axioms

Article Branching Functions for Admissible Representations of Affine Lie Algebras and Super-Virasoro Algebras

Namhee Kwon Department of Mathematics, Daegu University, Gyeongsan, Gyeongbuk 38453, Korea; [email protected]



Received: 2 May 2019; Accepted: 17 July 2019; Published: 19 July 2019


Abstract: We explicitly calculate the branching functions arising from the tensor product decompositions between level 2 and principal admissible representations over slb2. In addition, investigating the characters of the minimal series representations of super-Virasoro algebras, we present the tensor product decompositions in terms of the minimal series representations of super-Virasoro algebras for the case of principal admissible weights.

Keywords: branching functions; admissible representations; characters; affine Lie algebras; super-Virasoro algebras

1. Introduction One of the basic problems in representation theory is to find the decomposition of a tensor product between two irreducible representations. In fact, the study of tensor product decompositions plays an important role in quantum mechanics and in string theory [1,2], and it has attracted much attention from combinatorial representation theory [3]. In addition, recent studies reveal that tensor product decompositions are also closely related to the representation theory of Virasoro algebra and W-algebras [4–6]. In [6], the authors extensively study decompositions of tensor products between integrable representations over affine Lie algebras. They also investigate relationships among tensor products, branching functions and Virasoro algebra through integrable representations over affine Lie algebras. In the present paper we shall follow the methodology appearing in [6]. However, we will focus on admissible representations of affine Lie algebras. Admissible representations are not generally integrable over affine Lie algebras, but integrable with respect to a subroot system of the root system attached to a given affine Lie algebra. Kac and Wakimoto showed that admissible representations satisfy several nice properties such as Weyl-Kac type character formula and modular invariance [5,7]. In their subsequent works, they also established connections between admissible representations of affine Lie algebras and the representation theory of W-algebras [4,8]. In addition, Kac and Wakimoto expressed in ([9], Theorem 3.1) the branching functions arising from the tensor product decompositions between principal admissible and integrable representations as the q-series involving the associated dominant integral weights and string functions. One of the main results of this paper is the explicit calculations of the branching functions appearing in ([9], Theorem 3.1). We are particularly interested in the calculations of the branching functions obtained from certain tensor product decompositions of level 2 integrable and principal admissible representations over slb2 (see Theorem4). We shall see that these branching functions connect the representation theory of affine Lie algebras with the representation theory of super-Virasoro algebras. We usually apply the theory of modular functions for calculations of string functions [10]. However, in the current work we shall not use the tools of modular functions for the calculations of the string functions appearing in ([9], Theorem 3.1). Instead, we shall use both the invariance properties of

Axioms 2019, 8, 82; doi:10.3390/axioms8030082 www.mdpi.com/journal/axioms Axioms 2019, 8, 82 2 of 19 string functions under the action of affine Weyl group and the character formula whose summation is taken over maximal weights (see Theorem5). It seems like that this approach provides a simpler way for computations of the string functions in our cases. We would like to point out that in ([5], Corollary 3(c)) the authors expressed the branching functions in terms of theta functions. We shall show that our expressions for the branching functions appearing in Theorem4 are actually same as those of ([ 5], Corollary 3(c)) through the investigations of the characters of the minimal series representations of super-Virasoro algebras. Comparing our calculations of the branching functions over slb2 with the characters of the minimal series representations of super-Virasoro algebras, we also present the tensor product decompositions between level 2 integrable and principal admissible representations in terms of the minimal series representations of super-Virasoro algebras (see Theorem6). This generalizes the decomposition formula appearing in ([6], Section 4.1(a)) to the case of principal admissible weights.

2. Preliminaries A = a
g Let ij 1≤i,j≤n be a symmetrizable generalized Cartan matrix and the Kac-Moody Lie algebra associated with A. Let h be a Cartan subalgebra of g. Fix the set of simple roots Π = ∨  ∨ ∨ ∗ {α1, ··· , αn} of h and simple coroots Π = α1 , ··· , αn of h , respectively. Assume that Π and ∨ ∨
Π satisfy the condition αj αi = aij. We denote by ( | ) the non-degenerate invariant symmetric bilinear form on g. Write ∆, ∆+ and ∆− for the set of all roots, positive roots and negative roots of g, respectively. Put ∆re = {α ∈ ∆ | (α|α) > 0} and ∆im = {α ∈ ∆ | (α|α) ≤ 0}. For each i = 1, ··· , n, we ∗ define the fundamental reflection rαi of h by

∨
∗ rαi (λ) = λ − λ αi αi (λ ∈ h ) .

The subgroup W of GL (h∗) generated by all fundamental reflections is called the Weyl group of g. Among symmetrizable Kac-Moody Lie algebras, the most important Lie algebras are affine Lie algebras whose associated Cartan matrices are called affine Cartan matrices. It is known that every affine Cartan matrix is a positive semidefinite of corank 1. Each affine Cartan matrix is in one-to-one (r) correspondence with the affine Dynkin diagram of type Xn , where X = A, B, C, D, E, F or G and r = 1, 2 or 3. The number r is called the tier number (see [11,12] for details). Given an affine Cartan A = a
(l + ) a∨
(a ) matrix ij 0≤i,j≤l , two 1 -tuples i 0≤i≤l and i 0≤i≤l of positive integers are uniquely determined by the conditions

1. a∨, a∨, ··· , a∨
A = O, 0 1  l a0    a1  2. A   = O,  .   .  al ∨ ∨ ∨
3. gcd a0 , a1 , ··· , al = gcd (a0, a1, ··· , al) = 1, ( ) ∨
where O is the zero vector. We call ai 0≤i≤l (resp. ai 0≤i≤l) the label (resp. colabel) of the affine matrix l ∨ l ∨ A. The corresponding positive integer h = ∑i=0 ai (resp. h = ∑i=0 ai ) is called the Coxeter number l ∨ ∨ (resp. dual Coxeter number). Notice that the element K = ∑i=0 ai αi satisfies αi(K) = 0 for 0 ≤ i ≤ l, and we call this element the central element. Through the non-degenerate bilinear form ( | ) defined on l ∗ g, the central element K corresponds to δ = ∑i=0 aiαi in h . g A = a
Suppose that is the affine Lie algebra associated to an affine Cartan matrix ij 0≤i,j≤l, and let h be a Cartan subalgebra of g. The Cartan subalgebra h is (l + 2)-dimensional, and we can decompose h and h∗ as follows: h = h ⊕ CK ⊕ Cd, ∗ ∗ h = h ⊕ Cδ ⊕ CΛ0, Axioms 2019, 8, 82 3 of 19

l ∨ ∗ l where h = ∑i=1 Cαi and h = ∑i=1 Cαi. l ∨ l ∨ The lattice Q = ∑i=0 Zαi and Q = ∑i=0 Zαi are called the root lattice and coroot lattice, respectively. Set ( ∨ ( ) Q if r = 1 or A = A 2 , M = 2l ≥ 6= (2) Q if r 2 and A A2l . ∗ For an element α ∈ Q, we define tα ∈ GL (h ) by ( ) |α|2 t (λ) = λ + (λ|δ) α − (λ|δ) + (λ|α) δ (λ ∈ h∗) . α 2

