###### Abstract

This paper is about the orbifold theory of affine and parafermion vertex operator algebras. It is known that the parafermion vertex operator algebra associated to the integrable highest weight modules for the affine Kac-Moody algebra is the building block of the general parafermion vertex operator for any finite dimensional simple Lie algebra and any positive integer . We first classify the irreducible modules of -orbifold of the simple affine vertex operator algebra of type and determine their fusion rules. Then we study the representations of the -orbifold of the parafermion vertex operator algebra , we give the quantum dimensions, and more technically, fusion rules for the -orbifold of the parafermion vertex operator algebra are completely determined.

Fusion rules for -orbifolds of affine and parafermion vertex operator algebras

Cuipo Jiang^{1}^{1}1Supported by China NSF grants No.11771281 and No.11531004.
and Qing Wang^{2}^{2}2Supported by
China NSF grants No.11622107 and No.11531004, Natural Science Foundation of Fujian Province
No.2016J06002.

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

## 1 Introduction

This paper is a continuation in a series of papers on the study of the orbifold theory of affine and parafermion vertex operator algebras. It is known that the parafermion vertex operator algebra is the commutant of a Heisenberg vertex operator subalgebra in the simple affine vertex operator algebra , where is the integrable highest weight module with the positive integer level for the affine Kac-Moody algebra associated to a finite dimensional simple Lie algebra over . We denote by and by in this paper. Since parafermion vertex operator algebras can be identified with -algebras [17], the orbifold theory of the parafermion vertex algebras corresponds to the orbifold theory of -algebras. Some conjectures in the physics literature about the orbifold -algebras have been studied and solved in [4], [3], [30]. These results about the orbifold -algebras are mainly structural aspects. Our interest is to study the representation theory of the orbifold parafermion vertex operator algebra from the point of vertex algebras. From [17], we know that the full automorphism group of the parafermion vertex operator algebra for is the group of order generated by the automorphism , which is determined by , , , where is a standard Chevalley basis of with brackets , and . We have classified the irreducible modules of the orbifold parafermion vertex operator algebra in [28], where is the fixed-point vertex operator subalgebra of under . A natural problem next is to determine the fusion rules for . Note that the vertex operator algebra can be viewed as a subalgebra of the orbifold affine vertex operator algebra , where is the fixed-point vertex operator subalgebra of under . In order to understand the representation theory of the orbifold parafermion vertex operator algebra better, we should first understand the representation theory of the orbifold affine vertex operator algebra first. For this purpose, we classify the irreducible modules of and determine the fusion rules for in Section 3. We obtain Theorem 3.22 that there are two kinds of irreducible modules for . One kind is the untwisted type modules coming from the irreducible -modules, and the other kind is the twisted type modules coming from the -twisted -modules. Furthermore, we determine the contragredient modules of all these irreducible -modules in Theorem 3.25. These results together with the symmetric property of fusion rules imply that we only need to determine two kinds of fusion products, one is the fusion product between the untwisted type modules and the untwisted type modules, and the other is the fusion product between the untwisted type modules and the twisted type modules. Our first step is to construct the intertwining operators among untwisted and twisted -modules. We use the -operator introduced by Li in [33]. Then the fusion products between the untwisted type modules and the twisted type modules can be obtained by applying the fusion rules for the affine vertex operator algebra and the intertwining operator constructed from the -operator. Furthermore, by observing the action of the automorphism on the -operator, the fusion products between the untwisted type modules and the untwisted type modules follow from the fusion products between the untwisted type modules and the twisted type modules.

The determination of the fusion rules for is much more complicated. We first determine the quantum dimensions of the irreducible -modules, which can help us to determine the fusion rules for . However it is far from the complete determination of the fusion rules for . Our strategy is to employ the lattice realization of the irreducible -modules [17] and the lowest weights of the irreducible -modules [28], together with the decomposition of the irreducible -modules viewed as the modules of the lattice vertex operator subalgebra [17] for . From the classification results of the irreducible modules of , there are two families of untwisted type -modules. One family is from the irreducible modules of , which are not irreducible as -modules. We call it the untwisted module of type . The other family is from the irreducible modules of , which are also irreducible as -modules. We call it the untwisted module of type . We would like to point out that the main difficulty to determine the fusion products between the untwisted type modules and the untwisted type modules of is to find which one of the irreducible -modules of type can survive in the decomposition of the fusion product, and to distinguish the inequivalent modules emerging in the decomposition of the fusion product. The fusion products between the untwisted type modules and the twisted type modules of are extremely complicated in the case that the level is even, because from [28], we know that in the level , there are two irreducible twisted modules of , and the lowest weight vector can be in the grade zero or in the grade of the -twisted module of . Thus as the -modules, there are four irreducible modules in the level , when it emerges in the decomposition of the fusion product between the untwisted type module and the twisted type module of . We need to distinguish which one can survive for certain cases. The strategy is that we come back to the lattice realization of the irreducible -modules for , [17], and we technically use another basis of the Lie algebra and apply the intertwining operator among the modules of the lattice vertex operator algebra, together with the analysis of the lowest weights of the irreducible -modules we obtained in [28]. Furthermore, we determine the contragredient modules of all the irreducible -modules, thus the fusion rules for are completely determined.

