Journal of Lie Theory Volume 11 (2001) 305–337 \begin{matrix}c \\ ◯ \end{matrix} 2001 Heldermann Verlag

Fusion Algebras for N = 1 Superconformal Field Theories through Coinvariants, II: N = 1 Super-Virasoro-Symmetry

Kenji Iohara and Yoshiyuki Koga^{∗}

Communicated by K.-H. Neeb

Abstract. We calculate fusion algebras of the N = 1 super-minimal models for the Neveu-Schwarz sector. We also state conjectures on the fusion rule and give some supporting evidences for the Ramond sector. Fusion algebras for the Neveu-Schwarz sector coincide with Verlinde algebras.

0. Introduction

This is the part II of the series of our paper [13]. In [13], we calculated the fusion algebras for ôsp(1|2), and here we compute those for the N = 1 super Virasoro algebras by a similar argument. The N = 1 super Virasoro algebra is the (N = 1) super-extension of the Virasoro algebra and it has 2-sectors, called the Neveu-Schwarz sector and the Ramond sector ( NS and R in short resp.). To be concrete, we obtain the complete description of the fusion NS×NS → NS (Theorem 3.4 and 3.5) and partial results of that R × NS → R and R × R → NS (Proposition 6.5) with the conjectures for general cases (Conjecture 3.6). Let us add a few words for vertex operator superalgebras (SVOA for short). It is known that for each central charge, the vacuum of the N = 1 super minimal model has a SVOA structure [16] and the rationality holds for the NS-sector of the N = 1 super minimal models [1]. Our results suggest that it might be more natural to include R-sector as ‘NS SVOA’-module in order to describe the whole theory.

Let us briefly explain, why this paper can be regarded as the continuation of the part I [13] and what motivates us to work on these subjects. For non-super case, there are the functors, called the quantized Drinfel’d-Sokolov functor [8], from the full subcategory of the category O of an affine Lie algebra 𝔤̂ consisting of offcritical representations to the category O of the corresponding W -algebra W(𝔤). Via these functors, some classes of admissible representations of 𝔰̂l_{2} correspond to the minimal series representations of the Virasoro algebra (Vir for short). What is remarkable is that by these functors, integrable 𝔰̂l_{2}-modules do not correspond to any minimal Vir-modules! It is widely believed that such correspondence generalizes to some classes of admissible representations of 𝔤̂ (called non-degenerate

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag principal admissible representations [11]) and the minimal series representations of W(𝔤). On the other hand, the fusion algebras for 𝔰̂l_{2} are partially computed in [3] and completely by [9]. The fusion algebras for Vir are also computed in [7] and the relations between the fusion algebras for 𝔰̂l_{2} and that for Vir are investigated in [10]. It is quite natural to consider the generalizations of these theories to the other cases (e.g. the higher rank cases). But unfortunately, some technical difficulties arise and higher rank generalizations are still unclear until now. But for super case, there are still computable examples other than 𝔰̂l_{2} and the Virasoro algebra, that is the affine Lie super algebra ôsp(1|2) and the N = 1 super Virasoro algebra. In fact, Kimura showed that these algebras are related, at least at the level of the algebras, via Hamiltonian reductions [14]. We expect that these reduction can be lifted to the functors as in [8] and the series of our papers ( [13] and this paper ) is an attempt toward understanding of these structures.

∗ Yukawa Fellow

This paper is organized as follows: In Section 1, we recall the definition of the N = 1 super Virasoro algebras and Verma modules over them. We also recall embedding diagrams of Verma modules and resolutions for the minimal series representations. In Section 2, we define fusion algebras through coinvariants of Lie superalgebras generated by tensor fields on ℂP^{1}. In section 3, we state our main result. We consider not only the minimal models but also the case of generic central charge. In both cases, we give the multiplication rules of the fusion algebras for the NS sector in terms of the finite dimensional representations of 𝔰l_{2} (Theorem 3.4 and 3.5). For the fusion rules for both NS and R sectors, we give a conjecture at the end of this section (Conjecture 3.6). In Section 4, we calculate a modular transformation of the (super-) characters of the minimal series representations, and show an isomorphism between the fusion algebras and Verlinde algebras for the NS sector. In Section 5, our main results Theorem 3.4 and 3.5 are proved, and in Section 6, some supporting evidences of Conjecture 3.6 and partial results are stated. In Section 7, we prove two propositions related with the rationality of the minimal models.

Acknowledgement. We would like to thank Dr. C. Shweigert for useful comments. Especially, we would like to thank Prof. B. Feigin for discussions. We also would like to thank the referee for his careful reading and suggestions.

1. N = 1 super Virasoro algebra

In this section, we recall some basic facts about the N = 1 super Virasoro algebras.

The N = 1 super Virasoro algebra V ir_{ϵ} (ϵ = \frac{1}{2}, 0) is the Lie superalgebra over ℂ with the basis {L_{i}, G_{j}, C|i ∈ ℤ, j ∈ ϵ + ℤ}. These generators have the ℤ_{2}-gradation

deg L_{i} = deg C = 0, deg G_{i} = 1,

and satisfy the commutation relations

\begin{matrix}[L_{i}, L_{j}] = (i − j)L_{i+j} + \frac{1}{12}(i^{3} − i)δ_{i+j,0}C, \\ [G_{i}, L_{j}] = (i − \frac{1}{2}j)G_{i+j}, \\ [G_{i}, G_{j}] = 2L_{i+j} + \frac{1}{3}(i^{2} − \frac{1}{4})δ_{i+j,0}C, \\ [C, V ir_{ϵ}] = 0. \end{matrix}  (1)

The N = 1 super Virasoro algebras V ir_{\frac{1}{2}} and V ir_{0} are called the Neveu-Schwarz and Ramond algebras respectively. Below we abbreviate them as NS algebra and R algebra.

We fix a triangular decomposition of V ir_{ϵ} as follows:

V ir_{ϵ} = V ir_{ϵ}^{+} ⊕ V ir_{ϵ}^{0} ⊕ V ir_{ϵ}^{−},

where

V ir_{ϵ}^{+} = ⨁_{i>0} ℂL_{i} ⊕ ⨁_{i>0} ℂG_{i},

V ir_{ϵ}^{0} = \{ \begin{matrix}ℂL_{0} ⊕ ℂC & if ϵ = \frac{1}{2}, \\ ℂL_{0} ⊕ ℂG_{0} ⊕ ℂC & if ϵ = 0, \end{matrix}

V ir_{ϵ}^{−} = ⨁_{i<0} ℂL_{i} ⊕ ⨁_{i<0} ℂG_{i}.

We also set

V ir_{ϵ}^{≥} = V ir_{ϵ}^{+} ⊕ V ir_{ϵ}^{0}.