We call tα (α ∈ Q) the translation operator. It is known that the Weyl group W of the affine Lie algebra g is also given by W n tM, where W = hrαi |1 ≤ i ≤ li and tM = {tα|α ∈ M} . For a non-twisted affine Lie algebra (i.e., r = 1), recall that

∆im = {nδ|n ∈ Z − {0}} and re  ∆ = nδ + α|n ∈ Z, α ∈ ∆ , where ∆ is the set of all roots of the finite-dimensional simple Lie algebra associated with the finite A = a
Cartan matrix ij 1≤i,j≤l. Set

 ∗ ∨
P = λ ∈ h |λ αi ∈ Z for 0 ≤ i ≤ l , Pm = {λ ∈ P|λ(K) = m} ,  ∗ ∨
P+ = λ ∈ h |λ αi ∈ Z≥0 for 0 ≤ i ≤ l , m m P+ = P ∩ P+.

An element in P (reps. P+) is called an integral weight (resp. a dominant integral weight). Let ρ be the ∨
dominant integral weight defined by ρ αi = 1 for 0 ≤ i ≤ l. The element ρ is called the Weyl vector of g. It is sometimes convenient to choose the Weyl vector satisfying the additional condition ρ(d) = 0, ∨ and we get ρ = ρ + h Λ0 in this case. ∗  ∨ Define the fundamental weights Λi ∈ h (0 ≤ i ≤ l) by Λi αj = δij (0 ≤ j ≤ l) and Λi (d) = 0. ∨ ∨
Similarly, we define the fundamental coweights Λi ∈ h (0 ≤ i ≤ l) by Λi |αj = δij (0 ≤ j ≤ l) and ∨
∨ ∨ ∗ l Λi |d = 0. Let Λi and Λi be the restrictions of Λi and Λi to h and h, respectively. Put P = ∑i=1 ZΛi ∨ l ∨ and P = ∑i=1 ZΛi , and let us introduce a lattice

( ∨ ( ) P if r = 1 or A = A 2 , Me = 2l ≥ 6= (2) P if r 2 and A A2l .

W = W t extended affine Weyl group g Then, the group e n Me is called the of .

3. Branching functions for admissible weights Let g be the Kac-Moody Lie algebra associated to a symmetrizable generalized Cartan matrix A, and h a Cartan subalgebra of g. An element λ ∈ h∗ satisfying conditions

∨ re re 1. hλ + ρ, α i ∈ Q − Z≤0 for all α ∈ ∆+ := ∆ ∩ ∆+,  re ∨ re 2. Q-span of α ∈ ∆+| hλ + ρ, α i ∈ Z = Q-span of ∆+ Axioms 2019, 8, 82 4 of 19 is called an admissible weight. When λ is an admissible weight, the corresponding irreducible highest weight g-module L (λ) is called an admissible g-module or admissible representation. Write

∨re  ∨ re ∨ ∆λ = α |α ∈ ∆ and λ + ρ, α ∈ Z≥1 .

∨re ∨ ∨ Then, it is easy to see that ∆λ forms a subroot system of the coroot system ∆ . We denote by Πλ a ∨ re ∨ base of ∆λ , and put Wλ = rα|α ∈ Πλ . ∨ ∨ An admissible weight λ is called a principal admissible weight if Πλ is isomorphic to Π . In general, the level of a principal admissible weight is a rational number. In fact, it is known from [7] that a v rational number m = u (u ∈ Z≥1, v ∈ Z, gcd(u, v) = 1) is the level of principal admissible weights if and only if it satisfies

1. gcd (u, r∨) = 1, 2. u (m + h∨) ≥ h∨, where r∨ is the tier number of the transposed generalized Cartan matrix At and h∨ denotes the dual Coxeter number of g. Henceforth, we assume that g is an affine Lie algebra with a simple coroot system Π∨ =  ∨ ∨ α0 , ··· , αl . ∨ ∨ Given u ∈ Z≥1, put γ0 = (u − 1)c + α0 and γi = αi (1 ≤ i ≤ l). Define S(u) = {γi | 0 ≤ i ≤ l}. ∨  l  ∨ Then, S(u) becomes a simple coroot system of ∆ ∩ ∑i=0 Zγi if gcd (u, r ) = 1 (see [13], Lemma 3.2.1). Moreover, the following theorems are known.

v   Theorem 1. Let m = u with u ∈ Z≥1, v ∈ Z and gcd(u, v) = 1. Assume that y ∈ We satisfies y S(u) ⊂ ∨ m ∨   ∆+. Write Pu,y for the set of all principal admissible weights λ of level m with Πλ = y S(u) . Then, we have

∨ ∨ m n  0 ∨
 0 u(m+h )−h o Pu,y = y λ − (u − 1) m + h Λ0 + ρ − ρ | λ ∈ P+ .

Proof. See ([7], Theorem 2.1) or ([9], Proposition 1.5).

v m Theorem 2. Let m = u with u ∈ Z≥1, v ∈ Z and gcd(u, v) = 1. Let P+ be the set of all principal admissible m S m weights of level m (we use the same notation as the case of dominant integral weights). Then, P+ = y Pu,y, n   ∨ o where y runs over y ∈ We |y S(u) ⊂ ∆+ .

Proof. See ([9], Proposition 1.5).

Let us now review branching functions and their connections with the Virasoro algebra. L Recall the Virasoro algebra is an infinite dimensional Lie algebra Vir = ( n∈Z C`n) ⊕ Cc with brackets [`m, c] = 0 for all m ∈ Z and m3 − m [`m, `n] = (m − n)`m+n + δm+n,0 c for all m, n ∈ Z. 12  −  Let g be a finite dimensional simple Lie algebra, and g = C t, t 1 ⊗ g ⊕ CK ⊕ Cd the non-twisted affine Lie algebra over g. Let V be the highest weight g-module of level m such that m + h∨ 6= 0. g Define the operators Ln (n ∈ Z) via

dimg g 1 Lg(z) = L z−n−2 = : ui(z)u (z) :, (1) ∑ n 2 (m + h∨) ∑ i n∈Z i=1 Axioms 2019, 8, 82 5 of 19

 i j
where u and {ui} are bases of g satisfying ui|u = δij. It is well-known that V becomes a Vir-module by letting g mdimg `n 7−→ Ln (n ∈ Z) and c 7−→ . (2) m + h∨ The Virasoro action (2) satisfies the following properties:

h j i j+n `n, t ⊗ X = −jt ⊗ X (X ∈ g, n ∈ Z − {0} , j ∈ Z) , (3) (Λ + 2ρ|Λ) ` = Id − d. (4) 0 2 (m + h∨)

