβ,coevi,iβ=coeviβ, and set also evi,iβ=(coevi,iβ)β,coevi,iβ=(evi,iβ)β. Then equations (1.52) and (1.53) are satisfied, which shows that Wiβ is also a left dual of Wiβ. In this case we say that Wiβ is dualizable. Note that evi,iβ determines the remaining three ev and coev. In general, we say that evi,iββHom(Wiββ Wiβ,W0β),evi,iββHom(Wiββ Wiβ,W0β) are evaluations (or simply ev) of Wiβ and Wiβ if equations (1.52) and (1.53) are satisfied when setting coevi,iβ=evi,iββ,coevi,iβ=evi,iββ. In the case that Wiβ is self-dual, we say that evi,iβ is an evaluation (ev) of Wiβ, if, by setting coevi,iβ=evi,iββ, we have (evi,iββ1iβ)(1iββcoevi,iβ)=1iβ. Taking adjoint, we also have (1iββevi,iβ)(coevi,iββ1iβ)=1iβ.
Assume that Wiβ,Wjβ are dualizable with duals Wiβ,Wjββ respectively. Choose evaluations evi,iβ,evi,iβ,evj,jββ,evjβ,jβ. Then Wiβ jβ:=Wiββ Wjβ is also dualizable with a dual Wjββ iβ:=Wjβββ Wiβ and evaluations
[TABLE]
Convention 2.4**.**
Unless otherwise stated, if the ev for Wiβ,Wiβ and Wjβ,Wjββ are chosen, then we always define the ev for Wiβ jβ,Wjββ iβ using equations (2.2).
Using ev and coev for Wiβ,Wjβ and their duals Wiβ,Wjββ, one can define for any FβHom(Wiβ,Wjβ) a pair of transposes Fβ¨,β¨F by
[TABLE]
Pictorially,
[TABLE]
One easily checks that β¨(Fβ¨)=F=(β¨F)β¨, (FG)β¨=Gβ¨Fβ¨,β¨(FG)=(β¨G)(β¨F),(Fβ¨)β=β¨(Fβ).
Standard evaluations
The evaluations defined above are not unique even up to unitaries. For any evi,iβ,evi,iβ of Wiβ,Wiβ, and any invertible KβEnd(Wiβ), evi,iβ:=evi,iβ(Kβ1iβ) and evi,iβ:=evi,iβ(1iββ(Kβ)β1) are also evaluations. Thus one can normalize evaluations to satisfy certain nice conditions. In Cβ-tensor categories, the ev that attract most interest are the so called standard evaluations. It is known that for dualizable objects, standard ev always exist and are unique up to unitaries, and that the two transposes defined by standard ev are equal. We refer the reader to [LR97, Yam04, BDH14] for these results. In the following, we review an explicit method of constructing standard evaluations following [Yam04]. Since the evaluations for VOA tensor categories defined in the previous chapter can be realized by this construction, these evaluations are standard (see proposition 2.5).
Define scalars TrLβ(F) and TrRβ(F) for each FβEnd(Wiβ) such that evi,iβ(Fβ1iβ)coevi,iβ=TrLβ(F)10β and evi,iβ(1iββF)coevi,iβ=TrRβ(F)10β. Then TrLβ is a positive linear functional on End(Wiβ). Moreover, TrLβ is faithful: if TrLβ(FβF)=0, then evi,iββ(Fβ1iβ) is zero (since its absolute value is zero). So F=0. Similar things can be said about TrRβ. We say that evi,iβ is a standard evaluation if TrLβ(F)=TrRβ(F) for all FβEnd(Wiβ). Since TrLβ(Fβ¨)=TrRβ(F)=TrLβ(Fβ¨) by easy graphical calculus, it is easy to see that evi,iβ is standard if and only if evi,iβ is so. Standard ev are unique up to unitaries: If Wiβ²ββWiβ,TβHom(Wiβ²β,Wiβ) is unitary, and evi,iβ²β,eviβ²,iβ are also standard, then we may find a unitary KβEnd(Wiβ) satisfying evi,iβ²β=evi,iβ(KβT),eviβ²,iβ=evi,iβ(TβK). (See [Yam04] lemma 3.9-(iii).)
If Wiβ is simple, a standard evaluation is easy to construct by multiplying evi,iβ by some nonzero constant Ξ» (and hence multiplying evi,iβ by Ξ»β1) such that TrLβ(1iβ)=TrRβ(1iβ). In general, if Wiβ is dualizable and hence semisimple, we have orthogonal irreducible decomposition Wiβββ¨sβSβ₯βWsβ where each irreducible subobject Wsβ is dualizable (with a dual Wsβ). Choose partial isometries {usββHom(Wiβ,Wsβ):sβS} and {vsββHom(Wiβ,Wsβ):sβS} satisfying utβusββ=Ξ΄s,tβ1sβ,vtβvsββ=Ξ΄s,tβ1sβ and βsβusββusβ=1iβ,βsβvsββvsβ=1iβ. Then we define
[TABLE]
(where evs,sβ and evs,sβ are standard for all s), define coev using adjoint. Then evi,iβ and evi,iβ are standard. (See [Yam04] lemma 3.9 for details.)333In [Yam04] the categories are assumed to be rigid. Thus any orthogonal subobject of Wiβ, i.e., any object Wsβ which is a associated with a partial isometry u:WiββWsβ satisfying uuβ=1sβ, is dualizable. This fact is also true without assuming C to be rigid. (Cf. [ABD04] lemma 4.20.) Here is one way to see this. Notice that we may assume TrLβ is tracial by multiplying evi,iβ by Kβ1iβ where K is a positive invertible element of End(Wiβ). (Cf. the proof of [ABD04] Thm. 4.12.) Thus, for any F,GβEnd(Wiβ), we have TrLβ(GF)=TrLβ(Fβ¨β¨β G) in general and TrLβ(FG)=TrLβ(GF) by tracialness, which shows Fβ¨β¨=F and hence Fβ¨=β¨F. Thus, (Fβ¨)β=(Fβ)β¨. So F is a projection iff Fβ¨ is so. Let p=uβu. Then pβ¨βEnd(Wiβ) is a projection. Thus, there exist an object Wsβ and vβHom(Wiβ,Wsβ) satisfying vvβ=1sβ,vβv=pβ¨. Then Wsβ is dual to Wsβ since one can choose evaluations evs,sβ:=evi,iβ(uββvβ),evs,sβ:=evi,iβ(vββuβ).
Proposition 2.5**.**
If V is a regular, CFT-type, and completely unitary VOA, then the ev and coev defined in chapter 1 (same as those in [Gui19a, Gui19b]) for any object Wiβ in Repu(V) and its contragredient module Wiβ are standard.
Proof.
First of all, assume Wiβ is irreducible. By [Gui19b] proposition 7.7 and the paragraph before that, we have TrLβ(1iβ)=diβ=diβ=TrRβ(1iβ). So evi,iβ is standard. Now, assume Wiβ is semisimple with finite orthogonal irreducible decomposition Wiβ=β¨sβ₯βWsβ. For each irreducible summand Wsβ, define usβ (resp. vsβ) to be the projection of Wiβ onto Wsβ (resp. Wiβ onto Wsβ). (Note that Wiβ and Wsβ are respectively the contragredient modules of Wiβ,Wsβ.) Using (1.40), it is easy to see that Yevi,iββ(w,z)=βsβYevs,sββ(usβw,z)vsβ for any wβWiβ. So the first equation of (2.4) is satisfied. Similarly, the second one of (2.4) holds.
β
Convention 2.6**.**
Unless otherwise stated, for any object Wiβ in Repu(V), Wiβ is always understood as the contragredient module of Wiβ, and the standard ev and coev for Wiβ,Wiβ are always defined as in chapter 1.
It is worth noting that standardness is preserved by tensor products: If standard ev are chosen for Wiβ,Wjβ and their dual Wiβ,Wjββ, then the ev of Wiβ jβ,Wjββ iβ defined by (2.2) are also standard.
Standard ev are also characterized by minimizing quantum dimensions. Define constants diβ,diβ satisfying evi,iβcoevi,iβ=diβ10β,evi,iβcoevi,iβ=diβ10β. Then standard ev are precisely those minimizing diβdiβ and satisfying diβ=diβ (cf. [LR97]). We will always assume diβ,diβ to be those defined by standard evaluations, and call them the quantum dimensions of Wiβ,Wiβ.
2.3 Unitarity of C-algebras and VOA extensions
Let Waβ be an object in C, choose ΞΌβHom(Waββ Waβ,Waβ),ΞΉβHom(W0β,Waβ), and asume that A=(Waβ,ΞΌ,ΞΉ) is an associative algebra in C (also called C-algebra444In [KO02, HKL15, CKM17], commutativity is required in the definition of C-algebras when C is braided. This is not assumed in our paper.), which means:
β’
(Associativity) ΞΌ(ΞΌβ1aβ)=ΞΌ(1aββΞΌ).
β’
(Unit) ΞΌ(ΞΉβ1aβ)=1aβ=ΞΌ(1aββΞΉ).
Since W0β is simple, we can choose DAβ>0 (called the quantum dimension of A) satisfying ΞΉβΞΌΞΌβΞΉ=DAβ10β. We say that A is
β’
haploid if dimHom(W0β,Waβ)=1;
β’
normalized if ΞΉβΞΉ=10β;
β’
special if ΞΌΞΌββC1aβ; in this case we set scalar dAβ>0 such that ΞΌΞΌβ=dAβ1aβ;
β’
standard if A is special, Waβ is dualizable (with quantum dimension daβ), and DAβ=daβ.
Note that any C-algebra A is clearly equivalent to a normalized one.
Assume that Waβ has a dual Waβ, together with (not necessarily standard) eva,aβ,eva,aβ. Define ev for Waββ Waβ and Waββ Waβ using (2.2). Assume also that Waβ is self-dual, i.e., WaββWaβ. Choose a unitary morphism \varepsilon=\vbox{\hbox{{
\begin{picture}(1.0,2.35805)\put(0.0,0.0){\includegraphics[width=10.82346pt,page=1]{reflection.pdf}}\end{picture}}}}~{}~{}\in\mathrm{Hom}(W_{a},W_{\overline{a}}), and write its adjoint as \varepsilon^{*}=\vbox{\hbox{{
\begin{picture}(1.0,2.35805)\put(0.0,0.0){\includegraphics[width=10.82346pt,page=1]{reflection-2.pdf}}\end{picture}}}}~{}~{}.
Write also \mu^{*}=\vbox{\hbox{{
\begin{picture}(1.0,1.01846)\put(0.0,0.0){\includegraphics[width=27.85576pt,page=1]{C-algebra-9.pdf}}\end{picture}}}}~{}~{}, \iota^{*}=\vbox{\hbox{{
\begin{picture}(1.0,1.59352)\put(0.0,0.0){\includegraphics[width=10.34084pt,page=1]{C-algebra-10.pdf}}\end{picture}}}}~{}~{}.
Definition 2.7**.**
A unitary Ξ΅βHom(Waβ,Waβ) is called a reflection operator of A (with respect to the chosen ev of Waβ,Waβ), if the following two equations are satisfied:
[TABLE]
Pictorially,
[TABLE]
Proposition 2.8**.**
The reflection operator Ξ΅ is uniquely determined by the dual Waβ and the ev of Waβ and Waβ.
Proof.
Apply ΞΉβ to the bottom of (2.12), and then apply the unit property, we have eva,aβ(Ξ΅β1aβ)=ΞΉβΞΌ, which, by rigidity, implies
[TABLE]
β
Definition 2.9**.**
Let C be a Cβ-tensor category with simple W0β. A C-algebra A=(Waβ,ΞΌ,ΞΉ) in C is called unitary, if Waβ is dualizable, and for a choice of Waβ dual to Waβ and ev of Waβ,Waβ, there exists a reflection operator Ξ΅βHom(Waβ,Waβ). If, moreover, the ev of Waβ,Waβ are standard, we say that A is s-unitary555The letter βsβ stands for several closely related notions: standard evaluations, spherical tensor categories, symmetric Frobenius algebras [FRS02]..
The definition of s-unitary C-algebras is independent of the choice of duals and standard ev, as shown below:
Proposition 2.10**.**
If A is s-unitary, then for any Waβ²β dual to Waβ, and any standard ev of Waβ,Waβ²β, there exists a reflection operator Ξ΅:WaββWaβ²β
Proof.