First we introduce the Verma module M_{ϵ}(c, h) over V ir_{ϵ} with highest weight (c, h) ∈ ℂ^{2}. For ϵ = \frac{1}{2} , let Vc,h^{\frac{1}{2}} = ℂ1c,h^{\frac{1}{2}} be the 1-dimensional representation of V ir≥_{\frac{1}{2}} given by

x.1c,h^{\frac{1}{2}} = 0  if x ∈ V ir+_{\frac{1}{2}},

C.1c,h^{\frac{1}{2}} = c1c,h^{\frac{1}{2}},

L_{0}.1c,h^{\frac{1}{2}} = h1c,h^{\frac{1}{2}}.

For ϵ = 0, let V_{c,h}^{0} = ⨁_{α=0,1} ℂ1_{c,h:α}^{0} be the 2-dimensional representation of V ir_{0}^{≥} given by

x.1_{c,h:α}^{0} = 0  if x ∈ V ir_{0}^{+},
C.1_{c,h:α}^{0} = c1_{c,h:α}^{0},
L_{0}.1_{c,h:α}^{0} = h1_{c,h:α}^{0},
G_{0}.1_{c,h:0}^{0} = 1_{c,h:1}^{0}.

We define the Verma module M_{ϵ}(c, h) with highest weight (c, h) by

M_{ϵ}(c, h) = Ind_{U(V ir_{ϵ}^{≥})}^{U(V ir_{ϵ})} V_{c,h}^{ϵ}.

The value c is called central charge. In the sequel, we denote the highest weight vectors 1 ⊗ 1c,h^{\frac{1}{2}} and 1 ⊗ 1_{c,h:0}^{0} by |c, h⟩_{\frac{1}{2}} and |c, h⟩_{0}. Since the choice of the parity of a highest weight vector is not essential for our fusion algebras, we choose it appropriately. Every highest weight module with highest weight (c, h) is a quotient of the Verma module M_{ϵ}(c, h). In particular, let us denote the irreducible quotient by L_{ϵ}(c, h).

For our calculation, we need the following two induced modules over the R algebra. From now on, for any ξ ∈ ℂ, let us fix the argument of ξ such that 0 ≤ arg ξ < 2π. Let W_{c,h}^{0} = ⨁_{α=0,1} ℂ˜1_{c,h:α}^{0} be the 2-dimensional representation of V ir_{0}^{≥} defined by

x.˜1_{c,h:α}^{0} = 0  if x ∈ V ir_{0}^{+},
C.˜1_{c,h:α}^{0} = c˜1_{c,h:α}^{0},
L_{0}.˜1_{c,h:α}^{0} = h˜1_{c,h:α}^{0},
G_{0}.˜1_{c,h:α}^{0} = (h − \frac{1}{24}c)^{\frac{1}{2}} ˜1_{c,h:1−α}^{0}.

We set

M′_{0}(c, h) = Ind_{U(V ir_{0}^{≥})}^{U(V ir_{0})} W_{c,h}^{0}.

Notice that M′_{0}(c, h) ≃ M_{0}(c, h) if and only if h ≠ \frac{1}{24}c. Further we define M̃_{0}^{±}(c, h) as follows. Notice that ℂ_{c,h}^{±} = ℂ{˜1_{c,h:1}^{0} ± ˜1_{c,h:0}^{0}} are V ir_{0}^{≥}-submodules of W_{c,h}^{0}. For τ = ±, we set

M̃_{0}^{τ}(c, h) = Ind_{U(V ir_{0}^{≥})}^{U(V ir_{0})}ℂ_{c,h}^{τ},

and call it the half Verma module with highest weight (c, h : τ). It is clear that M′_{0}(c, h) = ⨁_{τ=±} M̃_{0}^{τ}(c, h).

Here we remark on the injectivity of homomorphisms between Verma modules. A non-trivial homomorphism between Verma modules over the N = 1 super Virasoro algebras is not always injective. To be precise, for the NS algebra, such a homomorphism is always injective, but for the R algebra, there exists a nonzero homomorphism which is not injective. Even though all non-trivial homomorphism between Verma modules are not necessarily injective, we could construct Bernstein-Gel’fand-Gel’fand type resolutions for these algebras [12].

Next, we recall embedding diagrams of Verma modules and BGG type resolutions, which are necessary for our purpose. For t ∈ ℂ \ {0} and a, b ∈ ℤ_{≥0}, set

\begin{matrix}c(t) = \frac{15}{2} − 3(t + t^{−1}), \\ h_{a,b}^{ϵ}(t) = \frac{1}{8}(a^{2} − 1)t − \frac{1}{4}(ab − 1) + \frac{1}{8}(b^{2} − 1)t^{−1} + \frac{1}{16}(1 − 2ϵ). \end{matrix}  (2)

Note that if the Verma module M_{ϵ}(c, h) with central charge c = c(t) is reducible then h = h_{a,b}^{ϵ}(t) for some a, b ∈ ℤ_{≥0} such that a − b ∈ 2ϵ + 1 + 2ℤ. In this paper, we consider fusion algebras in the following two cases:

Case 1 Generic central charge i.e. t ∈ ℝ_{>0} \ ℚ_{>0}.

Case 2 Minimal central charge i.e. t = \frac{p}{q} for p, q ∈ ℤ_{>0} such that p, q ≥ 2, p − q ∈ 2ℤ, p ≠ q and (\frac{p−q}{2}, q) = 1.

Case 1. We describe resolutions of irreducible highest weight representations with generic (not rational) central charge.

Proposition 1.1. Suppose that t ∈ ℝ_{>0} \ ℚ_{>0}. If there exist a, b ∈ ℤ_{>0} such that a − b ∈ 2ϵ + 1 + 2ℤ, c = c(t) and h = h_{a,b}^{ϵ}(t), then the following sequence:

0 ⟶ M_{ϵ}(c, h + \frac{1}{2}ab) ⟶ M_{ϵ}(c, h) ⟶ L_{ϵ}(c, h) ⟶ 0

is exact.

Case 2. We state embedding diagrams of Verma modules and resolutions of the irreducible representations L_{ϵ}(c(\frac{p}{q}), h_{r,s}^{ϵ}(\frac{p}{q})) with r, s ∈ ℤ_{>0}. For our calculation, it is enough to consider the following two cases:

Case 2.1 r − s ∈ 2ϵ + 1 + 2ℤ and

\{ \begin{matrix}(1 ≤ r < q) ∧ (1 ≤ s < p) & if ϵ = \frac{1}{2}, \\ (1 ≤ r < q) ∧ (1 ≤ s < p) ∧ (r, s) ≠ (\frac{q}{2}, \frac{p}{2}) & if ϵ = 0. \end{matrix}

Case 2.2

r = \frac{1}{2}q ∧ s = \frac{1}{2}p ∧ (p, q) = 2 ∧ ϵ = 0.