Let p be a reductive subalgebra of g. Then, p is decomposed as p = p0 ⊕ p1 ⊕ · · · ⊕ ps, where p0 is the center of p and each pi (i = 1, ··· , s) is a simple Lie algebra. Assume that

s ! p0 ⊕ ∑ hi ⊂ h i=1 and s ∑ pi+ ⊂ g+, i=1 where hi (resp. h) is a Cartan subalgebra of pi (resp. g) and pi+ (resp. g+) is the sum of the positive −1
root spaces of pi (resp. g). Consider the affinization p = C[t, t ] ⊗ p ⊕ CK˙ ⊕ Cd of p. Since V is the highest weight g-module, V is also the highest weight p-module. However, the action of the central element K˙ on V is somewhat complicated. We refer to ([11], Chapter 12) for the details of the action of 0 the central element K˙ . Let m˙ be the level of V as a p-module, and write ( | ) for the standard bilinear form on p. Set dimp p 1 Lp = L z−n−2 = : u˙i(z)u˙ (z) :, (5) ∑ n + ˙ ∨
∑ i n∈Z 2 m˙ h i=1 0  i i
∨ where u˙ and {u˙i} are bases of p satisfying u˙ |u˙ j = δij and h˙ is the dual Coxeter number of p. Using (1) and (5), define g;p g p g;p −n−2 L (z) = L − L = ∑ Ln z . n∈Z Due to (3), it follows that

h g;p j i Ln , t ⊗ X = 0 for all X ∈ p, n ∈ Z − {0} and j ∈ Z. (6)

Applying the operator product expansions, we can verify that Lg;p(z) is, in fact, a Virasoro field with g;p mdimg mdim˙ p g;p the central charge c = ∨ − (see [13,14] for the details). We call the Virasoro field L (z) m+h m˙ +h˙ ∨ the coset Virasoro field. In the remaining part of this section, we assume that V = L (Λ) for a dominant integral weight Λ ˙  ˙ ∗ of level m. Let h be a Cartan subalgebra of p, and p+ the positive part of p. For ν ∈ h ⊕ CK˙ , set

g;p n  ˙ o Vν = v ∈ L (Λ) |Xv = 0 (∀ X ∈ p+) , Hv = ν (H) v ∀ H ∈ h ⊕ CK˙ .

g;p g;p g;p Due to (4) and (6), Vν is stable under the actions of Ln (n ∈ Z). So, Vν becomes a Vir-module. We call this module the coset Virasoro module. Notice that L (Λ) is decomposed as a Vir ⊕ [p, p]-module into M g;p
L (Λ) = Vν ⊗ L˙ (ν) , (7) ˙ ∗ ν∈h ⊕Cδ˙ mod Cδ˙ Axioms 2019, 8, 82 6 of 19 where L˙ (ν) is the irreducible [p, p]-module with highest weight ν and δ˙ is identified with K˙ via the non-degenerate bilinear form on p. From (7), we define a function

Λ −d
j  −δ c (q) = Tr g;p q = mult ν − jδ˙; p q q = e , (8) ν Vν ∑ Λ j∈Z≥0 where the multiplicity is defined as in ([6], Section 1.6). The function (8) is called the string function. Using the string function (8), the decomposition (7) yields the following formula for the character of L (Λ): Λ ˙ chL (Λ) = ∑ cν (q)chL (ν) . (9) ˙ ∗ ν∈h ⊕Cδ˙ mod Cδ˙ Let us now introduce the following numbers:

2 2 • = |Λ+ρ| − |ρ| mΛ 2(m+h∨) 2h∨ , | + |2 | |2 • m˙ = ν ρ˙ − ρ˙ , ν 2(m˙ +h˙ ∨) 2h˙ ∨ where ρ˙ is the Weyl vector associated with p. Λ mΛ−m˙ ν Λ 2πiτ Then, we define the branching function as bν (τ) = q cν (q) for q = e . By the strange Λ − 1 cg;p ` formula and (4), we see that the branching function also can be written as b (τ) = q 24 Tr g;p q 0 ν Vν (see [11] (Chapter 12) for the strange formula). 0 Recall that the normalized character ch L (Λ) is defined as

0 ch L (Λ) = e−mΛδchL (Λ) .

Introducing the coordinate (τ, z, t) for h = 2πi (−τd + z + tK) ∈ h, we obtain that 0 ( ) = mΛ ( ) chL(Λ) τ, z, t q chL(Λ) τ, z, t . So, the Formula (9) can be rewritten as

0 0 ( ) = Λ ( ) ( ) chL(Λ) τ, z, t ∑ bν τ chL˙ (ν) τ, z, t . ˙ ∗ ν∈h ⊕Cδ˙ mod Cδ˙

4. Tensor Product Decompositions  − 
In this section, we fix an affine Lie algebra g = C t, t 1 ⊗ g ⊕ CK ⊕ Cd over a finite dimensional simple Lie algebra g. We also fix a Cartan subalgebra h of g. For λ, µ ∈ h∗, let L (λ) and L (µ) be irreducible highest weight modules over g. We denote by πλ and πµ the representations 0 of g on L (λ) and L (µ), respectively. Put m = λ (K) and m = µ (K). Assume that m + h∨ 6= 0, 0 0 m + h∨ 6= 0 and m + m + h∨ 6= 0. It follows from (2) that the Virasoro algebra Vir acts on L (λ) and L (µ). The corresponding Virasoro fields are

1 dimg   Lλ(z) = : π (u (z)) π ui(z) : ( + ∨) ∑ λ i λ 2 m h i=1 and 1 dimg   µ( ) = ( ( )) i( ) L z 0 ∨
∑ : πµ ui z πµ u z :. 2 m + h i=1 Notice that the Virasoro algebra Vir acts on L (λ) ⊗ L (µ) via the tensor product action

λ,µ λ µ L (z) = L (z) ⊗ IdL(µ) + IdL(λ) ⊗ L (z)

0 mdimg m dimg with the central charge ∨ + 0 . m+h m +h∨ Axioms 2019, 8, 82 7 of 19

On the other hand, we may consider the whole tensor product L (λ) ⊗ L (µ) as the highest weight g-module. Applying (2) to the highest weight g-module L (λ) ⊗ L (µ), we get the associated Virasoro field

1 dimg   λ⊗µ( ) = ⊗
( ( )) ⊗
i( ) L z 0 ∨
∑ πλ πµ ui z πλ πµ u z 2 m + m + h i=1

 0  m+m dimg with the central charge 0 . m+m +h∨ Using (3), we have

h λ,µ n i n+m Lm , t ⊗ X = −nt ⊗ X (X ∈ g, m ∈ Z − {0} , n ∈ Z) ,

h λ⊗µ n i n+m Lm , t ⊗ X = −nt ⊗ X (X ∈ g, m ∈ Z − {0} , n ∈ Z) . (10)

λ,µ λ⊗µ −n−2 Set eL(z) = L (z) − L (z) = ∑n∈ eLnz . According to ([15], Proposition 10.3), the field eL(z) Z  0  0 + g mdimg m dimg m m dim yields the coset Virasoro field on L (λ) ⊗ L (µ) with central charge ∨ + 0 − 0 . m+h m +h∨ m+m +h∨ ∗ For µ ∈ h ⊕ Cδ, we define

λ,µ 
Vµ = v ∈ L (λ) ⊗ L (µ) |Xv = 0 (∀ x ∈ g+) , Hv = µ (H) v ∀ H ∈ h ⊕ CK .