Since A is s-unitary, we can choose Waβ dual to Waβ and standard ev of Waβ,Waβ such that there exists a reflection operator Ξ΅:WaββWaβ. Let us define Ξ΅=(ΞΉβΞΌβ1aβ)(1aββcoeva,aβ²β) and show that Ξ΅ is a reflection operator. We first show that Ξ΅ is unitary. Choose a unitary TβHom(Waβ²β,Waβ). Then by the up to unitary uniqueness of standard ev, there exists a unitary KβHom(Waβ,Waβ) such that eva,aβ²β=eva,aβ(KβT),evaβ²,aβ=eva,aβ(TβK). By (2.3), eva,aβ(Kβ1aβ)=eva,aβ(1aββKβ¨). Therefore eva,aβ²β=eva,aβ(1aββ(Kβ¨)T). Thus Ξ΅=(ΞΉβΞΌβ1aβ)(1aββTβ(Kβ¨)β)(1aββcoeva,aβ)=(10ββTβ(Kβ¨)β)(ΞΉβΞΌβ1aβ)(1aββcoeva,aβ), which, together with (2.20), implies Ξ΅=Tβ(Kβ¨)βΞ΅. Thus Ξ΅ is unitary since Ξ΅,Kβ¨,T are unitary.
Now, from the definition of Ξ΅, we see that evaβ²,aβ(Ξ΅β1aβ)=ΞΉβΞΌ. Therefore evaβ²,aβ(Ξ΅β1aβ)=eva,aβ(Ξ΅β1aβ). Using this fact, one can now easily check that \eqrefeq23 and \eqrefeq24 hold for Ξ΅ and the standard ev of Waβ,Waβ²β.
β
We now relate s-unitary C-algebras with unitary VOA extensions. First we need a lemma.
Lemma 2.11**.**
Let U be a preunitary CFT-type extension of V, AUβ=(Waβ,ΞΌ,ΞΉ) the associated Repu(V)-algebra, and Waβ the contragredient module of Waβ. Then U is a unitary VOA if and only if there exists a unitary Ξ΅βHom(Waβ,Waβ) satisfying for all w(a)βWaβ that
[TABLE]
Proof.
Suppose that such Ξ΅ exists, then by proposition 1.3 and the definition of adjoint and conjugate intertwining operators, U is unitary. Conversely, if U is unitary, then by proposition 1.3 there exists a unitary map Ξ΅:WaββWaβ such that (2.21) and (2.22) are true. Moreover, by proposition 1.4, Ξ΅ is a homomorphism of U-modules. So it is also a homomorphism of V-modules. This proves Ξ΅βHom(Waβ,Waβ).
β
The following is the main result of this section.
Theorem 2.12**.**
Let V be a regular, CFT-type, and completely unitary VOA. Let U be a CFT-type preunitary extension of V, and let AUβ=(Waβ,ΞΌ,ΞΉ) be the haploid commutative Repu(V)-algebra associated to U. Then U is unitary if and only if AUβ is s-unitary.
Proof.
Following convention 2.6, we let Waβ be the contragredient V-module of Waβ, and choose standard ev for Waβ,Waβ as in chapter 1. By lemma 2.11 and relation (1.2), the unitarity of U is equivalent to the existence of a unitary Ξ΅βHom(Waβ,Waβ) satisfying
[TABLE]
Now, by theorem 1.11, ΞΌβ =(eva,aββ1aβ)(1aββΞΌβ). Thus ΞΌβ (Ξ΅β1aβ)=(eva,aββ1aβ)(Ξ΅βΞΌβ). Therefore the first equation of (2.23) is equivalent to the first one of (2.5). By corollary 1.12, ΞΌβ=(\ssa,aβΞΌβ)β¨=((ΞΌ\ssa,aβ1β)β)β¨, which equals (ΞΌβ)β¨ by the commutativity of AUβ. Therefore the second equation of (2.23) is also equivalent to that of (2.5). We conclude that Ξ΅ satisfies (2.23) if and only if Ξ΅ is a reflection operator. Thus the unitarity of U is equivalent to the existence of a reflection operator under the standard ev, which is precisely the s-unitarity of AUβ.
β
2.4 Unitary C-algebras and Cβ-Frobenius algebras
Let A=(Waβ,ΞΌ,ΞΉ) be a C-algebra. A is called a Cβ-Frobenius algebra in C if (1aββΞΌ)(ΞΌββ1aβ)=ΞΌβΞΌ. By taking adjoint we have the equivalent condition ΞΌβΞΌ=(ΞΌβ1aβ)(1aββΞΌβ). Assume in this section that all line segments in the pictures are labeled by a. Then these two equations read
[TABLE]
A special (i.e. ΞΌΞΌββC1aβ) Cβ-Frobenius algebra is called a Q-system.666We warn the reader that in the literature there is no agreement on whether standardness is required in the definition of Q-systems. For example, the Q-systems in [BKLR15] are in fact standard Q-systems in our paper. We remark that the C-algebra A is a Q-system if and only if it is special. In other words, the Frobenius relations (2.24) are consequences of the unit property, the associativity, and the specialness of A; see [LR97] or [BKLR15] lemma 3.7.
The main goal of this section is to relate (s-)unitarity to Frobenius property. More precisely, we shall show
[TABLE]
Moreover, under the assumption of haploid condition, all these notions are equivalent. In the process of the proof we shall also see that (2.19) is a consequence of (2.12). Thus the definition of reflection operator can be simplified to assume only (2.12).
To begin with, let us fix a dual Waβ of Waβ together with evaluations eva,aβ,eva,aβ of Waβ,Waβ. Choose a unitary Ξ΅βHom(Waβ,Waβ).
Proposition 2.13**.**
If Ξ΅ satisfies (2.12), then eva,aβ(Ξ΅β1aβ)=eva,aβ(1aββΞ΅); equivalently,
[TABLE]
Proof.
Take the adjoint of (2.12) and apply (Ξ΅β1aβ) to the bottom, we have
[TABLE]
Bending the left legs to the top proves that ΞΌ equals
[TABLE]
We thus see that the right hand side of (2.12) equals (2.27). Finally, we apply ΞΉβ to their bottoms and use the unit property. This proves equation (2.26).
β
Corollary 2.14**.**
If Ξ΅ and the evaluations eva,aβ,eva,aβ of Waβ,Waβ satisfy (2.12), then there exists an evaluation eva,aβ for the self-dual object Waβ such that Ξ΅:=1aββEnd(Waβ) and eva,aβ also satisfy (2.12). Moreover, if eva,aβ,eva,aβ are standard, then one can also choose eva,aβ to be standard.
Proof.
We remind the reader that the definition of the evaluations of a self-dual object is given at the beginning of section 2.2. Assuming that Ξ΅:WaββWaβ satisfies (2.12), we simply define eva,aββHom(Waββ Waβ,W0β) to be the left and also the right hand side of (2.26), i.e.,
[TABLE]
Then one easily checks that (eva,aββ1aβ)(1aββcoeva,aβ)=1aβ where coeva,aβ:=(eva,aβ)β, and that 1aβ and eva,aβ also satisfy (2.12). If the ev for Waβ,Waβ are standard, then eva,aβ is also standard by the unitarity of Ξ΅.
β
We shall write eva,aβ as eva,aβ instead. Then the above corollary says that when (2.12) holds, we may well assume that a=a, eva,aβ=eva,aβ (which is written as eva,aβ), and Ξ΅=1aβ. Pictorially, we may remove the arrows and the β on the strings to simplify calculations. Moreover, by (2.12) and the unit property one has eva,aβ=ΞΉβΞΌ:
[TABLE]
We now prove the main results of this section. Recall that C is a Cβ-tensor category with simple W0β and A=(Waβ,ΞΌ,ΞΉ) is a C-algebra.
Theorem 2.15**.**
A* is a unitary C-algebra if and only if A is a Cβ-Frobenius algebra in C.*
Theorem 2.16**.**
If there exists Waβ dual to Waβ, evaluations eva,aβ,eva,aβ of Waβ,Waβ, and a unitary Ξ΅βHom(Waβ,Waβ) satisfying (2.12), then Ξ΅ also satisfies (2.19). Consequently, Ξ΅ is a reflection operator, and A is unitary.
We prove the two theorems simultaneously.
Proof.
Step 1. Suppose there exists a dual object Waβ, evaluations of Waβ,Waβ, and a unitary morphism Ξ΅:WaββWaβ satisfying (2.12). We
assume that a=a, eva,aβ:=eva,aβ=eva,aβ is an evaluation of Waβ, and Ξ΅=1aβ. Then
[TABLE]
where we have used successively equation (2.12), associativity, and again equation (2.12) in the above equations. This proves the first and hence also the second equation of (2.24). We conclude that A is a Cβ-Frobenius algebra.
Step 2. Assume that A is a Cβ-Frobenius algebra in C. We shall show that Waβ is self-dual, and construct a reflection operator. Define eva,aββHom(Waββ Waβ,W0β) to be eva,aβ=ΞΉβΞΌ (see (2.29)). One then easily verifies (eva,aββ1aβ)(1aββ(eva,aβ)β)=1aβ by applying respectively 1aββΞΉ and ΞΉββ1aβ to the top and the bottom of the second equation of (2.24), and then applying the unit property. This shows that Waβ is self-dual and eva,aβ=ΞΉβΞΌ is an evaluation of Waβ. Therefore we can also omit arrows. Apply ΞΉββ1aβ and 1aββΞΉβ to the bottoms of the second and the first equation of (2.24) respectively, and then use the unit property and equation (2.29), we obtain
[TABLE]
which proves that 1aβ and eva,aβ satisfy equation (2.12). Since the first and the third items of (2.30) are equal, we take the adjoint of them and bend their left or right legs to the top to obtain
[TABLE]
In other words, ΞΌ is invariant under clockwise and anticlockwise β1-click rotationsβ. Thus ΞΌ equals the clockwise 1-click rotation of the left hand side of (2.30), which proves (2.19) for 1aβ and eva,aβ. By (2.28), Ξ΅ and the original evaluations eva,aβ,eva,aβ also satisfy equation (2.19). Therefore Ξ΅ is a reflection operator of A with respect to Waβ and the given evaluations of Waβ,Waβ. This finishes the proof of the two theorems.
β
Corollary 2.17**.**
A* is a special unitary C-algebra if and only if A is a Q-system.*
Proof.
Q-systems are by definition special Cβ-Frobenius algebras.
β
We now relate s-unitarity and standardness. We have seen that if A is unitary then eva,aβ=ΞΉβΞΌ is an evaluation of Waβ. Therefore, setting coeva,aβ=eva,aββ, we have eva,aβcoeva,aβ=DAβ10β by the definition of DAβ. By the minimizing property of standard evaluations, we have DAββ₯daβ, with equality holds if and only if eva,aβ is standard. Note that 1aβ is a reflection operator with respect to eva,aβ. Therefore we have the following:
Proposition 2.18**.**
Let A be unitary. Then eva,aβ:=ΞΉβΞΌ is an evaluation of the self-dual object Waβ, and DAββ₯daβ. Moreover, we have DAβ=daβ if and only if A is s-unitary.
Corollary 2.19**.**
A* is a special s-unitary C-algebra if and only if A is a standard Q-system.*
Proof.
By theorem 2.15 and the definition of Q-systems, A is a standard Q-system if and only if A is a special unitary C-algebra satisfying DAβ=daβ. Thus the corollary follows immediately from the above proposition.
β
Thus weβve finished proving the relations (2.25) given at the beginning of this section.
Proposition 2.20**.**
Assuming haploid condition, the six notions in (2.25) are equivalent.
Proof.
Let A be a haploid Cβ-Frobenius algebra in C. Then by [BKLR15] lemma 3.3, A is special. The standardness follows from [LR97] section 6 (see also [MΓΌg03] remark 5.6-3, or [NY18a] theorem 2.9). Therefore A is a standard Q-system.
β
We can now restate the main result of the last section (theorem 2.12) in the following way:
Theorem 2.21**.**
Let V be a regular, CFT-type, and completely unitary VOA. Let U be a CFT-type preunitary extension of V, and let AUβ=(Waβ,ΞΌ,ΞΉ) be the haploid commutative Repu(V)-algebra associated to U. Then U is a unitary extension of V if and only if AUβ is a Cβ-Frobenius algebra. If this is true then AUβ is also a standard Q-system.