For simplicity, set

h_{i:ϵ} := \begin{cases}h_{iq+r,s}^{ϵ}(\frac{p}{q}) & i ∈ 2ℤ \\ h_{iq+r,p−s}^{ϵ}(\frac{p}{q}) & i ∈ 2ℤ + 1 \end{cases} , h′_{i:ϵ} := \begin{cases}h_{iq+(q−r),p−s}^{ϵ}(\frac{p}{q}) & i ∈ 2ℤ \\ h_{iq+(q−r),s}^{ϵ}(\frac{p}{q}) & i ∈ 2ℤ + 1 \end{cases} .

Definition 1.2. [15] L_{ϵ}(c(\frac{p}{q}), h_{0}^{ϵ}) in Case 2.1 and Case 2.2 are called minimal series representations.

Now we recall embedding diagrams of Verma modules in Case 2.1.

Proposition 1.3. There exist the following embedding diagrams of Verma modules:

⋯ → M_{ϵ}(c, h_{3:ϵ}) → M_{ϵ}(c, h_{2:ϵ}) → M_{ϵ}(c, h_{1:ϵ}) _{↘}
↘ ↗ ↘ ↗ ↘ ↗ M′_{ϵ}(c, h_{0:ϵ}).
⋯ → M_{ϵ}(c, h′_{3:ϵ}) → M_{ϵ}(c, h′_{2:ϵ}) → M_{ϵ}(c, h′_{1:ϵ}) ^{↗}

Next we state the resolution for the irreducible highest weight module L_{ϵ}(c, h_{0}).

Theorem 1.4. [12] The following resolutions exist:

Case 2.1

\begin{matrix}⋯ → M_{ϵ}(c, h_{k:ϵ}) ⊕ M_{ϵ}(c, h′_{k:ϵ}) → ⋯ \\ ⋯ → M_{ϵ}(c, h_{1:ϵ}) ⊕ M_{ϵ}(c, h′_{1:ϵ}) → M_{ϵ}(c, h_{0:ϵ}) → L_{ϵ}(c, h_{0:ϵ}) → 0, \end{matrix}

Case 2.2

\begin{matrix}⋯ → M_{0}(c, h_{k:ϵ})^{⊕2} → ⋯ \\ ⋯ → M_{0}(c, h_{1:ϵ})^{⊕2} → M′_{0}(c, h_{0:ϵ}) → L_{0}(c, h_{0:ϵ}) ⊕ ΠL_{0}(c, h_{0:ϵ}) → 0, \end{matrix}

where Π denotes the parity shift.

2. Fusion algebras

In this section, we define fusion algebras through coinvariants.

We denote by ℂ[[t]] and ℂ((t)) the completions of ℂ[t] and ℂ[t, t^{−1}] by the positive powers of t.

To compute fusion rules explicitly, the following loop module is important. Let θ be the Grassmann variable i.e. θ^{2} = 0. For ϵ = \frac{1}{2}, 0, σ = 0, 1 and λ, μ ∈ ℂ, let

F_{λ,μ}^{ϵ,σ} = ( ⨁_{i∈ℤ+σϵ}ℂF_{i}) ⊕ ( ⨁_{i∈ℤ+(1−σ)ϵ}ℂF_{i}θ)

be a V ir_{ϵ}-module with basis F_{i} and F_{i}θ, on which V ir_{ϵ} acts via

\begin{matrix}C.F_{i}θ^{γ} = 0, \\ L_{n}.F_{i}θ^{γ} = \{−i + μ + (n − 1)λ + \frac{1}{2}nγ(−1)^{2ϵ}\}F_{i+n}θ^{γ}, \\ G_{m}.F_{i}θ^{γ} = \{ \begin{matrix}δ_{γ,1}F_{i+m} + δ_{γ,0}\{−i + μ + (2m − 1)λ\}F_{i+m}θ & if ϵ = \frac{1}{2}, \\ δ_{γ,0}F_{i+m}θ + δ_{γ,1}\{−i + m + μ + (2m − 1)λ\}F_{i+m} & if ϵ = 0, \end{matrix} \end{matrix} (3)

for γ = 0, 1, n ∈ ℤ and m ∈ ℤ + ϵ. Remark that F_{λ,μ}^{0,0} = F_{λ,μ}^{0,1}.

Next, we introduce Verma modules and loop modules attached to a point on ℂP^{1}. For w ∈ ℂP^{1}, let

z_{w} = \{ \begin{matrix}z − w & if w ∈ ℂ, \\ z^{−1} & if w = ∞, \end{matrix}

be a local coordinate arround w. For f(z_{w}), g(z_{w}) ∈ ℂ((z_{w})), let

f(z_{w})∂_{z_{w}} and g(z_{w})z_{w}^{\frac{1}{2} −ϵ}∂z_{w}^{\frac{1}{2}},

be tensor fields over the formal neighborhood of w which satisfy the following commutation relations:

[p(z_{w})∂_{z_{w}}, q(z_{w})∂_{z_{w}}] = {p(z_{w})q′(z_{w}) − p′(z_{w})q(z_{w})}∂_{z_{w}},

[p(z_{w})∂_{z_{w}}, q(z_{w})∂z_{w}^{\frac{1}{2}}] = \{p(z_{w})q′(z_{w}) − \frac{1}{2}p′(z_{w})q(z_{w})\}∂z_{w}^{\frac{1}{2}},

[p(z_{w})∂z_{w}^{\frac{1}{2}}, q(z_{w})∂z_{w}^{\frac{1}{2}}] = 2p(z_{w})q(z_{w})∂_{z_{w}},

where p′(z_{w}) = \frac{d}{dz_{w}} p(z_{w}). We define the ℤ_{2}-gradation of these fields by

deg(f(z_{w})∂_{z_{w}}) = 0, deg(g(z_{w})z_{w}^{\frac{1}{2} −ϵ}∂z_{w}^{\frac{1}{2}}) = 1.

Here let us introduce L_{w;ϵ} as the Lie superalgebra of the tensor fields over the formal neighborhood of w spanned by the above tensor fields.