The paper is organized as follows. In Section 2, we recall some results about the parafermion vertex operator algebra , its orbifold vertex operator subalgebra and their irreducible modules. In Section 3, we classify the irreducible modules of the -orbifold of the affine vertex operator algebra and determine the fusion rules for . In Section 4, we give the quantum dimensions for irreducible -modules. In Section 5, we determine the fusion rules for the -orbifold of parafermion vertex operator algebra .

## 2 Preliminaries

In this section, we recall from [17], [19], [23], [5] and [28] some basic results on the parafermion vertex operator algebra associated to the irreducible highest weight module for the affine Kac-Moody algebra of level with being a positive integer and their -orbifolds. We first recall the notion of the parafermion vertex operator algebra.

We are working in the setting of [17]. Let be a standard Chevalley basis of with Lie brackets , , and the normalized Killing form , , . Let be the affine Lie algebra associated to . Let be an integer and

be the induced -module such that acts as and acts as on . Then is a vertex operator algebra generated by for such that

where , with the vacuum vector and the Virasoro vector

Let be the vertex operator subalgebra of generated by with the Virasoro element

of central charge .

The vertex operator algebra has a unique maximal ideal , which is generated by a weight vector [29]. The quotient algebra is a simple, rational vertex operator algebra as is a positive integer (cf. [27], [34]). Moreover, the image of in is isomorphic to and will be denoted by again. Set

Then which is the space of highest weight vectors with highest weight for is the commutant of in and is called the parafermion vertex operator algebra associated to the irreducible highest weight module for The Virasoro element of is given by

with central charge , where we still use to denote their images in . We denote by .

Set

in , and also denote its image in by . It was proved in [17](cf.[19], [22]) that the parafermion vertex operator algebra is simple and is generated by and . If , the parafermion vertex operator algebra in fact is generated by . The irreducible -modules for were constructed in [17]. Note that . It was also proved in [17, Theorem 4.4] that as -module . Theorem 8.2 in [5] showed that the irreducible -modules for constructed in [17] form a complete set of isomorphism classes of irreducible -modules. Moreover, is -cofinite [5] and rational [6] (see also [20]).

Let for be the irreducible modules for the rational vertex operator algebra with the top level which is an -dimensional irreducible module of the simple Lie algebra . The top level of is a one dimensional space spanned by for [17]. The following result was due to [17].

###### Lemma 2.1.

The operator acts on as follows:

(2.1) |

Let be an automorphism of Lie algebra defined by . can be lifted to an automorphism of the vertex operator algebra of order 2 in the following way:

for and . Then induces an automorphism of as preserves the unique maximal ideal , and the Virasoro element is invariant under . Thus induces an automorphism of the parafermion vertex operator algebra . In fact, .

###### Lemma 2.2.

[17] If , the automorphism group Aut is of order 2.

###### Remark 2.3.

If , . If , is generated by . Thus the automorphism group Aut is trivial for and . Therefore, by Lemma 2.2, we only need to consider the orbifold of parafermion vertex operator algebra under the automorphism for .

Let be the -orbifold vertex operator algebra, i.e., the fixed-point vertex operator subalgebra of under the automorphism . The following theorem gives the classification of the irreducible modules of for [28].

###### Theorem 2.4.

[28] If , , there are inequivalent irreducible modules of . If , , there are inequivalent irreducible modules of . More precisely, if , , the set

gives all inequivalent irreducible -modules. If , , the set

gives all inequivalent irreducible -modules.

###### Remark 2.5.

With the notations in Theorem 2.4, we call and twisted type modules and untwisted modules of type and type respectively.

## 3 Fusion rules for the -orbifold of the affine vertex operator algebra

In this section, we first recall the definition of weak -twisted modules, -twisted modules and admissible -twisted modules following [15, 16]. Let be the -orbifold vertex operator subalgebra of the affine vertex operator algebra , i.e., the fixed-point subalgebra of under . We then classify and construct the irreducible modules for . Furthermore, we determine the contragredient modules of irreducible -modules and the fusion rules for the vertex operator algebra .

Let be a vertex operator algebra (see [26], [34]) and an automorphism of with finite order . Let denote the space of -valued formal series in arbitrary complex powers of for a vector space . Denote the decomposition of into eigenspaces with respect to the action of by

where .