Let us give an application of this theorem.
Corollary 2.22**.**
The c<1 CFT-type unitary VOAs are in one to one correspondence with the irreducibile conformal nets with the same central charge c. Their classifications are given by [KL04] table 3.
Proof.
As shown in [KL04] proposition 3.5, c<1 irreducible conformal nets are precisely irreducible finite-index extensions of the Virasoro net Acβ with central charge c. Thus they are in 1-1 correspondence with the haploid commutative Q-systems in Repss(Acβ), where Repss(Acβ) is the unitary modular tensor category of the semisimple representations of Acβ. By [Gui20] theorem 5.1, Repss(Acβ) is unitarily equivalent to Repu(V), where V is the unitary Virasoro VOA with central charge c. By [HKL15] theorem 3.2, haploid commutative Repu(V)-algebras with trivial twist are in 1-1 correspondence with CFT-type extensions of V. Thus, by theorem 2.21 and the equivalence of unitary modular tensor categories, CFT-type unitary extensions of Vβ haploid commutative Q-systems in Repss(Acβ). (The trivial twist condition is redundant; see theorem 3.25.) But also CFT-type unitary extensions of Vβ unitary VOAs with central charge c by [DL14] theorem 5.1. This proves the desired result.
β
2.5 Strong unitarity of unitary VOA extensions
Starting from this section, A=(Waβ,ΞΌ,ΞΉ) is assumed to be a unitary C-algebra, or equivalently, a Cβ-Frobenius algebra in C. We say that (Wiβ,ΞΌLβ) (resp. (Wiβ,ΞΌRβ))777Later we will write ΞΌLβ and ΞΌRβ as ΞΌLiβ and ΞΌRiβ to emphasize the dependence of ΞΌLβ,ΞΌRβ on the Wiβ. is a left A-module (resp. right A-module), if Wiβ is an object in C, and ΞΌLββHom(Waββ Wiβ,Wiβ) (resp. ΞΌRββHom(Wiββ Waβ,Wiβ)) satisfies the unit property
[TABLE]
and the associativity:
[TABLE]
We write \mu_{L}=\vbox{\hbox{{
\begin{picture}(1.0,1.50253)\put(0.0,0.0){\includegraphics[width=21.75864pt,page=1]{module.pdf}}\end{picture}}}}~{}~{},~{}\mu_{L}^{*}=\vbox{\hbox{{
\begin{picture}(1.0,1.15379)\put(0.0,0.0){\includegraphics[width=26.44275pt,page=1]{module-2.pdf}}\end{picture}}}}~{}~{},~{}\mu_{R}=\vbox{\hbox{{
\begin{picture}(1.0,1.48859)\put(0.0,0.0){\includegraphics[width=21.75864pt,page=1]{module-3.pdf}}\end{picture}}}}~{}~{},~{}\mu_{R}^{*}=\vbox{\hbox{{
\begin{picture}(1.0,1.14538)\put(0.0,0.0){\includegraphics[width=27.06047pt,page=1]{module-4.pdf}}\end{picture}}}}~{}~{}.
If (Wiβ,ΞΌLβ) is a left A-module and (Wiβ,ΞΌRβ) is a right A-module, we say that (Wiβ,ΞΌLβ,ΞΌRβ) is an A-bimodule if the following bimodule associativity holds:
[TABLE]
We leave it to the reader to draw the pictures of associativity and unit property. We abbreviate (Wiβ,ΞΌLβ), (Wiβ,ΞΌRβ), or (Wiβ,ΞΌLβ,ΞΌRβ) to Wiβ when no confusion arises.
Set eva,aβ=ΞΉβΞΌ as in the last section. A left (resp. right) A-module (Wiβ,ΞΌLβ) (resp. (Wiβ,ΞΌRβ)) is called unitary, if
[TABLE]
An A-bimodule (Waβ,ΞΌLβ,ΞΌRβ) is called unitary if (Waβ,ΞΌLβ) is a unitary left A-module and (Waβ,ΞΌRβ) is a unitary right A-module. Unitarity can be stated for any evaluations and reflection operators:
Proposition 2.23**.**
Let Waβ be dual to Waβ, eva,aβ,eva,aβ evaluations of Wa,aβ,Wa,aβ, and Ξ΅:WaββWaβ a reflection operator. Then a left (resp. right) A-module (Waβ,ΞΌLβ) (resp. (Waβ,ΞΌRβ)) is unitary if and only if
[TABLE]
Graphically,
[TABLE]
Proof.
This is obvious since we have equations (2.28).
β
If Wiβ and Wjβ are left (resp. right) A-modules, then a morphism FβHom(Wiβ,Wjβ) of C is called a left (resp. right) A-module morphism, if
The category of unitary left A-modules (resp. right A-modules, A-bimodules) is a Cβ-category whose β-structure inherits from that of C. In particular, this category is closed under finite orthogonal direct sums and subobjects.
Proof.
If Wiβ,Wjβ are unitary left A-modules and FβHomA,ββ(Wiβ,Wjβ), one can easily check that FββHomA,ββ(Wjβ,Wiβ) using figures (2.49) and (2.67). Hence, the Cβ-ness of the category of left A-modules follows from that of C. Existence of finite orthogonal direct sums follow from that of C. If pβEndA,ββ(Wiβ) is a projection of the unitary left A-module (Wiβ,ΞΌLiβ), we choose an object Wkβ in C and a partial isometry uβHom(Wiβ,Wkβ) such that uuβ=1kβ,uβu=p. Then (Wkβ,uΞΌLiβuβ) is easily verified to be a unitary left A-module. Note that the fact that p intertwines the left action of A is used to verify the associativity.
The cases of right modules and bimodules are proved in a similar way.
β
A left A-module (resp. right A-module, A-bimodule) Wiβ is called C-dualizable if Wiβ is a dualizable object in C. Wiβ is called unitarizable if there exists a unitary left A-module (resp. right A-module, A-bimodule) Wjβ and an invertible FβHomA,ββ(Wiβ,Wjβ) (resp. FβHomβ,Aβ(Wiβ,Wjβ), FβHomAβ(Wiβ,Wjβ)).
Corollary 2.25**.**
The category of unitary C-dualizable left A-modules (resp. right A-modules, A-bimodules) is a semisimple Cβ-category whose β-structure inherits from that of C.
Proof.
Suppose Wiβ is a C-dualizable left A-module. Then EndAβ(Wiβ) is a Cβ-subalgebra of the finite dimensional Cβ-algebra E(Wiβ). Thus EndAβ(Wiβ) is a direct sum of matrix algebras which implies that Wiβ is a finite orthogonal direct sum of irreducible left A-modules. The other types of modules are treated in a similar way.
β
We are now going to prove the first main result of this section, that any C-dualizable module is unitarizable. First we need a lemma.
Lemma 2.26**.**
Let Wiβ,Wkβ be C-dualizable left A-modules (resp. right A-modules, A-bimodules). If Wkβ is unitary, and there exists a surjective FβHomA,ββ(Wkβ,Wiβ) (resp. FβHomβ,Aβ(Wkβ,Wiβ), FβHomAβ(Wkβ,Wiβ)), then Wiβ is unitarizable. In particular, Wiβ is semisimple as a left A-module (resp. right A-module, A-bimodule).
We remark that this lemma is obvious when Wiβ is already known to be semisimple as a left, right or bi A-module, which is enough for our application to representations of VOA extensions. (Indeed, the extension U of V considered in this paper is always regular, hence its modules are semisimple.) Those who are only interested in the application to VOAs can skip the following proof.
Proof.
We only prove this for left modules, since the other cases can be proved similarly. Write the two modules as (Wiβ,ΞΌLiβ),(Wkβ,ΞΌLkβ). Note that FβF is a positive element in the finite dimensional Cβ-algebra End(Wkβ). So limnβββ(FβF)1/n converges under the Cβ-norm to a projection PβEnd(Wkβ) which is the range projection of Fβ. Set G=PΞΌLkβ and H=PΞΌLkβ(1aββP) which are morphisms in Hom(Waββ Wkβ,Wkβ). Then using (2.67) and the fact that F=FP, we obtain FG=FH, since both equal FΞΌLkβ. Therefore (FβF)nG=(FβF)nH for any integer n>0, and hence (FβF)1/nG=(FβF)1/nH by polynomial interpolation. Thus G=PG=PH=H. We conclude
[TABLE]
This equation, together with (2.49), shows (1aββP)(ΞΌLkβ)β=(1aββP)(ΞΌLkβ)βP, whose adjoint is
[TABLE]
We can therefore combine (2.68) and (2.69) to get PΞΌLkβ=ΞΌLkβ(1aββP). In other words, P is a projection in EndA,ββ(Wkβ). Thus, by proposition 2.24, one can find a unitary left A-module Wjβ and a partial isometry KβHomA,ββ(Wjβ,Wkβ) satisfying KβK=1jβ and KKβ=P. Therefore the left A-module Wiβ is equivalent to Wjβ since FKβHomA,ββ(Wjβ,Wiβ) is invertible.
β
Theorem 2.27**.**
C-dualizable left A-modules, right A-modules, and A-bimodules are unitarizable.
In particular, when C is rigid, any left A-module, right A-module, or A-bimodule is unitarizable.
Proof.
For any C-dualizable object Wiβ in C the induced left A-module (Waββ Wiβ,ΞΌβ1iβ), abbreviated to Waββ Wiβ, is clearly C-dualizable. By the unitarity of A, one easily checks that Waββ Wiβ is a unitary left A-module. Now assume that (Wiβ,ΞΌLβ) is a left A-module. Then ΞΌLββHomA,ββ(Waββ Wiβ,Wiβ). Moreover, ΞΌLβ is surjective since (ΞΉβ1iβ)ΞΌLβ=1iβ is surjective. Therefore Wiβ is unitarizable by lemma 2.26. The case of right modules is proved in a similar way. In the case that (Wiβ,ΞΌLβ,ΞΌRβ) is a C-dualizable A-bimodule, we notice that (Waββ Wiββ Waβ,ΞΌβ1iββ1aβ,1aββ1iββΞΌ) is a unitary A-bimodule, and ΞΌRβ(ΞΌLββ1aβ)=ΞΌLβ(1aββΞΌRβ)βHomAβ(Waββ Wiββ Waβ,Wiβ) whose surjectivity follows again from the unit property. Thus, again, Wiβ is a unitarizable A-bimodule.
β
In the case that A is special, there is another proof of unitarizability due to [BKLR15] which does not require dualizability. To begin with, we say that a left A-module (Wiβ,ΞΌLβ) is standard if ΞΌLβΞΌLβββC1iβ. We now follow the argument of [BKLR15] lemma 3.22. For any left A-module (Wiβ,ΞΌLβ), Ξ:=(ΞΌLβΞΌLββ)1/2 is invertible. Therefore (Wiβ,ΞΌLβ) is equivalent to (Wiβ,ΞΌβLβ) where ΞΌβLβ=Ξβ1ΞΌLβ(1aββΞ). Using associativity and the fact that ΞΌΞΌβ=dAβ1aβ, one can check that (Wiβ,ΞΌβLβ) is standard and ΞΌβLβΞΌβLββ=dAβ1iβ. In particular, if (Wiβ,ΞΌLβ) is standard then Ξ is a constant and hence ΞΌβLβ=ΞΌLβ. Thus we must have ΞΌLβΞΌLββ=dAβ1iβ. This proves that any left A-module is equivalent to a standard left A-module, that any standard left A-module must satisfy ΞΌLβΞΌLββ=dAβ1aβ. Moreover, by [BKLR15] formula (3.4.5), any standard left A-module is unitary. Conversely, if (Wiβ,ΞΌLβ) is unitary, one can check that ΞΌLβΞΌLβββEndA,ββ(Wiβ) and hence ΞβEndA,ββ(Wiβ). This proves that ΞΌβLβ=ΞΌLβ and hence that (Wiβ,ΞΌLβ) is standard. Right A-modules can be proved in a similar way. When (Wiβ,ΞΌLβ,ΞΌRβ) is an A-bimodule, we define ΞΌLRβ=ΞΌRβ(ΞΌLββ1aβ)=ΞΌLβ(1aββΞΌRβ), and say that the A-bimodule Wiβ is standard if ΞΌLRβΞΌLRβββC1iβ (cf. [BKLR15] section 3.6). With the help of Ξ:=(ΞΌLRβΞΌLRββ)1/2 one can prove similar results as of left A-modules, with the only exception being that ΞΌLRβΞΌLRββ=dA2β1iβ. We summarize the discussion in the following theorems:
Theorem 2.28**.**
If A is a Q-system in C, then all left A-modules, right A-modules, and A-bimodules are unitarizable.