Moreover, we consider a central extension of the Lie superalgebras L_{w;ϵ}. Let us define the cocycle ⟨, ⟩_{w;ϵ} of L_{w;ϵ} by

⟨p(z_{w})∂_{z_{w}}, q(z_{w})∂_{z_{w}}⟩_{w;ϵ} = \frac{1}{12} Res_{z_{w}=0}(p′′′(z_{w})q(z_{w}))dz_{w},

⟨p(z_{w})∂_{z_{w}}, q(z_{w})∂z_{w}^{\frac{1}{2}}⟩_{w;ϵ} = 0,

⟨p(z_{w})∂z_{w}^{\frac{1}{2}}, q(z_{w})∂z_{w}^{\frac{1}{2}}⟩_{w;ϵ} = \frac{1}{3} Res_{z_{w}=0}(p′(z_{w})q′(z_{w}))dz_{w},

and let L̂_{w;ϵ} = L_{w;ϵ} ⊕ ℂC be the 1-dimensional central extension defined by the cocycle ⟨, ⟩_{w;ϵ}. Remark that for n ∈ ℤ and m ∈ ℤ + ϵ,

C ↦ C, L_{n} ↦ −z_{w}^{n+1}∂_{z_{w}}, G_{m} ↦ \sqrt{−1}z_{w}^{m+ \frac{1}{2}} ∂z_{w}^{\frac{1}{2}}  (4)

give an isomorphism V ir_{ϵ} ≃ L̂_{w;ϵ}. In the sequel, for w ∈ ℂP^{1}, we denote

L_{n}^{(w)} = −z_{w}^{n+1}∂_{z_{w}}, G_{m}^{(w)} = \sqrt{−1}z_{w}^{m+ \frac{1}{2}} ∂z_{w}^{\frac{1}{2}}.  (5)

Now we define the Verma module M_{ϵ}(c, h)(w) attached to w ∈ ℂP^{1}. Set

L̂_{w;ϵ}^{≥} = ℂ[[z_{w}]]z_{w}∂_{z_{w}} ⊕ ℂ[[z_{w}]]z_{w}^{\frac{1}{2} +ϵ}∂z_{w}^{\frac{1}{2}} ⊕ ℂC.

For (c, h) ∈ ℂ^{2}, let Vc,h^{\frac{1}{2}}(w) = ℂ1_{c,h}^{\frac{1}{2} :w} be the 1-dimensional L̂_{w; \frac{1}{2}}^{≥} -module defined by

z_{w}^{n+1}∂_{z_{w}}.1_{c,h}^{\frac{1}{2} :w} = 0  for n ∈ ℤ_{>0},
z_{w}^{m+ \frac{1}{2}} ∂z_{w}^{\frac{1}{2}}.1_{c,h}^{\frac{1}{2} :w} = 0  for m ∈ ℤ_{>0} − \frac{1}{2},
−z_{w}∂_{z_{w}}.1_{c,h}^{\frac{1}{2} :w} = h1_{c,h}^{\frac{1}{2} :w},
C.1_{c,h}^{\frac{1}{2} :w} = c1_{c,h}^{\frac{1}{2} :w},

and let V_{c,h}^{0}(w) = ⨁_{α=0,1} ℂ1_{c,h:α}^{0:w} be the 2-dimensional L̂_{w;0}^{≥}-module defined by

z_{w}^{n+1}∂_{z_{w}}.1_{c,h:α}^{0:w} = 0  for n ∈ ℤ_{>0},
z_{w}^{m+ \frac{1}{2}} ∂z_{w}^{\frac{1}{2}}.1_{c,h:α}^{0:w} = 0  for m ∈ ℤ_{>0},

−z_{w}∂_{z_{w}}.1_{c,h:α}^{0:w} = h1_{c,h:α}^{0:w},

\sqrt{−1}zw^{\frac{1}{2}}∂z_{w}^{\frac{1}{2}}.1_{c,h:0}^{0:w} = 1_{c,h:1}^{0:w},

C.1_{c,h:α}^{0:w} = c1_{c,h:α}^{0:w}.

We define the Verma module M_{ϵ}(c, h)(w) attached to w ∈ ℂP^{1} by

M_{ϵ}(c, h)(w) = Ind_{U(L̂_{w;ϵ}^{≥})}^{U(L̂_{w;ϵ})} V_{c,h}^{ϵ}(w).

We denote the irreducible quotient of M_{ϵ}(c, h)(w) by L_{ϵ}(c, h)(w).

We similarly define the induced module M_{ϵ}(c, h)(w) attached to w ∈ ℂP^{1}. For (c, h) ∈ ℂ^{2}, let W_{c,h}^{0}(w) = ⨁_{α=0,1} ℂ˜1_{c,h:α}^{0:w} be the following 2-dimensional representation of L̂_{w;0}^{≥}:

z_{w}^{n+1}∂_{z_{w}}.˜1_{c,h:α}^{0:w} = 0  for n ∈ ℤ_{>0},
z_{w}^{m+ \frac{1}{2}} ∂z_{w}^{\frac{1}{2}}.˜1_{c,h:α}^{0:w} = 0  for m ∈ ℤ_{>0},

−z_{w}∂_{z_{w}}.˜1_{c,h:α}^{0:w} = h˜1_{c,h:α}^{0:w},

\sqrt{−1}zw^{\frac{1}{2}}∂z_{w}^{\frac{1}{2}}.˜1_{c,h:α}^{0:w} = (h − \frac{1}{24}c)^{\frac{1}{2}} ˜1_{c,h:1−α}^{0:w},

C.˜1_{c,h:α}^{0} = c˜1_{c,h:α}^{0:w}.

We define the induced module M′_{0}(c, h)(w) attached to w ∈ ℂP^{1} by

M′_{0}(c, h)(w) = Ind_{U(L̂_{w;0}^{≥})}^{U(L̂_{w;0})} W_{c,h}^{0}(w).

Half Verma modules attached to w ∈ ℂP^{1} are defined as follows: If we set ℂ_{c,h}^{±}(w) = ℂ{˜1_{c,h:1}^{0:w} ± ˜1_{c,h:0}^{0:w}}, then ℂ_{c,h}^{±}(w) is a submodule of W_{c,h}^{0}(w). We define M̃_{0}^{τ}(c, h)(w) by

M̃_{0}^{τ}(c, h)(w) = Ind_{U(L̂_{w;0}^{≥})}^{U(L̂_{w;0})} ℂ_{c,h}^{τ}(w),

and call it the half Verma module with highest weight (c, h : τ) attached to w. For each Verma module M_{ϵ}(c, h)(w) attached to w and j ∈ (1 − ϵ)ℤ, we denote by M_{ϵ}(c, h)(w)_{j} the subspace of level j i.e.

M_{ϵ}(c, h)(w)_{j} = {v ∈ M_{ϵ}(c, h)(w)| (−z_{w}∂_{z_{w}}).v = (h + j)v}.  (6)

For the induced modules M′_{0}(c, h)(w) and M̃_{0}^{τ}(c, h)(w), we sometimes use similar notation.