λ,µ It follows from (10) that the space Vµ becomes a Vir-module via the coset Virasoro field eL(z). Notice that L (λ) ⊗ L (µ) is decomposed as a Vir ⊕ [g, g]-module into

λ,µ L (λ) ⊗ L (µ) = ∑ Vµ ⊗ L (µ) . (11) ∗ µ∈h ⊕Cδ mod Cδ

We obtain from (11) a string function

λ⊗µ −d j cν (q) = Tr λ,µ q = multλ⊗µ (µ − jδ; g) q . (12) Vµ ∑ j∈Z≥0

Using (11) and (12), we get

λ⊗µ ch (λ) ch (µ) = ∑ cν (q)chL (ν) . (13) ∗ µ∈h ⊕Cδ mod Cδ

If we define the normalized branching function by

⊗ ⊗ λ µ mλ+mµ−mν λ µ bν (τ) = q cν (q) , then the Formula (13) yields

0 0 λ⊗µ 0 chL(λ) (τ, z, t) chL(µ) (τ, z, t) = ∑ bν (τ) chL(ν) (τ, z, t) . (14) ν

Let Λ be a dominant integral weight and µ a principal admissible weight of the affine Lie algebra g. Then, the branching function of the tensor product L (Λ) ⊗ L (µ) can be expressed in terms of the string functions of L (Λ) as follows.

0 v Theorem 3. Let g be any affine Lie algebra and m ∈ Z≥0. Let m = u with u ∈ Z≥1, v ∈ Z and  0  gcd(u, v) = 1. Assume that Λ and µ0 are dominant integral weights of level m and u m + h∨ − h∨, Axioms 2019, 8, 82 8 of 19

| |2 m − ξ ∗ Λ ( ) Λ 2m Λ ( ) ∈ h ⊕ respectively. Write ceξ q for the modified string function q cξ,(g;g) q for ξ Cδ, where Λ ( ) (g g) p = g cξ,(g;g) q is the string function defined with respect to the pair ; (i.e., in (8)). Then, for a principal 0  0  0 ∨  m admissible weight µ = y µ − (u − 1) m + h Λ0 + ρ − ρ ∈ Pu,y, the following formula holds:

0 0 Λ⊗µ 0 chL(Λ) (τ, z, t) chL(µ) (τ, z, t) = ∑ bν (τ) chL(ν) (τ, z, t) , 0 m+m ν∈Pu,y s.t. ν≡Λ+µ mod Q where

   0 ∨ 0 ∨ 2 m +h m+m +h w(ν0+ρ) 0 − µ +ρ 2m 0 ∨ 0 ∨ Λ⊗µ m+m +h m +h Λ bν (τ) = e(w)q c 0 0 (q) . ∑ ey(w(ν +ρ)−(µ +ρ)−(u−1)mΛ0) w∈W

Proof. See ([9], Theorem 3.1).

Λ Λ In the next section, we simply write ceλ for ceλ (q) if no confusion seems likely to arise, and will calculate explicitly the branching functions for some specific cases.

5. Explicit Calculations of Branching Functions

Let Λ0 and Λ1 be the fundamental weights of slb2, and λ a principal admissible weight of slb2. In this section, we explicitly calculate the branching functions arising from the tensor product decompositions of (L (2Λ0) ⊕ L (2Λ1)) ⊗ L (λ) and L (ρ) ⊗ L (λ). Let us write Π = {α} for the simple root system of sl2. Then it is easy to check

1 h∨ = 2, ρ = Λ + Λ = ρ + h∨Λ and Λ = Λ + α (15) 0 1 0 1 0 2

v for slc2. Let m = u (u ∈ 2Z≥1, v ∈ 2Z + 1), and choose a principal admissible weight λ of level m 0 m 0 u(m+2)−2 satisfying λ = λ − (u − 1)(m + 2)Λ0 ∈ Pu,1 for λ ∈ P+ (see Theorems1 and2). Applying Theorem3 to the tensor product representations L (2Λ0) ⊗ L (λ) and L (2Λ1) ⊗ L (λ), we obtain

0 0 ch (τ, z, t) ch (τ, z, t) L(2Λ0) L(λ) 0 2Λ0⊗λ = ∑ bν (τ) chL(ν) (τ, z, t) , (16) m+2 ν∈Pu,1 s.t. ν≡2Λ0+λ mod Q where 2 w(ν0+ρ) 0 (m+2)(m+4) − λ +ρ 4 m+4 m+2 2Λ0⊗λ 2Λ0 bν (τ) = e(w)q c 0 0 ∑ ew(ν +ρ)−(λ +ρ)−2(u−1)Λ0 w∈W and

0 0 ch (τ, z, t) ch (τ, z, t) L(2Λ1) L(λ) ⊗ 0 = b2Λ1 λ (τ) ch (τ, z, t) , (17) ∑ νe L(νe) m+2 νe∈Pu,1 s.t. νe≡2Λ1+λ mod Q where 2 w(ν0+ρ) 0 (m+2)(m+4) e − λ +ρ 4 m+4 m+2 2Λ1⊗λ 2Λ1 b (τ) = e(w)q c 0 0 . νe ∑ ew(νe +ρ)−(λ +ρ)−2(u−1)Λ0 w∈W Axioms 2019, 8, 82 9 of 19

Similarly, if we apply Theorem3 to the tensor product representation L (ρ) ⊗ L (λ) then we have

0 0 chL(ρ) (τ, z, t) chL(λ) (τ, z, t) ρ⊗λ 0 = ∑ bν (τ) chL(ν) (τ, z, t) , (18) m+2 ν∈Pu,1 s.t. ν≡ρ+λ mod Q where 2 w(ν0+ρ) 0 (m+2)(m+4) − λ +ρ 4 m+4 m+2 ρ⊗λ ρ bν (τ) = e(w)q c 0 0 . ∑ ew(ν +ρ)−(λ +ρ)−2(u−1)Λ0 w∈W

0 u(m+2)−2 0 u(m+4)−2 For λ ∈ P+ and ν ∈ P+ , let us write

0 λ = (u(m + 2) − 2 − n) Λ0 + nΛ1, 0  0  0 ν = u(m + 4) − 2 − n Λ0 + n Λ1 (19)

0 for some n ∈ Z and n ∈ Z. Then, we can rewrite λ and ν in (16) as

0 λ = λ − (u − 1)(m + 2)Λ0 = (m − n)Λ0 + nΛ1 and 0  0  0 ν = ν − (u − 1)(m + 4)Λ0 = m − n + 2 Λ0 + n Λ1.