Theorem 2.29**.**
Let A be a Q-system in C, and (Wiβ,ΞΌLβ) (resp. (Wiβ,ΞΌRβ),(Wiβ,ΞΌLβ,ΞΌRβ)) a left A-module (resp. right A-module, A-bimodule). Then the following statements are equivalent.
β’
Wiβ* is unitary.*
β’
Wiβ* is standard.*
β’
ΞΌLβΞΌLββ=dAβ1iβ* (left A-module case), or ΞΌRβΞΌRββ=dAβ1iβ (right A-module case), or ΞΌLRβΞΌLRββ=dA2β1iβ where ΞΌLRβ=ΞΌRβ(ΞΌLββ1aβ)=ΞΌLβ(1aββΞΌRβ) (A-bimodule case).*
If C is braided with braid operator \ss, and if A is commutative, a left A-module (Wiβ,ΞΌLβ) is called single-valued if ΞΌLβ=ΞΌLβ\ss2 (more precisely, ΞΌLβ=ΞΌLβ\ssi,aβ\ssa,iβ). If (Wiβ,ΞΌLβ) is a single-valued left A-module, then (Wiβ,ΞΌRβ) is a right A-module where ΞΌRβ=ΞΌLβ\ssi,aβ=ΞΌLβ\ssa,iβ1β. Moreover, by the associativity of (Wiβ,ΞΌLβ), (Wiβ,ΞΌLβ,ΞΌRβ) is an A-bimodule. We summarize that when C is braided and A is commutative, any single-valued left A-module is an A-bimodule. Moreover, if Wiβ is a unitary single-valued left A-module, the it is also a unitary A-bimodule. The category of (unitary) single-valued left A-module is naturally a full subcategory of the (Cβ-)categories of (unitary) left A-modules, (unitary) right A-modules, and (unitary) A-bimodules.
To discuss the unitarizability of U-modules, the following is needed:
Theorem 2.30**.**
Assume that V is a CFT-type, regular, and completely unitary VOA, U is a CFT-type unitary extension of V, and Wiβ is a preunitary U-module. Then Wiβ is a unitary U-module if and only if Wiβ is a unitary left AUβ-module. If this is true then Wiβ is also a unitary AUβ-bimodule.
Proof.
Let Waβ be the contragredient module of Waβ, eva,aβ,eva,aβ the evaluations of Waβ,Waβ defined in chapter 1, and Ξ΅:U=WaββU=Waβ the reflection operator with respect to the chosen dual and evaluations. By equation (1.70) and the definition of adjoint intertwining operators, Wiβ is unitary if and only if YΞΌLββ(w(a),z)=YΞΌLβ ββ(Ξ΅w(a),z) for any w(a)βWaβ=U. From equation (1.2) we know that Wiβ is unitary if and only if ΞΌLβ=ΞΌLβ β(Ξ΅β1iβ). With the help of theorem 1.11, this equation is equivalent to ΞΌLβ=(eva,aββ1iβ)(Ξ΅βΞΌLββ), which by proposition 2.23 means precisely the unitarity of the left AUβ-module Wiβ. If we already have that Wiβ is a unitary left AUβ-module, then, since Wiβ is single-valued, it is also a unitary A-bimodule.
β
We now prove the strong unitarity of U.
Theorem 2.31**.**
If V is a CFT-type, regular, and completely unitary VOA, and U is a CFT-type unitary extension of V, then U is strongly unitary, i.e., any U-module is unitarizable.
Proof.
Since U-modules are clearly preunitarizable, we choose a preunitary U-module Wiβ. Then by either theorem 2.27 or theorem 2.28, Wiβ is unitarizable as a left AUβ-module. Thus, by equation (1.2) and theorem 2.30, Wiβ is unitarizable as a U-module.
β
3 Cβ-tensor categories associated to Q-systems and unitary VOA extensions
3.1 Unitary tensor products of unitary bimodules of Q-systems
In this chapter, C is a Cβ-tensor category with simple W0β as before, and A=(Waβ,ΞΌ,ΞΉ) is a Q-system (i.e., special Cβ-Frobenius algebra) in C. Set evaluation eva,aβ=ΞΉβΞΌ as usual. We suppress the label a in diagram calculus. Let BIMu(A) be the Cβ-category of unitary A-bimodules whose morphisms are A-bimodule morphisms. In this and the next sections, we review the construction of a Cβ-tensor structure on BIMu(A). See [NY16, KO02, CKM17] for reference. Note that our setting is slightly more general than that of [NY16], since we do not assume C is rigid or A is standard. Nevertheless, many ideas in [NY16] still work in our setting. To make our article self-contained, we include detailed proofs for all the relevant results.
Choose unitary A-bimodules (Wiβ,ΞΌLiβ,ΞΌRiβ),(Wjβ,ΞΌLjβ,ΞΌRjβ). Then Wiββ Wjβ is a unitary A-bimodule with left action ΞΌLiββ1jβ and right action 1iββΞΌRjβ. Define Ξ¨i,jββHomAβ(Wiββ Waββ Wjβ,Wiββ Wjβ) and Οi,jββEndAβ(Wiββ Wjβ) to be
[TABLE]
Definition 3.1**.**
Let Wiβ,Wjβ be unitary A-bimodules. We say that (Wijβ,ΞΌi,jβ) (abbreviated to Wijβ when no confusion arises) is a tensor product of Wiβ,Wjβ over A (cf. [CKM17]), if
β’
Wijβ=(Wijβ,ΞΌLijβ,ΞΌRijβ) is an A-bimodule, ΞΌi,jββHomAβ(Wiββ Wjβ,Wijβ),888One should not confuse iβ j and ij. By our notation, Wiβ jβ=Wiββ Wjβ is different from Wijβ. and ΞΌi,jβΞ¨i,jβ=0.
β’
(Universal property) If (Wkβ,ΞΌLkβ,ΞΌRkβ) is a unitary A-bimodule, Ξ±βHomAβ(Wiββ Wjβ,Wkβ), and Ξ±Ξ¨i,jβ=0,999Such Ξ± is called a categorical intertwining operator in [CKM17]. then there exists a unique Ξ±βHomAβ(Wijβ,Wkβ) satisfying Ξ±=Ξ±ΞΌi,jβ. In this case, we say that Ξ± is induced byΞ± via the tensor product Wijβ.
The tensor product (Wijβ,ΞΌi,jβ) is called unitary if Wijβ is a unitary A-bimodule and Οi,jβ=ΞΌi,jββΞΌi,jβ.
We write \mu_{i,j}=\vbox{\hbox{{
\begin{picture}(1.0,1.56885)\put(0.0,0.0){\includegraphics[width=21.96297pt,page=1]{tensor-product.pdf}}\end{picture}}}}~{}~{},~{}\mu_{i,j}^{*}=\vbox{\hbox{{
\begin{picture}(1.0,1.14148)\put(0.0,0.0){\includegraphics[width=26.37958pt,page=1]{tensor-product-2.pdf}}\end{picture}}}}~{}~{}. Then the equation Οi,jβ=ΞΌi,jββΞΌi,jβ reads
[TABLE]
which is a special case of the Frobenius relations for unitary tensor products to be proved later (theorem 3.14). (Note that the first equality of (3.16) follows from the unitarity of Wiβ and Wjβ.)
The existence of a tensor product is clear if one assumes moreover that C is abelian: Let Wijβ be a cokernel of Ξ¨i,jβ, and define the bimodule structure on Wijβ using that of Wiββ Wjβ. Then Wijβ becomes a tensor product over A of Wiβ and Wjβ; see [KO02] for more details. However, to make the tensor product unitary one has to be more careful when choosing the cokernel. In the following, we proceed in a slightly different way motivated by [BKLR15] section 3.7, and we do not require the abelianess of C. To begin with, using ΞΌΞΌβ=dAβ1aβ one verifies easily that Οi,j2β=dAβΟi,jβ. Therefore,
dAβ1βΟi,jββEndAβ(Wiββ Wjβ)* is a projection.*
By proposition 2.24, there exists a unitary A-bimodule Wijβ and a partial isometry ui,jββHomAβ(Wiββ Wjβ,Wijβ) satisfying ui,jβui,jββ=1ijβ and ui,jββui,jβ=dAβ1βΟi,jβ. Setting ΞΌi,jβ=dAββui,jβ, one obtains ΞΌi,jββΞΌi,jβ=Οi,jβ (equations (3.16)) and ΞΌi,jβΞΌi,jββ=dAβ1ijβ. We now show that (Wijβ,ΞΌi,jβ) is a unitary tensor product.
Proposition 3.3**.**
Let Wijβ be a unitary A-bimodule, and ΞΌi,jββHomAβ(Wiββ Wjβ,Wijβ). Then (Wijβ,ΞΌi,jβ) is a unitary tensor product of Wiβ,Wjβ over A if and only if ΞΌi,jββΞΌi,jβ=Οi,jβ and ΞΌi,jβΞΌi,jββ=dAβ1ijβ.
Proof.
βIfβ: Since ΞΌi,jββΞΌi,jβ=Οi,jβ and the unitarity of the A-bimodule Wijβ are assumed, it suffices to show that Wijβ is a tensor product. Since Οi,jβΞ¨i,jβ clearly equals [math], we compute
[TABLE]
If Wkβ is a unitary A-bimodule, Ξ±βHomAβ(Wiββ Wjβ,Wkβ), and Ξ±Ξ¨i,jβ=0, then one can set Ξ±=dAβ1βΞ±ΞΌi,jββ and compute
[TABLE]
where we have used Ξ±Ξ¨i,jβ=0 and
ΞΌRiβ(ΞΌRiβ)β=dAβ1iβ (theorem 2.29) respectively to prove the third and the fourth equalities. If there is another Ξ± satisfying also Ξ±=Ξ±ΞΌi,jβ, then Ξ±=dAβ1βΞ±ΞΌi,jβΞΌi,jββ=dAβ1βΞ±ΞΌi,jββ=Ξ±. Thus the universal property is checked.
βOnly ifβ: This will be proved after the next theorem.
β
Theorem 3.4**.**
Let Wiβ,Wjβ be unitary A-bimodules. Then unitary tensor products over A of Wiβ,Wjβ exist and are unique up to unitaries. More precisely, uniqueness means that if (Wijβ,ΞΌi,jβ) and (Wiβjβ,Ξ·i,jβ) are unitary tensor products of Wiβ,Wjβ over A, then there exists a (unique) unitary uβHomAβ(Wijβ,Wiβjβ) such that Ξ·i,jβ=uΞΌi,jβ.
Proof.
Existence has already been proved. We now prove the uniqueness. Since Ξ·i,jββHomAβ(Wiββ Wjβ,Wiβjβ) is annihilated by Ξ¨i,jβ, by the universal property for (Wijβ,ΞΌi,jβ) there exists a unique uβHomAβ(Wijβ,Wiβjβ) satisfying Ξ·i,jβ=uΞΌi,jβ. In other words u is the A-bimodule morphism induced by Ξ·i,jβ. It remains to prove that u is unitary.
We first show that u is invertible. By the universal property for (Wiβjβ,Ξ·i,jβ), there exists vβHomAβ(Wiβjβ,Wijβ) such that ΞΌi,jβ=vΞ·i,jβ. Thus Ξ·i,jβ=uvΞ·i,jβ. Therefore, uv is induced by Ξ·i,jβ via the tensor product Wiβjβ. But 1iβjβ is clearly also induced by Ξ·i,jβ via Wiβjβ. Therefore uv=1iβjβ. Similarly vu=1ijβ. This proves that u is invertible.
We now calculate
[TABLE]
By the universal property, ΞΌi,jββuβu and ΞΌi,jββ are equal since they are induced by the same morphism via Wijβ. So uβuΞΌi,jβ=ΞΌi,jβ. By the universal property again, we have uβu=1ijβ. Therefore u is unitary.
β
Proof of the βonly ifβ part of Proposition 3.3.