Now we define a loop module attached to w ∈ ℂP^{1}. To introduce it, we set F_{λ,μ}^{ϵ,σ}(w) = (⨁_{i∈ℤ+σϵ} ℂF_{i}) ⊕ (⨁_{i∈ℤ+(1−σ)ϵ} ℂF_{i}θ) and define the L̂_{w;ϵ}-module structure on this space by using the isomorphism (4) and the action (3). If we set

L̂_{w;ϵ}^{<} = ℂ[[z_{w}^{−1}]]∂_{z_{w}} ⊕ ℂ[[z_{w}^{−1}]]z_{w}^{ϵ− \frac{1}{2}} ∂z_{w}^{\frac{1}{2}},

and

H_{λ,μ}^{ϵ,σ}(w) = (⨁_{i<0} ℂF_{i}) ⊕ (⨁_{i<0} ℂF_{i}θ),

then H_{λ,μ}^{ϵ,σ}(w) is a L̂_{w;ϵ}^{<}-submodule of F_{λ,μ}^{ϵ,σ}(w). We regard the quotient space

G_{λ,μ}^{ϵ,σ}(w) = F_{λ,μ}^{ϵ,σ}(w) /H_{λ,μ}^{ϵ,σ}(w)  (7)

as a L̂_{w;ϵ}^{<}-module.

Now, we construct fusion algebras for the N = 1 super Virasoro algebras by using coinvariants of a Lie superalgebra of tensor fields on ℂP^{1}, following [7]. For a finite subset E = \{(w_{1}, ϵ_{1}), ⋯, (w_{m}, ϵ_{m})\} of ℂP^{1} ×\{\frac{1}{2}, 0\} such that w_{1}, ⋯, w_{m} are distinct, let L_{E} be the direct sum ⨁_{(w,ϵ)∈E} L_{w;ϵ}, and L̂_{E} = L_{E} ⊕ ℂC be the

central extension of L_{E} by the cocycle ⟨, ⟩_{E} = ∑_{(w,ϵ)∈E}⟨, ⟩_{w,ϵ}. Let L(E) be the Lie

superalgebra consists of the multi-valued meromorphic tensor fields F (z) which satisfy the following conditions:

(i) F (z) is holomorphic outside {w_{i}|(w_{i}, ϵ_{i}) ∈ E}.

(ii) For any w such that (w, ϵ) ∈ E, F (z) has the following Laurent expansion:

F (z) = f(z_{w})∂_{z_{w}} + g(z_{w})z_{w}^{\frac{1}{2} −ϵ}∂z_{w}^{\frac{1}{2}},

where f(z_{w}), g(z_{w}) ∈ ℂ((z_{w})).

Remark that, by the Riemann-Hurwitz formula, L(E) is trivial if ♯{i|ϵ_{i} = 0} ≢ 0 (mod 2). By the Laurent expansion, we have the Lie superalgebra homomorphism L(E) ⟶ L_{E}. It is easy to see that this lifts to a homomorphism

L(E) ⟶ L̂_{E}.  (8)

Let W_{ϵ}(w) be a highest weight representation of L̂_{w;ϵ}. We regard the tensor product ⨂_{(w,ϵ)∈E} W_{ϵ}(w) as a L̂_{E} -module via the homomorphism (8). We

define fusion algebras using the following coinvariants

H_{0}(L(E), ⨂_{(w,ϵ)∈E} W_{ϵ}(w)) = ⨂_{(w,ϵ)∈E} W_{ϵ}(w) /L(E)( ⨂_{(w,ϵ)∈E} W_{ϵ}(w)) .

Let us fix a positive number t ∈ ℝ_{>0} and set c = c(t). Let S_{c} be a subset (at most countable) of ℂ × \{\frac{1}{2}, 0\}, and A_{c} the ℤ-free module with a basis \{l_{h}^{ϵ}|(h, ϵ) ∈ S_{c}\}. Set A_{c}^{ℚ} = ℚ ⊗_{ℤ} A_{c}. First we introduce symmetric forms on A_{c}^{ℚ}. For a set of distinct points {w_{1}, ⋯, w_{m}} of ℂP^{1}, we define a ℚ-multilinear symmetric form Φ_{m} : ⨂^{m} A_{c}^{ℚ} → ℚ by

Φ_{m}(l_{h_{1}}^{ϵ_{1}}, ⋯, l_{h_{m}}^{ϵ_{m}}) = (\frac{1}{\sqrt{2}})^{♯\{i|ϵ_{i}=0\}} dim H_{0}(L(E), ⨂_{i=1}^{m} L′_{ϵ_{i}}(c, h_{i})(w_{i})),

where E = {(w_{1}, ϵ_{1}), ⋯, (w_{m}, ϵ_{m})} and

L′_{ϵ}(c, h)(w) = \{ \begin{matrix}L_{0}(c, h)(w) ⊕ ΠL_{0}(c, h)(w) & if ϵ = 0 and h = \frac{1}{24}c, \\ L_{ϵ}(c, h)(w) & otherwise . \end{matrix}

Note that Φ_{m} is well-defined as a form on A_{c}^{ℚ}, since L(E) = {0} if ♯{i|ϵ_{i} = 0} ≢ 0 (mod 2).

Remark 2.1. (i) Roughly speaking, the symbol l_{h}^{ϵ} corresponds to a primary field as follows:

l_{h}^{ϵ} ⇔ \{ \begin{matrix}ϕ_{L_{\frac{1}{2}} (c,h)} & if ϵ = \frac{1}{2}, \\ \frac{1}{\sqrt{2}}ϕ_{L_{0}(c,h)} & if ϵ = 0 and h ≠ \frac{1}{24}c, \\ \frac{1}{\sqrt{2}} (ϕ_{L_{0}(c,h)} + ϕ_{ΠL_{0}(c,h)}) & if ϵ = 0, and h = \frac{1}{24}c, \end{matrix}

where ϕ_{L_{ϵ}(c,h)} denotes the primary field attached to the irreducible representation L_{ϵ}(c, h).

(ii) In the R sector, the fields which have physical meaning correspond to L′_{0}(c, h) rather than L_{0}(c, h). Mathematically, L′_{0}(c, h) naturally appears in the (ℤ_{2}- graded) BGG type resolution (see Theorem 1.4).

For the form Φ_{2}, we have the following lemma.