0 Since 2Λ0 − (ν − λ) ∈ Q and 2Λ1 − 2Λ0 = α, we should have n ≡ n (mod 2). 0  00  00 u(m+4)−2 Similarly, for νe = u(m + 4) − 2 − n Λ0 + n Λ1 ∈ P+ , we obtain νe =  00  00  00  m − n + 2 Λ0 + n Λ1 n ∈ Z . From the condition 2Λ1 − (νe − λ) ∈ Zα, we have the same 00 condition n ≡ n (mod 2) as the case of ν. For this reason, we shall identify νe with ν in the following Theorem4. The same argument yields that the condition ν ≡ ρ + λ mod Q in (18) is equivalent to the 0 condition n ≡ n + 1 (mod 2) in (19).

v 0 Theorem 4. Let m = u for u ∈ 2Z≥1 and v ∈ 2Z + 1, and let p = u(m + 4) and p = u(m + 2). 1. Suppose that 0 λ = (u(m + 2) − 2 − n) Λ0 + nΛ1 and 0  0  0 ν = u(m + 2) − 2 − n Λ0 + n Λ1

0 0 for some n ∈ 4Z and n ∈ Z satisfying n ≡ n (mod 2). Then, the branching functions in (16) and (17) are explicitly given by

2 1  0  0  0  0 2pp j+ n +1 p −(n+1)p 2Λ0⊗λ 8pp bν (τ) = ∑ q A j∈Z 2 1  0  0  0  0 2pp j− n +1 p −(n+1)p − ∑ q 8pp B (20) j∈Z Axioms 2019, 8, 82 10 of 19

and

2 1  0  0  0  0 2pp j+ n +1 p −(n+1)p 2Λ1⊗λ 8pp bν (τ) = ∑ q B j∈Z 2 1  0  0  0  0 2pp j− n +1 p −(n+1)p − ∑ q 8pp A, (21) j∈Z

( 0 A = c2Λ0 , B = c2Λ0 if n ≡ 0 (mod 4) e2Λ0 e2Λ1 where 0 A = c2Λ0 , B = c2Λ0 if n ≡ 2 (mod 4) . e2Λ1 e2Λ0 2. Assume that 0 λ = (u(m + 2) − 2 − n) Λ0 + nΛ1 and 0  0  0 ν = u(m + 2) − 2 − n Λ0 + n Λ1

0 for some n ∈ 4Z and n ∈ 4Z + 1. Then, the branching function in (18) is explicitly given by

ρ⊗λ bν (τ) 2 2 1  0  0  0  1  0  0  0  ! 0 2pp j+ n +1 p −(n+1)p 0 2pp j− n +1 p −(n+1)p 8pp 8pp ρ = ∑ q − q ceρ. (22) j∈Z

Proof. We first prove (20) and (21).  Recall that the Weyl group W of slc2 is given by tjα, tjαrα|j ∈ Z . By (15) and (19), we have

0 n + 1 ν0 + ρ = u(m + 4)Λ + α 0 2 and n + 1 λ0 + ρ = u(m + 2)Λ + α. 0 2 So, we get

 0   0  tjα ν + ρ − λ + ρ − 2(u − 1)Λ0 0 ! n − n   0   = 2Λ + u(m + 4)j + α − u(m + 4)j2 + n + 1 j δ (23) 0 2 and

 0   0  tjαrα ν + ρ − λ + ρ − 2(u − 1)Λ0 0 ! n + n + 2   0   = 2Λ + u(m + 4)j − α + u(m + 4)j2 − n + 1 j δ. (24) 0 2

Notice from ([11], (12.7.9)) that we have

2Λ0 2Λ0 c 0 = c 0 (25) ew(λ )+2γ+aδ eλ

0 ∗ for λ ∈ h , w ∈ W, γ ∈ Zα and a ∈ C. Since W = {1, rα}, we see from (25) that

2Λ0 = 2Λ0 = 2Λ0 ceλ+(2n+1)α+aδ ce(λ+α)+2nα+aδ ceλ+α Axioms 2019, 8, 82 11 of 19 and c2Λ0 = c2Λ0 = c2Λ0 . erα(λ)+(2n+1)α+aδ erα(λ+α)+(2n+2)α+aδ eλ+α Hence, in any case we obtain 2Λ0 = 2Λ0 cew(λ)+(2n+1)α+aδ ceλ+α (26) for w ∈ W. Since u is even, we have

0 0 n − n n − n u(m + 4)j + ≡ (mod2) 2 2 and 0 0 n + n + 2 n + n + 2 u(m + 4)j − ≡ − (mod 2) . 2 2 0 0 0 0 Since n ∈ 4Z and n ≡ n (mod 2), we obtain n ≡ 0 (mod 4) or n ≡ 2 (mod 4). If n ≡ 0 (mod 4), 0 0 n −n n +n+2 then 2 ≡ 0 (mod 2) and − 2 ≡ 1 (mod 2). Thus, by (23), (24), (25) and (26) we get

2Λ0 2Λ0 c 0 0 = c (27) etjα(ν +ρ)−(λ +ρ)−2(u−1)Λ0 e2Λ0 and 2Λ0 2Λ0 2Λ0 c 0 0 = c = c . (28) etjαrα(ν +ρ)−(λ +ρ)−2(u−1)Λ0 e2Λ0+α e2Λ1 0 0 0 n −n n +n+2 Similarly, if n ≡ 2 (mod 4), then 2 ≡ 1 (mod 2) and − 2 ≡ 0 (mod 2). So, in this case we have 2Λ0 2Λ0 2Λ0 c 0 0 = c = c (29) etjα(λ +ρ)−(µ +ρ)−2(u−1)Λ0 e2Λ0+α e2Λ1 and 2Λ0 2Λ0 c 0 0 = c . (30) etjαrα(λ +ρ)−(µ +ρ)−2(u−1)Λ0 e2Λ0 2 (m+2)(m+4) w(ν0+ρ) λ0+ρ We now compute the exponent − of q in (16) and (17). 4 m+4 m+2 Since p = u(m + 4) in assumption, we see that