By the βifβ part of proposition 3.3 and the paragraph before that, there exists a unitary tensor product (Wijβ,ΞΌi,jβ) satisfying ΞΌi,jβΞΌi,jββ=dAβ1ijβ. By uniqueness up to unitaries, this equation holds for any unitary tensor product.
β
In the remaining part of this section, we generalize the notion of unitary tensor product to more than two unitary A-bimodules. For simplicity we only discuss the case of three bimodules. The more general cases can be treated in a similar fashion and are thus left to the reader.
Choose unitary A-bimodules Wiβ,Wjβ,Wkβ with left actions ΞΌLiβ,ΞΌLjβ,ΞΌLkβ and right actions ΞΌRiβ,ΞΌRjβ,ΞΌRkβ respectively. Then (Wiββ Wjββ Wkβ,ΞΌLiββ1jββ1kβ,1iββ1jββΞΌRkβ) is a unitary A-bimodule.
Lemma 3.5**.**
Οi,jββ1kβ* and 1iββΟj,kβ commute. Define Οi,j,kββEndAβ(Wiββ Wjββ Wkβ) to be their product. Then dAβ2βΟi,j,kβ is a projection.*
Proof.
The commutativity of these two morphisms is verified using the commutativity of the left and right actions of Wjβ. Thus dAβ1βΟi,jββ1kβ and dAβ1β1iββΟj,kβ are commuting projections, whose product is therefore also a projection.
β
Definition 3.6**.**
(Wijkβ,ΞΌi,j,kβ) (or Wijkβ for short) is called a unitary tensor product of Wiβ,Wjβ,Wkβ over A, if
β’
Wijkβ=(Wijkβ,ΞΌLijkβ,ΞΌRijkβ) is a unitary A-bimodule, ΞΌi,j,kββHomAβ(Wiββ Wjββ Wkβ,Wijkβ), and ΞΌi,j,kβ(Ξ¨i,jββ1kβ)=ΞΌi,j,kβ(1iββΞ¨j,kβ)=0.
β’
(Universal property) If (Wlβ,ΞΌLlβ,ΞΌRlβ) is a unitary A-bimodule, Ξ±βHomAβ(Wiββ Wjββ Wkβ,Wlβ), and Ξ±(Ξ¨i,jββ1kβ)=Ξ±(1iββΞ¨j,kβ)=0, then there exists a unique Ξ±βHomAβ(Wijkβ,Wlβ) satisfying Ξ±=Ξ±ΞΌi,j,kβ. In this case, we say that Ξ± is induced byΞ± via the tensor product Wijkβ.
β’
(Unitarity) Οi,j,kβ=ΞΌi,j,kβΞΌi,j,kββ.
Proposition 3.7**.**
Let Wijkβ be a unitary A-bimodules, and ΞΌi,j,kββHomAβ(Wiββ Wjββ Wkβ,Wijkβ). Then (Wijkβ,ΞΌi,j,kβ) is a unitary tensor product of Wiβ,Wjβ,Wkβ over A if and only if ΞΌi,j,kββΞΌi,j,kβ=Οi,j,kβ and ΞΌi,j,kβΞΌi,j,kββ=dA2β1ijkβ.
Theorem 3.8**.**
Unitary tensor products of Wiβ,Wjβ,Wkβ exist and are unique up to unitaries.
We omit the proofs of these two results since they can be proved in a similar way as proposition 3.3 and theorem 3.4.
3.2 Cβ-tensor categories associated to Q-systems
We are now ready to define the unitary tensor structure on the Cβ-category BIMu(A) of unitary A-bimodules. The tensor bifunctor β Aβ is defined as follows. For any unitary A-bimodules Wiβ,Wjβ, we choose a unitary tensor product (Wijβ,ΞΌi,jβ). Then Wiββ AβWjβ is just the unitary A-bimodule Wijβ. To define tensor product of morphisms, we choose another pair of unitary A-bimodules Wiβ²β,Wjβ²β, and choose any FβHomAβ(Wiβ,Wiβ²β) and GβHomAβ(Wjβ,Wjβ²β). Of course, there is also a chosen unitary tensor product (Wiβ²jβ²β,ΞΌiβ²,jβ²β) of Wiβ²β,Wjβ²β over A. Since FβG:Wiββ WjββWiβ²ββ Wjβ²β is clearly an A-bimodule morphism, we have ΞΌiβ²,jβ²β(FβG)βHomAβ(Wiββ Wjβ,Wiβ²jβ²β), and one can easily show that ΞΌiβ²,jβ²β(FβG)Ξ¨i,jβ=0. Therefore, by universal property, there exists a unique morphism in HomAβ(Wijβ,Wiβ²jβ²β), denoted by FβAβG, such that
[TABLE]
This defines the tensor product of F and G in BIMu(A). We now show that β Aβ is a β-bifunctor. Notice that FββAβGβ is defined by ΞΌi,jβ(FββGβ)=(FββAβGβ)ΞΌiβ²,jβ²β. Therefore, using (FβG)β=FββGβ we compute
[TABLE]
To construct associativity isomorphisms we need the following:
Proposition 3.9**.**
Let Wiβ,Wjβ,Wkβ be unitary A-bimodules. Then (W(ij)kβ,ΞΌij,kβ(ΞΌi,jββ1kβ)) and (Wi(jk)β,ΞΌi,jkβ(1iββΞΌj,kβ)) are unitary tensor products of Wiβ,Wjβ,Wkβ over A.
Note that here W(ij)kβ is understood as the unitary tensor product of Wijβ and Wkβ over A, and Wi(jk)β is understood similarly.
Proof.
The two cases can be treated in a similar way. So we only prove the first one. Set ΞΌi,j,kβ=ΞΌij,kβ(ΞΌi,jββ1kβ). By proposition 3.7, it suffices to prove ΞΌi,j,kββΞΌi,j,kβ=Οi,j,kβ and ΞΌi,j,kβΞΌi,j,kββ=dA2β1ijkβ. The second equation follows directly from that ΞΌi,jβΞΌi,jββ=dAβ1ijβ and ΞΌij,kβΞΌij,kββ=dAβ1(ij)kβ. To prove the first one, we compute (recalling that we have suppressed the label a)
[TABLE]
β
Corollary 3.10**.**
For any unitary A-bimodules Wiβ,Wjβ,Wkβ there exists a (unique) unitary Ai,j,kββHomAβ(W(ij)kβ,Wi(jk)β) satisfying
[TABLE]
We define the unitary associativity isomorphismW(ij)kββWi(jk)β to be Ai,j,kβ.
Proof.
This follows immediately from the above proposition and theorem 3.8.
β
Proposition 3.11** (Pentagon axiom).**
Let Wiβ,Wjβ,Wkβ,Wlβ be unitary A-bimodules. Then
[TABLE]
Proof.
One can define unitary tensor products of Wiβ,Wjβ,Wkβ,Wlβ over A in a similar way as those of three unitary A-bimodules. Moreover, using the argument of proposition 3.9 one shows that (W((ij)k)lβ,ΞΌ(ij)k,lβ(ΞΌij,kβ(ΞΌi,jββ1kβ)β1lβ)) is a unitary tensor product of Wiβ,Wjβ,Wkβ,Wlβ over A. We now compute
[TABLE]
and
[TABLE]
Thus equation (3.20) holds when both sides are multiplied by ΞΌ(ij)k,lβ(ΞΌij,kβ(ΞΌi,jββ1kβ)β1lβ). Hence equation (3.20) is true by the universal property for the unitary tensor products of Wiβ,Wjβ,Wkβ,Wlβ.
β
We choose the vacuum bimodule (Waβ,ΞΌ,ΞΌ) to be the identity object of BIMu(A). Then by proposition 3.3, for any unitary A-bimodule (Wiβ,ΞΌLiβ,ΞΌRiβ), we have that (Wiβ,ΞΌLiβ) is a unitary tensor product of Waββ Wiβ and (Wiβ,ΞΌRiβ) is a unitary tensor product of Wiββ Waβ. By uniqueness up to unitaries, there exist unique unitary liββHomAβ(Waiβ,Wiβ) and riββHomAβ(Wiaβ,Wiβ) satisfying
[TABLE]
Proposition 3.12** (Triangle axiom).**
For any unitary A-bimodules Wiβ,Wjβ we have
[TABLE]
Proof.
Similar to (and simpler than) the proof of pentagon axiom, we show that
[TABLE]
which proves triangle axiom by universal property.
β
We conclude:
Theorem 3.13**.**
With the β-bifunctor β Aβ, the associativity isomorphisms, the unit object, and the left and right multiplications by unit defined above, BIMu(A) is a Cβ-tensor category.
In the following, we identify different ways of unitary tensor products via associativity isomorphisms, and identify Waiβ with Wiβ and Wiaβ with Wiβ via liβ and riβ respectively. Then BIMu(A) can be treated as if it is a strict Cβ-tensor category. We have (ij)k=i(jk), both denoted by ijk, and also ai=i=ia. Thus the Ai,j,kβ,liβ,riβ are all identity morphisms. Therefore ΞΌLiβ=ΞΌa,iβ,ΞΌRiβ=ΞΌi,aβ, and in particular ΞΌ=ΞΌa,aβ. Moreover, equation (3.19) now reads
[TABLE]
which means that the left action iβ·j and the right action jβΆk commute. These two actions indeed commute adjointly, as indicated below.
Theorem 3.14** (Frobenius relations).**
[TABLE]
Proof.
Let F and G be the first and the second item of (3.54). One computes
[TABLE]
Therefore F=dAβ1βF(ΞΌi,jββ1kβ)(ΞΌi,jββ1kβ)β=dAβ1βG(ΞΌi,jββ1kβ)(ΞΌi,jββ1kβ)β=G, which proves the first equation. The second one is the adjoint of the first one.
β
The above Frobenius relations are the decisive property that makes a tensor product theory unitary. They are indeed closely related to the locality axiom of the categorical extensions of conformal nets [Gui20] where the adjoint commutativity of left and right actions plays a central role. In subsequent works we will relate the Cβ-tensor categories of conformal net extensions and unitary VOA extensions using Frobenius relations.
We close this section by showing that the Cβ-tensor structure of BIMu(A) is independent of the choice of unitary tensor products. Suppose that we have two systems of unitary tensor products: for any objects Wiβ,Wjβ in BIMu(A) we have unitary tensor products (WiΓjβ,ΞΌi,jβ),(Wiβjβ,Ξ·i,jβ) of Wiβ,Wjβ over A, which define (strict) Cβ-tensor categories (BIMu(A),β Aβ),(BIMu(A),β‘Aβ). Tensor products of morphisms are written as βAβ,βAβ respectively. By uniqueness up to unitaries, there exists a unique unitary Ξ¦i,jββHomAβ(WiΓjβ,Wiβjβ) such that
[TABLE]
Proposition 3.15**.**
Ξ¦* is functorial: for any unitary A-modules Wiβ,Wiβ²β,Wjβ,Wjβ²β and any FβHomAβ(Wiβ,Wiβ²β),GβHomAβ(Wjβ,Wjβ²β),*
[TABLE]
Proof.
We compute
[TABLE]
Thus the desired equation is proved by universal property.
β
Theorem 3.16**.**
Ξ¦* induces an equivalence of Cβ-tensor categories (BIMu(A),β Aβ)β(BIMu(A),β‘Aβ). More precisely, for any unitary A-bimodules Wiβ,Wjβ,Wkβ,*
β’
The following diagram commutes.
[TABLE]
β’
The following two morphisms equal 1iβ.
[TABLE]
Proof.
To prove the first condition, we calculate
[TABLE]
and also
[TABLE]
This proves (3.57) since (WiΓjΓkβ,ΞΌi,jΓkβ(1iββΞΌj,kβ)) is a unitary tensor product of Wiβ,Wjβ,Wkβ over A by proposition 3.9.
Let ΞΌLiβ,ΞΌRiβ be the left and right actions of Wiβ. Then under the identifications i=aΓi=iΓa=aβi=iβa, we know by equations (3.21) that ΞΌa,iβ,Ξ·a,iβ both equal ΞΌLiβ, and ΞΌi,aβ,Ξ·i,aβ both equal ΞΌRiβ. Thus, by (3.55), Ξ¦a,iβ=1iβ=Ξ¦i,aβ.