Lemma 2.2. Let w_{1}, w_{2} be distinct points in ℂP^{1}. Suppose that ϵ_{1}, ϵ_{2} ∈ \{\frac{1}{2}, 0\} and E = \{(w_{1}, ϵ_{1}), (w_{2}, ϵ_{2})\}. For c = c(t) such that t ∈ ℝ_{>0} and h_{1}, h_{2} ∈ ℂ, we have

\begin{matrix}(\frac{1}{\sqrt{2}})^{♯\{i|ϵ_{i}=0\}} dim H_{0}(L(E), L′_{ϵ_{1}}(c, h_{1})(w_{1}) ⊗ L′_{ϵ_{2}}(c, h_{2})(w_{2})) \\ = δ_{h_{1},h_{2}}δ_{ϵ_{1},ϵ_{2}} × \{ \begin{matrix}1 & if ϵ_{1} = \frac{1}{2}, \\ 2^{δ_{h_{1}, \frac{1}{24} c}} & if ϵ_{1} = 0. \end{matrix} \end{matrix}

Remark that the form Φ_{2} is non-degenerate, but unlike the Virasoro algebra case, the basis {l_{h}^{ϵ}|(h, ϵ) ∈ S_{c}} is not self-dual with respect to the form Φ_{2}. Hence we define a fusion algebra on A_{c} as follows: We first introduce an algebraic structure on A_{c}^{ℚ} by using the form Φ_{3} : (A_{c}^{ℚ})^{⊗^{3}} → ℚ and the isomorphism A_{c}^{ℚ} ≃ (A_{c}^{ℚ})^{∗} (restricted dual) defined by Φ_{2}. More precisely, the multiplication in A_{c}^{ℚ} is given by

l_{h_{1}}^{ϵ_{1}} ∘ l_{h_{2}}^{ϵ_{2}} = ∑_{(h_{3},ϵ_{3})∈S_{c}} N_{(h_{1},ϵ_{1}),(h_{2},ϵ_{2})}^{(h_{3},ϵ_{3})}l_{h_{3}}^{ϵ_{3}},

where

\begin{aligned}N_{(h_{1},ϵ_{1}),(h_{2},ϵ_{2})}^{(h_{3},ϵ_{3})} & = \frac{Φ_{3}(l_{h_{1}}^{ϵ_{1}}, l_{h_{2}}^{ϵ_{2}}, l_{h_{3}}^{ϵ_{3}})}{Φ_{2}(l_{h_{3}}^{ϵ_{3}}, l_{h_{3}}^{ϵ_{3}})} \\ & = Φ_{3}(l_{h_{1}}^{ϵ_{1}}, l_{h_{2}}^{ϵ_{2}}, l_{h_{3}}^{ϵ_{3}}) × \{ \begin{matrix}1 & if ϵ_{3} = \frac{1}{2}, \\ 2^{−δ_{h_{3}, \frac{1}{24} c}} & if ϵ_{3} = 0. \end{matrix} \end{aligned}

A priori, we do not see whether N_{(h_{1},ϵ_{1}),(h_{2},ϵ_{2})}^{(h_{3},ϵ_{3})} ∈ ℤ_{≥0} or not. As a consequence of Theorem 3.4, Theorem 3.5, Remark 6.4 and Proposition 6.5, we see that N_{(h_{1},ϵ_{1}),(h_{2},ϵ_{2})}^{(h_{3},ϵ_{3})} ∈ ℤ_{≥0}. Therefore, the multiplication ∘ restricts to an operation on the ℤ-module A_{c} and we call it fusion algebra. Here we set

A_{c}^{NS} = ⨁_{(h, \frac{1}{2} )∈S_{c}} ℤlh^{\frac{1}{2}} and A_{c}^{R} = ⨁_{(h,0)∈S_{c}} ℤl_{h}^{0}

and call them the NS and R sectors respectively. By the definition, A_{c}^{NS} is a subalgebra of the fusion algebra A_{c} and A_{c}^{R} becomes an A_{c}^{NS} -module.

3. Main results

In this section, we will calculate fusion rules explicitly for two classes, a minimal case and a generic case.

Case 1 Generic central charge i.e. c = c(t) and t ∈ ℝ_{>0} \ ℚ_{>0}. Recall that the Verma module M_{ϵ}(c, h) is reducible if and only if h ∈ Λ_{ϵ}^{c}, where the set Λ_{ϵ}^{c} is given by

Λ_{ϵ}^{c} = {h_{a,b}^{ϵ}(t) |(a, b) ∈ (ℤ_{>0})^{2}, a − b ≡ (1 − 2ϵ) (mod 2)} .

Case 2 Minimal central charge i.e. c = c(t) and t = \frac{p}{q} ∈ ℚ_{>0} for p, q ∈ ℤ_{>0} such that p, q ≥ 2, p − q ∈ 2ℤ, p ≠ q and (\frac{p−q}{2}, q) = 1. In this case, we

define the set Λ_{ϵ}^{c} by

Λ_{ϵ}^{c} = {h_{a,b}^{ϵ}(t) |(a, b) ∈ ℤ^{2}, 1 ≤ a < q, 1 ≤ b < p, a − b ≡ (1 − 2ϵ) (mod 2)} .

Remark 3.1. By the definition 1.2, L_{ϵ}(c, h) is a minimal series representation if and only if c = c(\frac{p}{q}) satisfies the condition in Case 2 and h ∈ Λ_{ϵ}^{c}. For each case, the following theorem characterize the set S_{c}. We first state for Case 1. Set E = {(w_{1}, ϵ_{1}), (w_{2}, ϵ_{2}), (w_{3}, ϵ_{3})} with distinct points w_{1}, w_{2}, w_{3} ∈ ℂP^{1} and ϵ_{1}, ϵ_{2}, ϵ_{3} ∈ \{0, \frac{1}{2}\} such that ♯\{i|ϵ_{i} = 0\} ≡ 0 (mod 2).

Theorem 3.2. Suppose that c = c(t) satisfies the condition in Case 1. For fixed h_{1}, h_{2} ∈ ℂ,

(i) ♯{h_{3}|Φ_{3}(l_{h_{1}}^{ϵ_{1}}, l_{h_{2}}^{ϵ_{2}}, l_{h_{3}}^{ϵ_{3}}) ≠ 0} < ∞ if and only if h_{i} ∈ Λ_{ϵ_{i}}^{c} (i = 1, 2).

(ii) If there exists i ∈ {1, 2} such that h_{i} ∉ Λ_{ϵ_{i}}^{c}, then ♯{h_{3}|Φ_{3}(l_{h_{1}}^{ϵ_{1}}, l_{h_{2}}^{ϵ_{2}}, l_{h_{3}}^{ϵ_{3}}) ≠ 0} is uncountable.

Next for Case 2, set E_{ϵ} = \{(w_{1}, \frac{1}{2}), (w_{2}, ϵ), (w_{3}, ϵ)\} (ϵ = \frac{1}{2}, 0). Then the following theorem holds Theorem 3.3. Suppose that c = c(\frac{p}{q}) satisfies the condition in Case 2. If

H_{0}(L(E), L_{\frac{1}{2}} (c, 0)(w_{1}) ⊗ L_{ϵ}(c, h_{2})(w_{2}) ⊗ L_{ϵ}(c, h_{3})(w_{3})) ≠ 0,

then h_{2} = h_{3} and h_{2} ∈ Λ_{ϵ}^{c}.