0 !   n + 1   0   t ν0 + ρ = pΛ + pj + α − pj2 + n + 1 j δ. (31) jα 0 2 and 0 !   n + 1   0   t r ν0 + ρ = pΛ + pj − α − pj2 − n + 1 j δ. (32) jα α 0 2

0 It also follows from the assumption p = u(m + 2) that

2 ( + )( + ) w ν0 + ρ
0 + m 2 m 4 λ ρ − 4 m + 4 m + 2 2 2( + )( + ) w ν0 + ρ
0 + u m 2 m 4 λ ρ = − (33) 4 u(m + 4) u(m + 2) 0 2 w ν0 + ρ
0 + pp λ ρ = − 0 . 4 p p Axioms 2019, 8, 82 12 of 19

Notice from (31) and (32) that

0
0 tj ν + ρ λ + ρ α − p p0 0 ! !   n + 1 1 0 n + 1 = Λ0 + j + α − 0 p Λ0 + α mod Cδ 2p p 2 0 ! n + 1 n + 1 = j + − 0 α mod Cδ 2p 2p and

0
0 tj rα ν + ρ λ + ρ α − p p0 0 ! !   n + 1 1 0 n + 1 = Λ0 + j − α − 0 p Λ0 + α mod Cδ 2p p 2 0 ! n + 1 n + 1 = j − − 0 α mod Cδ. 2p 2p

Thus, we obtain

2 0
0 tjα ν + ρ λ + ρ 1  0  0  0 2 − = 2pp j + n + 1 p − (n + 1)p , 0 0 2 p p 2 pp
2 0
0 tjαrα ν + ρ λ + ρ 1  0  0  0 2 − = 2pp j − n + 1 p − (n + 1)p . (34) 0 0 2 p p 2 pp

0 Hence, if n ≡ 0 (mod 4), then we obtain from (27), (28), (33) and (34) that

2 1  0  0  0  ⊗ 0 2pp j+ n +1 p −(n+1)p b2Λ0 λ (τ) = q 8pp c2Λ0 ν ∑ e2Λ0 j∈Z 2 1  0  0  0  0 2pp j− n +1 p −(n+1)p − q 8pp c2Λ0 . ∑ e2Λ1 j∈Z

0 If n ≡ 2 (mod 4) then we also obtain from (29), (30), (33) and (34)

2 1  0  0  0  ⊗ 0 2pp j+ n +1 p −(n+1)p b2Λ0 λ (τ) = q 8pp c2Λ0 ν ∑ e2Λ1 j∈Z 2 1  0  0  0  0 2pp j− n +1 p −(n+1)p − q 8pp c2Λ0 . ∑ e2Λ0 j∈Z

The Formula (20) now follows. 2Λ1⊗λ Applying the same argument as above to the case of bν (τ), we obtain

0  2Λ1 2Λ1 c 0 0 = c if n ≡ 0 (mod 4) etjα(ν +ρ)−(λ +ρ)−2(u−1)Λ0 e2Λ0 0 (35) 2Λ1 2Λ1 c 0 0 = c if n ≡ 0 (mod 4) . etjαrα(ν +ρ)−(λ +ρ)−2(u−1)Λ0 e2Λ1 Axioms 2019, 8, 82 13 of 19 and 0  2Λ1 2Λ1 c 0 0 = c if n ≡ 2 (mod 4) etjα(ν +ρ)−(λ +ρ)−2(u−1)Λ0 e2Λ1 0 (36) 2Λ1 2Λ1 c 0 0 = c if n ≡ 2 (mod 4) . etjαrα(ν +ρ)−(λ +ρ)−2(u−1)Λ0 e2Λ0

MΛ +NΛ NΛ +MΛ Notice that we have c 0 1 = c 0 1 due to the outer automorphism of slb . emΛ0+nΛ1 enΛ0+mΛ1 2 Hence, we obtain that c2Λ1 = c2Λ0 and c2Λ1 = c2Λ0 (37) e2Λ0 e2Λ1 e2Λ1 e2Λ0 0 Therefore, if n ≡ 0 (mod 4) then we get from (35), (36), (33), (34) and (37) that

2 1  0  0  0  ⊗ 0 2pp j+ n +1 p −(n+1)p b2Λ1 λ (τ) = q 8pp c2Λ0 ν ∑ e2Λ1 j∈Z 2 1  0  0  0  0 2pp j− n +1 p −(n+1)p − q 8pp c2Λ0 . ∑ e2Λ0 j∈Z

0 Similarly, if n ≡ 2 (mod 4) then we obtain that

2 1  0  0  0  ⊗ 0 2pp j+ n +1 p −(n+1)p b2Λ1 λ (τ) = q 8pp c2Λ0 ν ∑ e2Λ0 j∈Z 2 1  0  0  0  0 2pp j− n +1 p −(n+1)p − q 8pp c2Λ0 . ∑ e2Λ1 j∈Z

The Formula (21) now follows. Let us now prove (22). The proof is exactly the same as those of (20) and (21) except for calculations of the string function ρ 0 c 0 0 . Recall from the assumption that n ∈ 4Z and n ∈ 4Z + 1. Then, by (23)–(25) ew(ν +ρ)−(λ +ρ)−2(u−1)Λ0 we obtain ρ ρ ρ ρ c 0 0 = c 0 = c 1 = cρ etjα(ν +ρ)−(λ +ρ)−2(u−1)Λ0 e n −n e2Λ0+ α e 2Λ0+ 2 α 2 and ρ ρ ρ ρ c 0 0 = c 0 = c 1 = cρ. etjαrα(ν +ρ)−(λ +ρ)−2(u−1)Λ0 e n +n+2 e2Λ0−2α+ α e 2Λ0− 2 α 2 The result now follows.

It is immediate from Theorem4 that the branching function of (L (2Λ0) ⊕ L (2Λ1)) ⊗ L (λ) for slb2 is given by

2Λ0⊗λ 2Λ1⊗λ bν (τ) + bν (τ) 2 1  0  0  0  0 2pp j+ n +1 p −(n+1)p   = q 8pp c2Λ0 + c2Λ0 ∑ e2Λ0 e2Λ1 j∈Z 2 1  0  0  0  0 2pp j− n +1 p −(n+1)p   − q 8pp c2Λ0 + c2Λ0 . ∑ e2Λ0 e2Λ1 j∈Z

ρ In the following theorem, we explicitly calculate c2Λ0 + c2Λ0 and c in terms of the Dedekind e2Λ0 e2Λ1 eρ eta function.

2Λ 2Λ η(τ) ρ η(2τ) 1 0 0 24 ∞ n Theorem 5. ce2Λ + ce2Λ = τ and ceρ = 2 , where η (τ) = q Πn=1 (1 − q ). 0 1 η( 2 )η(2τ) η(τ) Axioms 2019, 8, 82 14 of 19

Proof. It follows from the Weyl-Kac character formula that

1 = ( ) w(2Λ0+ρ) chL (2Λ0) ρ ∑ e w e , (38) e R w∈W where  W = tjα, tjαrα|j ∈ Z and ∞ n  −α n−1 α n R = Πn=1 (1 − q ) 1 − e q (1 − e q ) .