β
3.3 Dualizable unitary bimodules
Let (Wiβ,ΞΌLiβ,ΞΌRiβ) be a unitary A-bimodule as usual. Recall that Wiβ is called C-dualizable if it is dualizable as an object in C. The notion of BIMu(A)-dualizability is understood in a similar way. In [NY16] section 6.2, it was shown that if Wiβ is C-dualizable, then it is BIMu(A)-dualizable. The converse is also true by proposition 6.13 of [NY16].101010Note that although A is assumed in [NY16] to be standard, the results there also apply to the non-standard case since any Cβ-Frobenius algebra is isomorphic to a standard Q-system by [NY18a] theorem 2.9. In this section, we give a slightly different proof of this result; see theorem 3.18.
We first assume that Wiβ is C-dualizable. Our proof of the BIMu(A)-dualizability is motivated by [KO02] lemma 1.16 and [CKM17] proposition 2.77. Notice that Waβ iβ aβ=Waββ Wiββ Waβ is naturally a unitary A-bimodule with left action ΞΌβ1iββ1aβ and right action 1aββ1jββΞΌ. Moreover, ΞΌLRiβ:=ΞΌRiβ(ΞΌLiββ1aβ)=ΞΌLiβ(1aββΞΌRiβ):Waββ Wiββ WaββWiβ is an A-bimodule morphism, and dAβ1βΞΌLRiβ is a partial isometry with range 1iβ. Therefore Wiβ is a sub A-bimodule of Waββ Wiββ Waβ, and hence it suffices to show that Waββ Wiββ Waβ is BIMu(A)-dualizable.
Let Wiβ be a dual object of Wiβ, and choose evi,iβ,evi,iβ of Wiβ,Wiβ. Then a natural candidate of dual bimodule of Waββ Wiββ Waβ is Waββ Wiββ Waβ. Let us first understand their unitary tensor product over A. For this purpose, we choose a general unitary A-bimodule (Wjβ,ΞΌLjβ,ΞΌRjβ), and check easily using proposition 3.3 that (Waββ Wiββ Waββ Wjββ Waβ,1aββ1iββΞΌβ1iββ1aβ) is a unitary tensor product of Waββ Wiββ Waβ and Waββ Wjββ Waβ over A. We thus define the unitary tensor product of Waββ Wiββ Waβ and Waββ Wiββ Waβ over A in this way. Briefly, (aβ iβ a)(aβ iβ a)=aβ iβ aβ iβ a, and similarly, (aβ iβ a)(aβ iβ a)=aβ iβ aβ iβ a.
We now define evaβ iβ a,aβ iβ aAββHomAβ(Waβ iβ aβ iβ aβ,Waβ) and evaβ iβ a,aβ iβ aAββHomAβ(Waβ iβ aβ iβ aβ,Waβ) by
[TABLE]
Since we also have (aβ iβ a)(aβ iβ a)(aβ iβ a)=aβ iβ aβ iβ aβ iβ a, we check using (3.18) and the associativity of A that
[TABLE]
and that evaβ iβ a,aβ iβ aAβ satisfies similar relations. Using these equations it is straightforward to check that evaβ iβ a,aβ iβ aAβ and evaβ iβ a,aβ iβ aAβ are evaluations in BIMu(A) of Waββ Wiββ Waβ and Waββ Wiββ Waβ, which proves that Waββ Wiββ Waβ and hence Wiβ are BIMu(A)-dualizable.
To prove the inverse direction we need the following lemma.
Lemma 3.17**.**
Let WiΛβ be a unitary A-module, not yet known to be dual to Wiβ. Suppose that we have morphisms evi,iΛAββHomAβ(WiiΛβ,Waβ) and eviΛ,iAββHomAβ(WiΛiβ,Waβ). Set
[TABLE]
and also set coevi,iΛAβ=(evi,iΛAβ)β,coeviΛ,iAβ=(eviΛ,iAβ)β,coevi,iΛβ=(evi,iΛβ)β,coeviΛ,iβ=(eviΛ,iβ)β. Then
[TABLE]
We say that evi,iβ and evi,iAβ, evi,iβ and evi,iAβ are correlated if they satisfy (3.60).
Proof.
The two equations can be proved in a similar way, so we only prove the first one. Let L,R be respectively the left and right hand sides of (3.61). Then
[TABLE]
β
Now if Wiβ is BIMu(A)-dualizable, then we can find a unitary A-bimodule Wiβ dual to Wiβ, and evaluations evi,iAβ,evi,iAβ of Wiβ,Wiβ in BIMu(A). Define evi,iβ,evi,iβ by equations (3.60). Then equations (3.61) and (3.62) imply that evi,iβ,evi,iβ are evaluations of Wiβ,Wiβ in C. Thus Wiβ is C-dualizable. This finishes the proof of the following theorem.
Theorem 3.18**.**
If Wiβ is a unitary A-bimodule, then Wiβ is C-dualizable if and only if Wiβ is BIMu(A)-dualizable. Moreover, one can choose correlated evaluations ev,evA in C and BIMu(A).
By the above theorem, we will no longer distinguish between C- and BIMu(A)-dualizability. Using the same argument as lemma 3.17 one also proves that under correlated evaluations, the C-transposes and BIMu(A)-transposes of a unitary A-bimodule morphism F are equal. Therefore the symbols β¨F and Fβ¨ are defined unambiguously. Compare [NY16] lemma 6.10.
Proposition 3.19**.**
Let Wiβ,Wjβ be dualizable unitary A-bimodules. Choose dual objects Wiβ,Wjββ, and evaluations ev,evA (with suitable subscripts) in C and BIMu(A) respectively. Assume that ev,evA are correlated. Then for any FβHomAβ(Wiβ,Wjβ), its transposes in C are the same as those in BIMu(A). More precisely, we have
[TABLE]
Recall that for any FβEnd(Wiβ), one can define scalars TrLβ(F),TrRβ(F) such that evi,iβ(Fβ1iβ)coevi,iβ=TrLβ(F)10β and evi,iβ(1iββF)coevi,iβ=TrRβ(F)10β. If A is simple in the sense that EndAβ(Waβ)=C1aβ, and FβEndAβ(Wiβ), one can similarly define scalars TrLAβ(F),TrRAβ(F) such that evi,iAβ(FβAβ1iβ)coevi,iAβ=TrLAβ(F)1aβ and evi,iAβ(1iββAβF)coevi,iAβ=TrRβ(F)1aβ. In the case that evA and ev are correlated, these two traces satisfy very simple relations:
Proposition 3.20**.**
If A is a simple Q-system, Wiβ,Wiβ are mutually dual unitary A-bimodules, and the ev and evA for Wiβ are correlated, then for any FβEndAβ(Wiβ), we have
[TABLE]
As a consequence, evA are standard if the correlated ev are so.
Proof.
We only prove the relation for left traces.
[TABLE]
β
Note that a simple Cβ-Frobenius algebra is always a simple Q-system, since ΞΌΞΌβ is in EndAβ(Waβ), which must be a scalar and hence proves the specialness. Examples of simple Q-systems include haploid Cβ-Frobenius algebras, since in general we have dimHom(W0β,Waβ)=dimEndA,ββ(Waβ)β₯dimEndAβ(Waβ) (cf. [NY18a] remark 2.7-(1)). Recall that haploid Cβ-Frobenius algebras are also standard (DAβ=daβ) by proposition 2.20. As a consequence, the C-algebra AUβ associated to a unitary VOA extension U is haploid and hence a simple standard Q-system.
Construction of dual bimodules and correlated evA
In the remaining part of this section we assume that A is standard. Then for a dualizable unitary A-bimodule (Wiβ,ΞΌLiβ,ΞΌRiβ) one can explicitly construct the dual bimodule and evA following [KO02] figures 9-11 or [NY18b] section 4.1. This construction will be used in the next section to understand the ribbon structure of the unitary representation category of A.
Since A is now standard, eva,aβ:=ΞΉβΞΌ is a standard evaluation of Waβ. Choose an object Wiβ in C dual to Wiβ, and choose standard ev for Wiβ,Wiβ. Recall convention 2.4. Motivated by corollary 1.12, we define
[TABLE]
Then using graphical calculus it is not hard to verify that (Wiβ,ΞΌLiβ,ΞΌRiβ) is a unitary A-bimodule. (Note that the standardness is used to verify the unitarity.) Moreover, using the above definition, and noting that (β )β¨=β¨(β ), one checks that
[TABLE]
and that ei,iβΞ¨i,iβ=0=ei,iβΞ¨i,iβ. Therefore there exist evi,iAββHomAβ(Wi,iβ,Waβ),evi,iAββHomAβ(Wi,iβ,Waβ) satisfying
[TABLE]
By unit property, evi,iAβ and evi,iβ, evi,iAβ and evi,iβ are correlated. Therefore, by lemma 3.17, evi,iAβ and evi,iAβ are evaluations of Wiβ,Wiβ in BIMu(A). If, moreover, A is simple, then evA are standard, and TrLβ(F)=TrRβ(F)=daβTrLAβ(F)=daβTrRAβ(F) for any FβEndAβ(Wiβ) by proposition 3.20. By the uniqueness up to unitaries of standard evaluations, the values of traces are independent of the choice of standard evaluations. Therefore we have (cf. [KO02] theorem 1.18 and [NY16] proposition 6.9.):
Theorem 3.21**.**
If A is a simple and standard Q-system, Wiβ is a dualizable unitary A-bimodule, and Tr:=TrLβ=TrRβ and TrA:=TrLAβ=TrRAβ are defined using (not necessarily correlated) standard ev and standard evA respectively. Then Tr(F)=daβTrA(F), where daβ is the (C-)quantum dimension of Waβ. In particular, the C-quantum dimension of Wiβ equals daβ multiplied by the BIMu(A)-quantum dimension of Wiβ.
3.4 Braiding and ribbon structures
In this section, C is a braided Cβ-tensor category with (unitary) braiding \ss and simple W0β, and A is a commutative Q-system in C. Let Repu(A) be the Cβ-category of single-valued unitary left A-modules. As discussed in section 2.5, single-valued unitary left A-modules admits a canonical unitary bimodule structure, Repu(A) is a full Cβ-subcategory of BIMu(A), and HomA,ββ,Homβ,Aβ,HomAβ are the same for Repu(A). If Wiβ,Wjβ are in Repu(A), (Wkβ,ΞΌLkβ,ΞΌRkβ) is in BIMu(A), and Ξ±βHomAβ(Wiββ Wjβ,Wkβ) satisfies Ξ±Ξ¨i,jβ=0, then one can show easily using graphical calculus that ΞΌLkβ\ssk,aβ(Ξ±β1aβ)=ΞΌRkβ(Ξ±β1aβ). Now we choose a unitary tensor product (Wijβ,ΞΌi,jβ) of Wiβ,Wjβ over A, where Wijβ is a unitary A-bimodule with left and right actions ΞΌLijβ,ΞΌRijβ. Set Wkβ=Wijβ,Ξ±=ΞΌi,jβ. Then we have ΞΌLijβ\ssij,aβ=ΞΌRijβ since (Ξ±β1aβ)(Ξ±β1aβ)β=(ΞΌi,jββ1aβ)(ΞΌi,jββ1aβ)β=dAβ1ijββ1aβ. Similar argument shows ΞΌLijβ\ssa,ijβ1β=ΞΌRijβ. Therefore Wijβ is single-valued with left and right actions related by \ss. We conclude that Repu(A) is closed under unitary tensor products. In other words, Repu(A) is a full Cβ-tensor subcategory of BIMu(A).
If A is standard, and Wiβ is an object in Repu(A), then Wiβ is Repu(A)-dualizable if and only if Wiβ is BIMu(A)-dualizable (equivalently, C-dualizable).
Proof.
We have seen in theorem 3.18 that BIMu(A)-dualizability and C-dualizability are the same. Repu(A)-dualizability clearly implies BIMu(A)-dualizability. Now assume that Wiβ is C-dualizable. In section 3.3 we have constructed a unitary A-bimodule Wiβ dual to Wiβ. It is easy to check that the left and right actions of Wiβ defined by (3.66) are related by the braiding \ss of C. In particular, Wiβ is an object in Repu(A). Thus Wiβ is Repu(A)-dualizable.