Case 1 (The fusion algebra at a generic central charge) First we fix the central charge c = c(t), where t ∈ ℝ_{>0} \ ℚ_{>0}. We take

\{(h, ϵ)|h ∈ Λ_{ϵ}^{c} for ϵ = 0, \frac{1}{2}\}

as the index set S_{c} of the fusion algebra A_{c}. For j ∈ ℤ_{≥0}, let V_{j} be the irreducible (j + 1)-dimensional representation of sl_{2}. If h = h_{a,b}^{ϵ}(t) ∈ Λ_{ϵ}^{c}, then we assign the symbol (V_{a−1}, V_{b−1}) to l_{h}^{ϵ} ∈ A_{c}. Further we extend this notation by ℤ-bilinearlity. Note that this is a one-to-one correspondence between the set {l_{h}^{ϵ}|(h, ϵ) ∈ S_{c}} and the set of the symbols {(V_{i}, V_{j})|i, j ∈ ℤ_{≥0}}. Remark that if i ≡ j (mod 2) (resp. i ≢ j (mod 2)) then (V_{i}, V_{j}) corresponds to an element of the NS sector (resp. R sector). The multiplication rule of the NS sector A_{c}^{NS} can be explicitly described as follows:

Theorem 3.4. For the NS sector (i.e. in the case of i_{α} ≡ j_{α} (mod 2) for α = 0, 1) the structure of the fusion algebra at a generic central charge c = c(t) can be written as follows:

(V_{i_{0}}, V_{j_{0}}) ∘ (V_{i_{1}}, V_{j_{1}}) = (V_{i_{0}} ⊗ V_{i_{1}}, V_{j_{0}} ⊗ V_{j_{1}}).

Case 2 (The fusion algebra of minimal models) We fix a central charge c = c(\frac{p}{q}), where p, q ∈ ℤ_{>0} such that p, q ≥ 2, p − q ∈ 2ℤ, p ≠ q and (\frac{p−q}{2}, q) = 1. In this case, we take \{(h, ϵ)|h ∈ Λ_{ϵ}^{c} for ϵ = 0, \frac{1}{2}\} as the index set S_{c}. Define an equivalence relation ≈ on the set of (V_{i}, V_{j}) as follows (cf. [FM2]):

(V_{i}, V_{j}) ≈ (V_{−i+q−2}, V_{−j+p−2}) for 0 ≤ i ≤ q − 2, 0 ≤ j ≤ p − 2.

Let us denote the equivalence class of (V_{i}, V_{j}) by [(V_{i}, V_{j})]. We assign [(V_{a−1}, V_{b−1})] such that a − b ≡ (1 − 2ϵ) (mod 2) to l_{h}^{ϵ} with h = h_{a,b}^{ϵ}(\frac{p}{q}). This is a one-to-one correspondence between the set {l_{h}^{ϵ}|(h, ϵ) ∈ S_{c}} and the set of the equivalence classes {(V_{i}, V_{j})|0 ≤ i ≤ q − 2, 0 ≤ j ≤ p − 2}/ ≈. By using the following Kazhdan-Lusztig fusion functor:

V_{i} ˙⊗_{k}V_{j} ≃ V_{|i−j|} ⊕ V_{|i−j|+2} ⊕ V_{|i−j|+4} ⊕ ⋯ ⊕ V_{min{2k−i−j, i+j}},  (9)

we can state one of the main theorems of this paper in the following way:

Theorem 3.5. For the NS sector (i.e. in the case of i_{α} ≡ j_{α} (mod 2) for α = 0, 1) the structure of the fusion algebra of the minimal model with central charge c = c(\frac{p}{q}) can be written as follows:

[(V_{i_{0}}, V_{j_{0}})] ∘ [(V_{i_{1}}, V_{j_{1}})] = [(V_{i_{0}} ˙⊗_{q−2}V_{i_{1}}, V_{j_{0}} ˙⊗_{p−2}V_{j_{1}})].

Conjecture 3.6. Theorem 3.4 and 3.5 are valid for any sectors (i.e. without restriction of the parity of i_{α} − j_{α})

4. Verlinde algebras

In this section we list up the characters of minimal series for V ir_{ϵ} and calculate their modular transformations. As an application, we calculate Verlinde’s algebra [21]. Consequently we see that the Verlinde’s algebra coincides with the fusion algebra defined by coinvariants in the previous section.

To write down the characters of the minimal series for V ir_{ϵ}, we recall some holomorphic modular forms of weight \frac{1}{2} . Let η(τ) be the Dedekind η-function

η(τ) = q^{\frac{1}{24}} ∏_{n∈ℤ_{>0}}(1 − q^{n}),

and Θ_{n,m}(τ) be the classical theta function defined for m ∈ \frac{1}{2}ℤ_{>0} and n ∈ ℤ/2mℤ by

Θ_{n,m}(τ) = ∑_{k∈ℤ+ \frac{n}{2m}}e^{2πimk^{2}τ},

where q = e^{2πiτ}.

Through this section, let us fix c = c(\frac{p}{q}) as in Case 2 (minimal central charge) and set h = h_{r,s} = h_{r,s}(\frac{p}{q}). Since by definition

h_{q−r,p−s} = h_{r,s},

it is reasonable to define the sets Δ_{p,q}^{NS} and Δ_{p,q}^{R}, which parameterize the minimal series, as follows. For ϵ = \frac{1}{2}, 0, we set

KT_{p,q}^{ϵ} = {(r, s) ∈ ℤ^{2}|1 ≤ r < q, 1 ≤ s < p, r − s ≡ 1 − 2ϵ (mod 2)},

and define the equivalence relation ∼ on KT_{p,q}^{ϵ} by

(r, s) ∼ (r′, s′) if and only if r + r′ = q and s + s′ = p.

We sometimes denote KT_{p,q}^{NS} (resp. KT_{p,q}^{R}) for KTp,q^{\frac{1}{2}} (resp. KT_{p,q}^{0} ). Let us define Δ_{p,q}^{NS} and Δ_{p,q}^{R} as the set of the equivalence classes with respect to ∼ i.e.

Δ_{p,q}^{∗} = KT_{p,q}^{∗}/ ∼, for ∗ ∈ {NS, R},

and let π be the canonical projection

π : KT_{p,q}^{NS} ∐ KT_{p,q}^{R} ⟶ Δ_{p,q}^{NS} ∐ Δ_{p,q}^{R}.

For (r, s) ∈ KT_{p,q}^{ϵ}, we set

[r, s] := π{(r, s)}.