Calculating w (2Λ0 + ρ) for w ∈ W, we obtain from (38) ! ( )2+ (− − )2+(− − ) 1 2Λ +ρ+4jα− 4j 4j δ 2Λ +ρ+(−4j−1)α− 4j 1 4j 1 δ chL (2Λ ) = e 0 4 − e 0 4 . (39) 0 eρR ∑ ∑ j∈Z j∈Z

Similarly, we can evaluate chL (2Λ1) as follows: ! ( + )2+( + ) (− − )2+(− − ) 1 − 1 2Λ +ρ+(4j+1)α− 4j 1 4j 1 δ 2Λ +ρ+(−4j−2)α− 4j 2 4j 2 δ q 2 e 0 4 − e 0 4 . (40) eρR ∑ ∑ j∈Z j∈Z

Using (39), (40) and the Jacobi triple product identity, we have

1 chL (2Λ0) − q 2 chL (2Λ1) 2+ 1 j 2Λ +ρ+jα− j j δ = (−1) e 0 4 eρR ∑ j∈Z

2Λ0 2+ e j jα j j = (−1) e q 4 R ∑ j∈Z

2Λ0 ∞ e  n   α n   −α n−1  = ∏ 1 − q 2 1 − e q 2 1 − e q 2 (41) R n=1 ∞ 2  2n−1   α 2n−1   −α 2n−1  = e Λ0 ∏ 1 − q 2 1 − e q 2 1 − e q 2 n=1 ∞ n− 1 ∞ 2 1 − q 2 n  α 2n−1   −α 2n−1  = e Λ0 (1 − q ) 1 − e q 2 1 − e q 2 ∏ − n ∏ n=1 1 q n=1 1 ∞ n− 2 2 1 − q 2 j jα j = e Λ0 (−1) e q 2 . ∏ 1 − qn ∑ n=1 j∈Z

Recall from ([11], (12.7.1)) that

= 2Λ0 λ chL (2Λ0) ∑ cλ e . (42) λ∈max(2Λ0) and = 2Λ1 λ chL (2Λ1) ∑ cλ e . (43) λ∈max(2Λ1)

2Λ 1 From (42) and (43), the coefficient of e 0 in chL (2Λ0) − q 2 chL (2Λ1) should be equal to

2Λ 1 2Λ c 0 − q 2 c 1 . (44) 2Λ0 2Λ0 Axioms 2019, 8, 82 15 of 19

Comparing (44) with the coefficient of e2Λ0 in (41), we obtain

∞ n− 1 2Λ 1 2Λ 1 − q 2 c 0 − q 2 c 1 = . (45) 2Λ0 2Λ0 ∏ − n n=1 1 q

1 By substituting x = q 2 , we obtain from (45)

∞ 1 − x2n−1 c2Λ0 − xc2Λ1 = . 2Λ0 2Λ0 ∏ − 2n n=1 1 x

By letting x 7−→ −x, we get ∞ 1 + x2n−1 c2Λ0 + xc2Λ1 = , 2Λ0 2Λ0 ∏ − 2n n=1 1 x and this implies ∞ n− 1 2Λ 1 2Λ 1 + q 2 c 0 + q 2 c 1 = . (46) 2Λ0 2Λ0 ∏ − n n=1 1 q

2 2 − |2Λ0| = − 1 − |2Λ0| = 7 On the other hand, it is easy to check that m2Λ0 4 16 and m2Λ1 4 16 , and these 2Λ − 1 2Λ 2Λ 7 2Λ yield that c 0 = q 16 c 0 and c 1 = q 16 c 1 . So, (46) gives rise to e2Λ0 2Λ0 e2Λ0 2Λ0

∞ n− 1 1  2Λ 2Λ  1 + q 2 q 16 c 0 + c 1 = . (47) e2Λ0 e2Λ0 ∏ − n n=1 1 q

Thus,

η (τ) τ
η 2 η (2τ) − 1 q 16 =  n  ∞ ∞ n ∏n=1 1 − q 2 ∏n=1 (1 + q ) − 1 ∞  n  q 16 ∏n=1 1 + q 2 = ∞ n ∞ n ∏n=1 (1 − q ) ∏n=1 (1 + q ) − 1 ∞  n− 1  q 16 ∏n=1 1 + q 2 = ∞ n ∏n=1 (1 − q ) = c2Λ0 + c2Λ1 (see (47)) . e2Λ0 e2Λ0

ρ Next, we compute ceρ. Replacing all positive roots α by kα (k ∈ Z≥1), we obtain from the denominator identity that

 mult(α) ekρ ∏ 1 − e−kα = ∑ e(w)ew(kρ). α∈∆+ w∈W Axioms 2019, 8, 82 16 of 19

Thus, it follows from the Jacobi triple identity that

chL (ρ) 1 = ( ) w(2ρ) ρ ∑ e w e e R w∈W 1  mult(α) = e2ρ 1 − e−2α eρR ∏ α∈∆+ 2j
 −2α 2(j−1) 2α 2j
∞ 1 − q 1 − e q 1 − e q   = eρ q = e−δ (48) ∏ j
−α (j−1)
α j
j=1 1 − q 1 − e q 1 − e q ∞ 1 + qj
∞       = ρ − j + −α (j−1) + α j e ∏ j
∏ 1 q 1 e q 1 e q j=1 1 − q j=1

∞ j
2− 1 + q ρ−jα j j = e q 2 ∏ − qj
∑ j=1 1 j∈Z

∞ j
2− 1 + q ρ−jα− j j δ = e 2 . ∏ − qj
∑ j=1 1 j∈Z

On the other hand, we get from ([11], (12.7.1)) that

= ρ λ chL (ρ) ∑ cλe . (49) λ∈max(ρ)

Comparing the coefficients of eρ in (48) and (49), we have

∞ 1 + qj ρ = cρ ∏ j . j=1 1 − q

|ρ|2 ρ ρ Moreover, it is easy to check mρ − 4 = 0 which implies ceρ = cρ. The result now follows.