β
Braiding
We now define braiding for Repu(A). Let Wiβ,Wjβ be objects in Repu(A), and let (Wijβ,ΞΌi,jβ) and (Wjiβ,ΞΌj,iβ) be respectively the unitary tensor products over A of Wiβ,Wjβ and Wjβ,Wiβ used to define the tensor structure of BIMu(A). Since the braiding of C is unitary, using proposition 3.3 one easily shows that (Wijβ,ΞΌj,iβ\ssi,jβ) is also a unitary tensor product of Wjβ,Wiβ over A. Hence there exists a unique unitary \ssi,jAββEndAβ(Wijβ,Wjiβ) such that
[TABLE]
Theorem 3.23**.**
(Repu(A),β Aβ,\ssA)* is a braided Cβ-tensor category.*
Proof.
The hexagon axioms
[TABLE]
(for all Wiβ,Wjβ,Wkβ in Repu(A)) can be proved in a similar way as pentagon axiom (proposition 3.11): one shows that both sides are equal when multiplied from the right by ΞΌi,jkβ(1iββΞΌj,kβ)=ΞΌij,kβ(ΞΌi,jββ1kβ).
β
Now assume that we have two systems of unitary tensor products (WiΓjβ,ΞΌi,jβ),(Wiβjβ,Ξ·i,jβ) which define two braided Cβ-tensor categories (Repu(A),β Aβ,\ssA) and (Repu(A),β‘Aβ,ΟA). By theorem 3.16, the functorial unitary Ξ¦ defined by (3.55) induces an equivalence of the Cβ-tensor categories. Indeed, it also preserves the braidings:
Theorem 3.24**.**
The functorial unitary Ξ¦ defined by (3.55) induces an equivalence of the braided Cβ-tensor categories (Repu(A),β Aβ,\ssA)β(Repu(A),β‘Aβ,ΟA), which means that Ξ¦ satisfies the two conditions of theorem 3.16, together with the condition that for any objects Wiβ,Wjβ in Repu(A),
[TABLE]
Proof.
One verifies that Ξ¦j,iβ\ssi,jAβΞΌi,jβ=Οi,jAβΞ·i,jβ=Οi,jAβΞ¦i,jβΞΌi,jβ.
β
Ribbon structures
Let us now assume that C is rigid, which means that any object of C is dualizable. By [MΓΌg00] proposition 2.4, there is a canonical twist operator Ο=ΟiββEnd(Wiβ) for any object Wiβ in C: Choose Wiβ dual to Wiβ, standard evi,iβ,evi,iβ and corresponding coev for Wiβ,Wiβ. Then by standardness of ev one can show
[TABLE]
(Note that by uniqueness up to unitaries of standard ev, Οiβ is independent of the choice of standard evaluations.) By this relation, Ο is unitary. Moreover, Ο defines a ribbon structure on C (i.e., Ο commutes with all morphisms, Οiβ jβ=(ΟiββΟjβ)\ssj,iβ\ssi,jβ, and Οiβ¨β=Οiβ). Then (C,β ,\ss,Ο) is a rigid Cβ-ribbon category. Using the definition of Οiβ, one easily shows
[TABLE]
which completely determines the morphism Οiβ. In the case of Repu(V), we have shown in [Gui19b] section 7.3 (especially equation (7.30), which relies on [Gui19a] formula (1.41)) that e2iΟL0β satisfies the above equation. Thus the twist Ο=e2iΟL0β defined in the end of section 1.1 is the canonical twist of the braided Cβ-fusion category (unitary braided fusion category) Repu(V).
Suppose now that A is haploid. By the commutativity of A, haploidness is equivalent to simpleness since EndA,ββ(Waβ)=EndAβ(Waβ). A is also standard by proposition 2.20. Therefore, by theorem 3.22, Repu(A) is rigid. Thus Repu(A) also admits a canonical twist ΟA under which Repu(A) becomes a Cβ-ribbon category. The twist satisfies
[TABLE]
where Wiβ is an object of Repu(A) dual to Wiβ, and evi,iAβ,evi,iAβ are standard evaluations for Wiβ,Wiβ. We now show that the ribbon structures of C and Repu(A) are compatible.
Theorem 3.25**.**
Suppose that C is a rigid braided Cβ-tensor category, A is a haploid commutative Q-system in C, and Ο and ΟA are the canonical unitary twists of C and Repu(A) respectively. Then Οiβ=ΟiAβ for any object Wiβ in Repu(A).
As an immediate consequence, Waβ has trivial (C-)twist since ΟaAβ=1aβ.
Proof.
Choose any Wiβ in Repu(A). Using the definition of twist one checks easily that ΟiββEndAβ(Wiβ). Let Wiβ be a dual object in C, equip Wiβ with a unitary A-bimodule structure by (3.66), and use equations (3.71), (3.76), and (3.77) to define standard Repu(A)-evaluations evA correlated to standard C-evaluations ev for Wiβ,Wiβ. We now prove Οiβ=ΟiAβ by showing
[TABLE]
We compute
[TABLE]
β
A modular tensor category is called unitary if it is a (rigid) braided Cβ-tensor category, and if its twist is the canonical unitary twist associated to the rigid braided Cβ-tensor structure.
Corollary 3.26**.**
Let (C,β ,\ss,Ο) be a unitary modular tensor category. If A is a haploid commutative Q-system in C, then (Repu(A),β Aβ,\ssA,Ο) is also a unitary modular tensor category.
Proof.
We have shown that A has trivial C-twist. Thus by [KO02] theorem 4.5, Repu(A) is a modular tensor category.111111Our Repu(A) is written as Rep0(A) in [KO02]. By the above theorem, when restricted to Repu(A), Ο is the canonical unitary twist ΟA of Repu(A). Thus Repu(A) is a unitary modular tensor category.
β
3.5 Complete unitarity of unitary VOA extensions
Recall that V is a CFT-type, regular, and completely unitary VOA. Let U be a CFT-type unitary extension of V, and let AUβ=(Waβ,ΞΌ,ΞΉ) be the corresponding standard commutative Q-system. (Note that the trivial twist condition for A is now redundant by theorem 3.25.) Recall by theorem 2.30 that any unitary U-module is naturally a single-valued unitary left AUβ-module, and any single-valued unitary left AUβ-module can be regarded as a unitary U-module. Thus Repu(U) is naturally equivalent to Repu(AUβ) as Cβ-categories. Note that U is also regular (equivalently, rational and C2β-cofinite [ABD04]) by the proof of [McR20] theorem 4.13.121212Although [McR20] theorem 4.13 only discusses orbifold type extensions, the argument there is quite general and clearly applies to the general case. Note that the nonzeroness of quantum dimensions required in that theorem is obvious in the unitary case. Thus, just as V, the tensor category of U-modules is (rigid and) modular. In the following, we shall show that U is completely unitary, which implies that Repu(U) is a unitary modular tensor category. Moreover, we shall show that the unitary modular tensor categories Repu(U) and Repu(AUβ) are naturally equivalent.
Let Wiβ,Wjβ,Wkβ be unitary U-modules, which can be regarded respectively as unitary AUβ-bimodules with left actions ΞΌLiβ,ΞΌLjβ,ΞΌLkβ and right actions ΞΌRiβ,ΞΌRjβ,ΞΌRkβ related by \ss. Recall (1.1). Then YΞΌLiββ,YΞΌLjββ,YΞΌLkββ are the vertex operators of U on Wiβ,Wjβ,Wkβ respectively. We let VUβ(iΒ jkβ) be the vector space of type (iΒ jkβ) intertwining operators of U. Again, V(iΒ jkβ) denotes the vector space of type (iΒ jkβ) intertwining operators of V. Since any intertwining operator of U is also an intertwining operator of V, VUβ(iΒ jkβ) is a subspace of V(iΒ jkβ). We give a categorical interpretation of VUβ(iΒ jkβ). Note that Wiββ Wjβ is naturally a unitary AUβ-bimodule, with left and actions defined by the left action of Wiβ and the right action of Wjβ.
Lemma 3.27**.**
Any Ξ±βHomAUββ(Wiββ Wjβ,Wkβ) satisfies Ξ±Ξ¨i,jβ=0.
Proof.
This is easy to prove using graphical calculus and the fact that the left actions of Wiβ,Wjβ,Wkβ are related by \ss to the right ones.
β
The map Y:Hom(Wiββ Wjβ,Wkβ)ββV(iΒ jkβ),Β Ξ±β¦YΞ±β (see section 1.1) restricts to an isomorphism Y:HomAUββ(Wiββ Wjβ,Wkβ)ββVUβ(iΒ jkβ).
Proof.
We sketch the proof here; details can be found in the reference provided. Choose any Ξ±βHom(Wiββ Wjβ,Wkβ). Then YΞ±β being an intertwining operator of U means precisely that YΞ±β satisfies the Jacobi identity with the vertex operator of U. By contour integrals, the Jacobi identity is well known to be equivalent to the fusion relations
[TABLE]
for any uβU=Waβ,w(i)βWiβ, where 0<β£zβΞΆβ£<β£ΞΆβ£<β£zβ£ and arg(zβΞΆ)=argΞΆ=argz in the first equation, and 0<β£ΞΆβzβ£<β£zβ£<β£ΞΆβ£ and argz=argΞΆ=Ο+arg(zβΞΆ) in the second one. (See for instance [Gui19a] proposition 2.13.) By [Gui19a] proposition 2.9, (3.85) is equivalent to the fusion relation
[TABLE]
The categorical interpretations of (3.84) and (3.86) are respectively Ξ±βHomAUβ,ββ(Wiββ Wjβ,Wkβ) and Ξ±Ξ¨i,jβ=0, which are clearly equivalent to that Ξ±βHomAUββ(Wiββ Wjβ,Wkβ).
β
Recall the definition of VOA modules in section 1.1. Recall by theorem 2.31 that any irreducible U-module admits a unitary structure. We choose a representative Wtβ for each equivalence class [Wtβ] of irreducible unitary U-modules (equivalently, irreducible unitary single-valued left AUβ-modules), and let all these Wtβ form a set EUβ. That WtββEUβ is abbreviated to tβEUβ. We also assume that the vacuum U-module Waβ is in EUβ. Then the tensor product of U-modules Wiβ,Wjβ is
[TABLE]
We choose an inner product ΞUβ for any VUβ(iΒ jtβ)β, and assume that the above direct sum is orthogonal. The vertex operator for Wijβ is β¨tβ1βYΞΌLtββ, where ΞΌLtββHomAUββ(Waββ Wtβ,Wtβ) is the left action of the AUβ-bimodule Wtβ.
Define a U-intertwining operator YΞΌi,jββ of type (iΒ jijβ)=(WiβWjβWijββ), such that for any w(i)βWiβ,w(j)βWjβ,tβEUβ,YΞ±ββVUβ(iΒ jtβ), and w(t)βWtβ (the contragredient unitary U-module of Wtβ),
[TABLE]
To write the above definition more explicitly, we choose a basis Ξ₯i,jtβ of the vector space HomAUββ(Wiββ Wjβ,Wtβ). Then {YΞ±β:Ξ±βΞ₯i,jkβ} is a basis of VUβ(iΒ jtβ) whose dual basis is denoted by {YβΞ±:Ξ±βΞ₯i,jkβ}. Then
[TABLE]
Note that ΞΌi,jββHomAUββ(Wiββ Wjβ,Wijβ). Then the above relation can also be written as
[TABLE]
By lemma 3.27 we have ΞΌi,jβΞ¨i,jβ=0 . We claim that YΞΌi,jββ satisfies the universal property that for any unitary U-module Wkβ and any YΞ±β in VUβ(iΒ jkβ) (equivalently, Ξ±βHomAUββ(Wiββ Wjβ,Wkβ)) there exists a unique U-module homomorphism Ξ±:WijββWkβ (equivalently, Ξ±βHomAUββ(Wijβ,Wkβ)) such that YΞ±β=Ξ±YΞΌi,jββ (equivalently, Ξ±=Ξ±ΞΌi,jβ by (1.2)). Indeed, since the vector space HomUβ(Wijβ,Wkβ) of U-module morphisms from Wijβ to Wkβ is naurally identified with HomAUββ(Wijβ,Wkβ), just as (1.1), we have a natural isomorphism of vector spaces
[TABLE]
It is easy check for any Ξ±βHomAUββ(Wijβ,Wkβ) that
[TABLE]
which by (1.2) also equals YΞ±ΞΌi,jββ. Now, by proposition 3.28, for any Ξ±βHomAUββ(Wiββ Wjβ,Wkβ), we can find an Ξ± satisfying
[TABLE]
Thus we have YΞ±β=YβΞ±β=Ξ±YΞΌi,jββ=YΞ±ΞΌi,jββ, which shows Ξ±=Ξ±ΞΌi,jβ. Recalling definition 3.1, we conclude that (Wijβ,ΞΌi,jβ) is a tensor product of the AUβ-bimodules Wiβ,Wjβ over AUβ. Indeed, under suitable choice of ΞUβ the tensor products become unitary:
Theorem 3.29**.**
There exists for each tβEUβ a unique inner product ΞUβ on the vector space VUβ(iΒ jtβ)β such that (Wijβ,ΞΌi,jβ) becomes a unitary tensor product of the AUβ-bimodules Wiβ,Wjβ over AUβ. Moreover, ΞUβ is the invariant sesquilinear form of U (cf. section 1.3).