###### Definition 3.1.

A *weak -twisted -module* is
a vector space with a linear map

which satisfies the following conditions for , :

where .

The following identities are the consequences of the twisted-Jacobi identity [15] (see also [2], [11]).

(3.2) |

(3.3) |

where .

###### Definition 3.2.

A *-twisted -module* is a weak -twisted -module*
* which carries a -grading
where and is one of the coefficient operators of
Moreover we require
that is finite and for fixed
for all small enough integers

###### Definition 3.3.

An *admissible -twisted -module*
is a -graded weak -twisted module
such that
for homogeneous and

If , we have the notions of weak, ordinary and admissible -modules [15].

###### Definition 3.4.

A vertex operator algebra is called *-rational*
if the admissible -twisted module category is semisimple.

###### Remark 3.5.

The following lemma about -rational vertex operator algebras is well known [15].

###### Lemma 3.6.

If is -rational, then

(1) Any irreducible admissible -twisted -module is a -twisted -module, and there exists a such that where And is called the conformal weight of

(2) There are only finitely many irreducible admissible -twisted -modules up to isomorphism.

Let be an admissible -twisted -module, the contragredient module is defined as follows: , where The vertex operator is defined for via

(3.4) |

where is the natural paring

###### Remark 3.7.

is an admissible -twisted -module [25]. One can also define the contragredient module for a -twisted -module . In this case, is a -twisted -module. Moreover, is irreducible if and only if is irreducible.

Now we recall from [25] the notions of intertwining operators and fusion rules.

###### Definition 3.8.

Let be a vertex operator algebra and
let and be
-modules. An *intertwining operator* of type is a linear map

satisfying:

(1) for any and , for sufficiently large;

(2) ;

(3) (Jacobi identity) for any

The space of all intertwining operators of type is denoted by

Let . These integers are usually called the
*fusion rules*.

###### Definition 3.9.

Let be a vertex operator algebra, and
be two -modules. A module , where is called a *tensor product* (or fusion product) of
and if for any -module and there is a unique -module homomorphism such
that As usual, we denote by

###### Remark 3.10.

It is well known that if is rational, then for any two irreducible -modules and the fusion product exists and

where runs over the set of equivalence classes of irreducible -modules.

Fusion rules have the following symmetric property [25].

###### Proposition 3.11.

Let be -modules. Then

We will use the following lemma from [14] later.

###### Lemma 3.12.

Let be a vertex operator algebra, and let and be irreducible -modules and a -module. If is a nonzero intertwining operator of type , then for any nonzero vectors and .

We fix some notations. Let be irreducible -modules. In this section, we use to denote the space of all intertwining operators of type , and use to denote the fusion product for simplicity. We recall the fusion rules for the affine vertex operator algebra of type [36] for later use.

###### Lemma 3.13.

where

We notice that since is rational, is rational, and thus is -rational. Then from [16], we have the following result.

###### Proposition 3.14.

There are precisely inequivalent irreducible -twisted modules of .

###### Proof.

Since is -rational, from [16], we know that the number of inequivalent irreducible -twisted modules of is precisely the number of -stable irreducible untwisted modules of . Notice that for exhaust all the irreducible modules for with the top level . By direct calculation, we have

(3.5) |

We see that these lowest weights are pairwise different for , which shows that for are -stable irreducible modules. Thus there are totally inequivalent irreducible -twisted modules of . ∎

Recall from [28] that is a standard Chevalley basis of with brackets , , . Set

Then is a -triple. Let

Note that for are all the irreducible modules for the rational vertex operator algebra . From [32], we have the following result.

###### Lemma 3.15.

For , are irreducible -twisted -modules.

As in [28], for such that , , , we use the notation and respectively to distinguish the action of the elements in on -twisted modules and untwisted modules as follows

Recall that the top level of for is an -dimensional irreducible module for . Let

then is the lowest weight vector with weight in -dimensional irreducible module for , that is, and , and we have:

###### Lemma 3.16.

[28] For the positive integer , and ,

By Lemma 3.16, we have

###### Lemma 3.17.

(3.6) |

We can now construct the inequivalent irreducible -twisted modules of .

###### Theorem 3.18.

for are inequivalent irreducible -twisted modules of generated by .

###### Proof.

We just need to notice that is the lowest weight vector of the -twisted module , and

We now classify all the irreducible modules of the orbifold vertex operator algebra . Set

(3.7) |

By applying the results in [15], we have:

###### Proposition 3.19.

For , let and be the -modules generated by and respectively. Then and for are irreducible modules of with the lowest weights

###### Proposition 3.20.

For , we have

where for is an irreducible module of generated by with weight , and for is an irreducible module of