6. Super-Virasoro algebras In this section, we shall investigate relationships between our results on branching functions and the representation theory of super-Virasoro algebras. As by-products, we generalize the tensor product decomposition formulas ([6], (4.1.2a) and (4.1.2b)) to the case of principal admissible weights.  −  Let us first review the theta functions associated to an affine Lie algebra g = C t, t 1 ⊗ g ⊕ CK ⊕ Cd and its Cartan subalgebra h. m For λ ∈ P (m ∈ Z≥0), the theta function θλ is defined as

|λ|2 − δ tαλ θλ = e 2m ∑ e , α∈Q where Q is the root lattice of g. Using the coordinate (τ, z, t) for the Cartan subalgebra h, we get

2πimt m |γ|2 2πim(γ|z) θλ (τ, z, t) = e ∑ q 2 e , λ γ∈Q+ m where λ is the projection of λ onto h. Axioms 2019, 8, 82 17 of 19

1 m In particular, if we take λ = md + 2 nα + rK ∈ P for g = slb2 then the corresponding theta function is 2πimt mk2 2πim(kα|z) θλ (τ, z, t) = e ∑ q e . (50) n k∈Z+ m Evaluating (50) at (τ, 0, 0), we have

2 m(j+ n )  2πiτ θ (τ, 0, 0) = ∑ q 2m q = e . (51) j∈Z

For convenience, we shall simply write θn,m for (51) in the remaining part of this section.

 1  1 Next, we review the super-Virasoro algebras Vire e = 0, 2 . (For e = 0 or 2 , Vire is called the Ramond and Neveu-Schwarz superalgebra, respectively.) The super-Virasoro algebra Vire is the complex superalgebra with a basis  c, `j, gm|j ∈ Z and m ∈ e + Z , and it satisfies commutation relations

  1 3
1. `i, `j = (i − j)`i+j + i − i δi+j,0c,   12 2. c, `j = 0, n
3. [gm, `n] = m − 2 gm+n, 4. [gm, c] = 0, 1  2 1  5. {gm, gn} = 2`m+n + 3 m − 4 δm+n,0c, where { , } denotes an anti-commutator bracket between two odd elements. Recall that every minimal series irreducible module of Vire corresponds to the pair of numbers    0 !      0 2 !  0  0 p,p 0 0 2 p−p p,p p,p p,p p,p 3 z , h . Here, z is the central charge equals z = 1 − 0 , and h r,s;e 2 pp r,s;e

 0   0 2  0 2 p,p pr−p s − p−p 1 0 is the minimal eigenvalue of ` equals hr s = 0 + (1 − 2e) for p, p , r, s ∈ , 0 , ;e 8pp 16 Z  0  0 0 p−p 0 0 2 ≤ p < p, p − p ∈ 2Z, gcd 2 , p = 1, 1 ≤ r ≤ p − 1, 1 ≤ s ≤ p − 1 and r − s ∈ 2Z (we refer to ([16], Theorem 5.2) for the details).  0   0 ! p,p p,p Write Ve z , hr,s;e for the minimal series module over Vire corresponding to

 0   0 ! p,p p,p z , hr,s;e . According to [17,18], it follows that

   0 ! 0  0  p,p p,p   p,p 1 z = 24 0 0 − 0 0 chVe z , hr,s;e q ηe (τ) θ pr−p s pp θ pr+p s pp , 2 , 2 2 , 2

 ( ) η 2τ if e = 0  ( )2 where η (τ) = η τ e η(τ) 1  τ if e = 2 . η( 2 )η(2τ) By (51), we see that

2 1  0 0  0 2pp j+pr−p s 0 0 8pp θ − = q , pr p s , pp ∑ 2 2 j∈Z 2 1  0 0  0 2pp j+pr+p s 0 0 8pp θ + = q . pr p s , pp ∑ 2 2 j∈Z Axioms 2019, 8, 82 18 of 19

 0   0 ! p,p p,p So, the normalized character of Ve z , hr,s;e is

 0  p,p χr,s;e (τ) 2 2 1  0 0  1  0 0  ! 0 2pp j+pr−p s 0 2pp j+pr+p s 8pp 8pp = ηe (τ) ∑ q − ∑ q , (52) j∈Z j∈Z

 0   0   0 ! p,p p,p  0  p,p − 1 z p,p where χr,s;e (τ) = q 24 chVe z , hr,s;e .

 0  Let r = −(n + 1) and s = n + 1 in (52). Then, by Theorem4, Theorem5 and (52), we obtain the following result.

v Proposition 1. Let m = u (u ∈ 2Z≥1, v ∈ 2Z + 1). Suppose that λ is a principal admissible weight of 0 m 0 u(m+2)−2 slc2 such that λ = λ − (u − 1)(m + 2)Λ0 ∈ Pu,1 for λ ∈ P+ . Then, the branching function 2Λ0⊗λ 2Λ1⊗λ ρ bν (τ) + bν (τ) (resp. bν (τ)) of (L (2Λ0) ⊕ L (2Λ1)) ⊗ L (λ) (resp. L (ρ) ⊗ L (λ)) is the same as  0   0  p,p p,p the normalized character χ 0 1 (τ) (resp. χ 0 (τ)) of the Neveu-Schwarz (resp. Ramond) −(n+1),n +1; 2 −(n+1),n +1;0 superalgebra.

It follows from Section4 that   2Λ0,λ 2Λ1,λ (L (2Λ0) ⊕ L (2Λ1)) ⊗ L (λ) = ∑ Vν ⊕ Vν ⊗ L (ν) (53) ν and ρ,λ  0  L (ρ) ⊗ L (λ) = ∑ V 0 ⊗ L ν , (54) 0 ν ν 0 m+2 0 where ν and ν are taken over Pu,1 such that ν ≡ 2Λ0 + λ mod Q and ν ≡ ρ + λ mod Q, respectively. According to [17] the coset Virasoro action introduced in Section 4 can be extended to the action of super-Virasoro algebras, and (53) and (54) can be considered as decompositions of Ve ⊕ [g, g]-module. 2Λ0 2Λ1 ρ Thus, (14) and Proposition1 imply that Vν ⊕ Vν (resp. Vν ) should be isomorphic to the  0 0   0 0  p,p p,p p,p p,p minimal series module V1 z , h 0 1 (resp. V0 z , h 0 ) as Vir 1 -modules 2 −(n+1),n +1; 2 −(n+1),n +1;0 2 (resp. Vir0-modules). Hence, we obtain the following theorem.

Theorem 6. Let m and λ be the same as Proposition1. Then, we have

 0 0  p,p p,p (L (2Λ0) ⊕ L (2Λ )) ⊗ L (λ) = V1 z , h 0 ⊗ L (ν) 1 ∑ −(n+1),n +1; 1 ν 2 2 and  0 0  p,p p,p  0  L (ρ) ⊗ L (λ) = ∑ V0 z , h 0 ⊗ L ν , 0 −(n+1),n +1;0 ν 0 m+2 0 where ν and ν are taken over Pu,1 such that ν ≡ 2Λ0 + λ mod Q and ν ≡ ρ + λ mod Q, respectively.

Funding: This research was supported by the Daegu University Research Grant, 2016. Conflicts of Interest: The author declares no conflict of interest. Axioms 2019, 8, 82 19 of 19

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