Proof.
Let (Wiβjβ,Ξ·i,jβ) be a unitary tensor product of Wiβ,Wjβ over AUβ. By the first half of the proof of theorem 3.4, tensor products of unitary bimodules of a Q-system are unique up to multiplications by invertible morphisms. Thus there exists an invertible KβHomAUββ(Wijβ,Wiβjβ) such that Ξ·i,jβ=KΞΌi,jβ. Therefore, the decomposition of Wiβjβ into irreducible single-valued left AUβ-modules is the same as that of Wijβ, which takes the form (3.87). Now, using linear algebra, one can easily find an inner product ΞUβ on any VUβ(iΒ jtβ)β, such that K becomes unitary. Then the tensor product (Wiβjβ,Ξ·i,jβ) defined by such ΞUβ is clearly unitary. This proves the existence of ΞUβ. The uniqueness of ΞUβ follows from the uniqueness up to unitaries of the unitary tensor products of AUβ-bimodules (theorem 3.4).
Now assume that (Wijβ,ΞΌi,jβ) is a unitary tensor product. We show that ΞUβ is the invariant sesquilinear form. Assume that for each tβEUβ, Ξ₯i,jtβ is chosen in such a way that {YβΞ±:Ξ±βΞ₯i,jkβ} is an orthonormal basis of VUβ(iΒ jtβ)β under ΞUβ. Then by Οi,jβ=ΞΌi,jββΞΌi,jβ and equation (3.90), we have
[TABLE]
and hence
[TABLE]
which by theorem 1.11 implies for any w1(i)β,w2(i)ββWiβ the fusion relation
[TABLE]
Recall that YΞΌLjββ is the vertex operator of U on Wjβ. Since ΞΌRiβ=ΞΌLiβ\ssa,iβ, we have YΞΌRiββ=B+βYΞΌLiββ by (1.38), which shows that YΞΌRiβββVUβ(iΒ aiβ) is the creation operator of the U-module Wiβ. (See section (1.1) for the definition of creation and annihilation operators). Thus by (1.75), Y(ΞΌRiβ)β ββVUβ(iΒ iaβ) is the annihilation operator of the U-module Wjβ. Therefore, by definition 1.7, we see that (3.118) is the fusion relation that defines invariant sesquilinear forms for U. This shows that {YβΞ±:Ξ±βΞ₯i,jkβ} is also an orthonormal basis of VUβ(iΒ jtβ)β under the invariant sesquilinear form, which proves that the later is positive definite and equals ΞUβ.
β
Theorem 3.30**.**
Let V be a CFT-type, regular, and completely unitary VOA, and let U be a CFT-type unitary VOA extension of V. Then:
β’
U* is also (regular and) completely unitary.*
β’
Under the natural identification of Repu(U) and Repu(AUβ) as Cβ-categories, the monoidal, braiding, and ribbon structures of Repu(U) agree with those of (Repu(AUβ),β AUββ,\ssAUβ,ΟAUβ) defined by the system of unitary tensor products (Wijβ,ΞΌi,jβ) (for any Wiβ,Wjβ in Repu(AUβ)) as constructed in (3.87), (3.88) under the invariant inner product ΞUβ.
By βnatural identificationβ, we mean that each unitary U-module Wiβ is identified with the corresponding single-valued unitary left AUβ-module Wiβ; a homomorphism F:WiββWjβ of unitary U-modules is identified with F, considered as a homomorphism of AUβ-bimodules.
Proof.
As mentioned at the beginning of this section, the regularity of U is proved in [McR20]. By theorems 2.31 and 3.29, U is completely unitary. Hence Repu(U) is a unitary modular tensor category. That Repu(U) and Repu(AUβ) share the same tensor and braiding structures is proved in [CKM17]. In order for this paper to be self-contained, we sketch the proof as follows.
For any objects Wiβ,Wjβ,Wkβ, let Ai,j,kβ:W(ij)kβ=(Wiββ UβWjβ)β UβWkββWi(jk)β=Wiββ Uβ(Wjββ UβWkβ) be the associativity isomorphism of Repu(U). It is shown in [Gui20] proposition 4.3 that under the identification of W(ij)kβ and Wi(jk)β via Ai,j,kβ, one has the fusion relation
[TABLE]
for any w(i)βWiβ,w(j)βWjβ. This means that relation (3.33) holds under the identification via Ai,j,kβ. But we know that due to equation (3.19), the same relations also hold under the identification via Ai,j,kβ, the associativity isomorphism of Repu(AUβ). Thus Ai,j,kβ=Ai,j,kβ.
Next, we know that in Repu(U), the identification WaiββWiβ is via the vertex operator YΞΌLiββ of U on Wiβ, and the identification WiaββWiβ is via the creation operator of the U-module Wiβ, which is YΞΌRiββ, as argued at the end of the proof of theorem 3.29. Define liββHomAUββ(Waiβ,Wiβ) and riββHomAUββ(Wiaβ,Wiβ) using equations (3.21). Then, from section 3.2, we know that liβ and riβ define respectively the equivalences WaiββWiβ and WiaββWiβ in Repu(AUβ) (as a full Cβ-tensor subcategory of BIMu(AUβ)). On the other hand, by (3.93) we have YΞΌLiββ=Yβliββ. Therefore liβ:Waiβ=Waββ UβWiββWiβ is the U-module homomorphism corresponding to the vertex operator of the U-module Wiβ. Thus, by the definition of the monoidal stuctures of VOA tensor categories (see section 1.1), liβ also defines the equivalence WaiββWiβ in Repu(U). Similarly, riβ is the U-module morphism corresponding to the creation operator of Wiβ. Hence it defines the equivalence WiaββWiβ in Repu(U). We have now proved that the (Cβ-)monoidal structure of Repu(U) agrees with that of Repu(AUβ).
Let \ssi,jUββHomUβ(Wijβ,Wjiβ) be the braiding of Wiββ UβWjβ in RepU. We want to show that \ssU equals the braiding \ssAUβ of Repu(AUβ). By (3.78) it suffices to check that for any object Wkβ in Repu(AUβ) and any Ξ±βHomAUββ(Wijβ,Wkβ),
[TABLE]
where \ss is the braiding of Repu(V). Set Ξ±=Ξ±ΞΌj,iββHomAUββ(Wiββ Wjβ,Wkβ). Then by (1.38), YΞ±\ssi,jββ=B+βYΞ±β, and similarly YβΞ±\ssi,jUββ=B+βYβΞ±β. Note that the braiding B+β defined for V-intertwining operators and for U-intertwining operators are the same since U and V have the same Virasoro operators. We now compute
[TABLE]
Finally, for both categories the twists are defined by the rigid braided Cβ-tensor structures. Therefore the ribbon structures agree.
β
3.6 Applications
To use theorem 3.30 in its full power, we first prove the complete unitarity for another type of extensions (which do not preserve conformal vectors).
Proposition 3.31**.**
Let V and V be CFT-type and regular VOAs. Then VβV is (regular and) completely unitary if and only if both V and V are completely unitary. If this is true then Repu(VβV) is the tensor product of Repu(V) and Repu(V).
Proof.
Clearly VβV is CFT-type. Note that VβV is also regular by [DLM97] proposition 3.3. Assume first of all that V and V are completely unitary. By [DL14] proposition 2.9, VβV is unitary. By [FHL93] theorem 4.7.4, any irreducible VβV-module is the tensor product of an irreducible V-module and an irreducible V-module, which by the strong unitarity of V and V are unitarizable. Therefore the VβV-module is also unitarizable, and hence VβV is strongly unitary.
We now show that VβV is completely unitary. Choose unitary V-modules Wiβ,Wjβ and unitary V-modules Wiβ,Wjββ. Choose Wtβ in E. Let also E be a complete set of representatives of irreducible V-modules, and choose any Wtβ in E. Choose bases Ξi,jtβ of V(iΒ jtβ) and Ξi,jβtβ of V(iΒ jβtβ) (the vector space of type (iΒ jβtβ) intertwining operators of V) so that their dual bases {YβΞ±:Ξ±βΞi,jtβ} and {YβΞ±:Ξ±βΞi,jβtβ} are orthonormal under the invariant inner products in V(iΒ jtβ)β and V(iΒ jβtβ)β respectively. Then equation (1.77) holds for V with Ξ(YβΞ±β£YβΞ²)=δα,Ξ²β, and a similar relation holds for V. Let Yjβjββ be the vertex operator of the VβV-module WjββWjββ, and let Yeviβiβ,iβiββ be the annihilation operator of WiββWiβ. Then Yjβjββ=YjββYjββ and Yeviβiβ,iβiββ=Yevi,iβββYevi,iββ. Set YΞ±βΞ±β=YΞ±ββYΞ±β, which is a type (iβiΒ jβjβtβtβ) intertwining operator of VβV if Ξ±βΞi,jtβ,Ξ±βΞi,jβtβ. Then for any w1(i)β,w2(i)ββWiβ,w3(i)β,w4(i)ββWiβ we have the fusion relation
[TABLE]
From this relations, we see that the invariant sesquilinear form Ξ on V(iβiΒ jβjβtβtβ)β is positive. Moreover, by the non-degeneracy of this Ξ, Ξi,jtβΓΞi,jβtβ is a basis of V(iβiΒ jβjβtβtβ) whose dual basis is therefore orthonormal in V(iβiΒ jβjβtβtβ)β. Thus V(iβiΒ jβjβtβtβ)=V(iΒ jtβ)βV(iΒ jβtβ), and the Ξ on V(iβiΒ jβjβtβtβ)β equals ΞβΞ on V(iΒ jtβ)ββV(iΒ jβtβ)β. That Repu(VβV)=Repu(V)βRepu(V) now follows easily. (It also follows from [ADL05] theorem 2.10.)
The above proposition implies a strategy of proving the completely unitarity of a non-conformal unitary extension U of V. Let Vc be the commutant of V in Vc (the coset subalgebra) which is unitary by [CKLW18]. Then U is a unitary (conformal) extension of VβVc by [Ten19a] proposition 2.21. Now it suffices to show the regularity and complete unitarity of V and Vc, the proof of which might require a similar trick applied to V and Vc.
Corollary 3.32**.**
Let V be a (finite) tensor product of c<1 unitary Virasoro VOAs, affine unitary VOAs, and even lattice VOAs. Let U be a CFT-type unitary extension of V. Then U is regular and completely unitary. Consequently, the category of unitary U-modules is a unitary modular tensor category.
Proof.
The affine unitary VOAs of these types are regular by [DLM97] and completely unitary by [Gui19b] theorem 8.4 , [Gui19c] theorem 6.1, and [Ten19b] theorem 5.5. The c<1 unitary Virasoro VOAs (resp. even lattice VOAs) are regular also by [DLM97] and completely unitary by [Gui19b] theorem 8.1 (resp. [Gui20] theorem 5.8). Therefore, by theorem 3.30 and proposition 3.31, CFT-type unitary extensions of their tensor products are also regular and completely unitary.
β
The above corollary is by no means in the most general form. For example, we know that W-algebras in discrete series of type A and E are completely unitary by [Ten19b] theorem 5.5. So one can definitely add these examples to the list in that corollary.
Corollary 3.33**.**
Let U be a CFT-type unitary VOA with central charge c<1. Then U is completely unitary. Consequently, the category of unitary U-modules is a unitary modular tensor category.
Proof.
By [DL14] theorem 5.1, U is a unitary extension of the unitary Virasoro VOA L(c,0).
β
\printindex
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