Let

χ_{ϵ}(c, h)(τ) = \frac{1}{2(1 − ϵ)} tr_{L_{ϵ}(c,h)} q^{L_{0}− \frac{1}{24} c} = q^{− \frac{1}{24} c} ch L_{ϵ}(c, h),

χ̃_{ϵ}(c, h)(τ) = \frac{1}{2(1 − ϵ)} str_{L_{ϵ}(c,h)} q^{L_{0}− \frac{1}{24} c} = q^{− \frac{1}{24} c} sch L_{ϵ}(c, h),

be the normalized (super-)character of the irreducible representation L_{ϵ}(c, h). For simplicity, we set χ_{[r,s]}^{NS}(τ) = χ_{\frac{1}{2}} (c, h_{r,s})(τ), χ̃_{[r,s]}^{NS}(τ) = χ̃_{\frac{1}{2}} (c, h_{r,s})(τ) and χ_{[r,s]}^{R}(τ) = (1 + δ_{h, \frac{c}{24}} )χ_{0}(c, h_{r,s})(τ). The following character formulae are immediate consequences of Theorem 1.4.

K. Iohara and Y. Koga 317

Lemma 4.1.

1. Characters for the NS algebra

χ_{[r,s]}^{NS}(τ) = \frac{η(τ)}{η(\frac{1}{2}τ)η(2τ)} \{Θ_{\frac{rp−sq}{2} , \frac{pq}{2}} (τ) − Θ_{\frac{rp+sq}{2} , \frac{pq}{2}} (τ)\} .

2. Characters for the R algebra

χ_{[r,s]}^{R}(τ) = \frac{η(2τ)}{η(τ)^{2}} \{Θ_{\frac{rp−sq}{2} , \frac{pq}{2}} (τ) − Θ_{\frac{rp+sq}{2} , \frac{pq}{2}} (τ)\} .

3. Super-characters for the NS algebra

\begin{matrix}χ̃_{[r,s]}^{NS}(τ) = \frac{η(\frac{1}{2}τ)}{η(τ)^{2}} \{(Θ_{rp−sq,2pq}(τ) − Θ_{rp−sq+2pq,2pq}(τ)) \\ −(−1)^{rs} (Θ_{rp+sq,2pq}(τ) − Θ_{rp+sq−2pq,2pq}(τ))\} , \end{matrix}

if p ≡ q ≡ 1 (mod 2), and

χ̃_{[r,s]}^{NS}(τ) = \frac{η(\frac{1}{2}τ)}{η(τ)^{2}} \{Θ_{\frac{rp−sq}{2} , \frac{pq}{2}} (τ) − (−1)^{rs}Θ_{\frac{rp+sq}{2} , \frac{pq}{2}} (τ)\} ,

if p ≡ q ≡ 0 (mod 2). Next, we will show how the characters of the minimal series transform under the action of SL(2, ℤ). By using the following modular transformation laws

\begin{matrix}η(τ + 1) = e^{\frac{πi}{12}} η(τ), & η(−\frac{1}{τ}) = \sqrt{\frac{τ}{i}} η(τ), \\ Θ_{n,m}(τ + 1) = e^{\frac{πin^{2}}{2m}} Θ_{n,m}(τ), & Θ_{n,m}(−\frac{1}{τ}) = \sqrt{\frac{τ}{2mi}} ∑_{n′∈ℤ/2mℤ} e^{− \frac{πinn′}{m}} Θ_{n′,m}(τ), \end{matrix} ; for m ∈ ℤ_{>0}, n ∈ ℤ/2mℤ,

we obtain the following generalization of a result in [17]. Proposition 4.2. Modular transformations of the (super)characters of the minimal series are given as follows:

1. For τ ↦ τ + 1, we have

\begin{matrix}χ_{[r,s]}^{NS}(τ + 1) = e^{πi\{ \frac{(rp−sq)^{2}}{4pq} − \frac{1}{8} \}}χ̃_{[r,s]}^{NS}(τ), χ̃_{[r,s]}^{NS}(τ + 1) = e^{πi\{ \frac{(rp−sq)^{2}}{4pq} − \frac{1}{8} \}}χ_{[r,s]}^{NS}(τ), \\ χ_{[r,s]}^{R}(τ + 1) = e^{πi \frac{(rp−sq)^{2}}{4pq}} χ_{[r,s]}^{R}(τ). \end{matrix}

2. For τ ↦ −\frac{1}{τ}, we have

\begin{aligned}χ_{[r,s]}^{NS}(−\frac{1}{τ}) & = ∑_{[r′,s′]∈Δ_{p,q}^{NS}}S_{[r,s],[r′,s′]}^{NS,NS}χ_{[r′,s′]}^{NS}(τ), \\ \sqrt{2}χ_{[r,s]}^{R}(−\frac{1}{τ}) & = ∑_{[r′,s′]∈Δ_{p,q}^{NS}}S_{[r,s],[r′,s′]}^{R,NS}χ̃_{[r′,s′]}^{NS}(τ), \\ χ̃_{[r,s]}^{NS}(−\frac{1}{τ}) & = ∑_{[r′,s′]∈Δ_{p,q}^{R}}S_{[r,s],[r′,s′]}^{NS,R}\sqrt{2}χ_{[r′,s′]}^{R}(τ), \end{aligned}

where

S_{[r,s],[r′,s′]}^{NS,NS} = \frac{4}{\sqrt{pq}} sin \frac{πrr′}{2q} (p − q) sin \frac{πss′}{2p} (p − q),

S_{[r,s],[r′,s′]}^{R,NS} = (−1)^{\frac{1}{2} (r′−s′)} \frac{4}{\sqrt{pq}} sin \frac{πrr′}{2q} (p − q) sin \frac{πss′}{2p} (p − q),

S_{[r,s],[r′,s′]}^{NS,R} = \begin{cases}(−1)^{\frac{1}{2} (r−s)} \frac{2}{\sqrt{pq}} sin \frac{πrr′}{2q} (p − q) sin \frac{πss′}{2p} (p − q) & if [r′, s′] = [\frac{q}{2}, \frac{p}{2}], \\ (−1)^{\frac{1}{2} (r−s)} \frac{4}{\sqrt{pq}} sin \frac{πrr′}{2q} (p − q) sin \frac{πss′}{2p} (p − q) & otherwise. \end{cases}

We remark that by the definition of the matrices (S_{[r,s],[r′,s′]}^{∗, ∗′}) (∗, ∗′ ∈ {NS, R}), the following formulae hold:

Unitarity  ∑_{[r′,s′]∈Δ_{p,q}^{∗′}} S_{[r,s],[r′,s′]}^{∗, ∗′}S_{[r′,s′],[r′′,s′′]}^{∗′, ∗} = δ_{[r,s],[r′′,s′′]}.  (10)

Now we define the Verlinde’s algebra of N = 1 super minimal series as follows. Let V_{p,q} be the ℤ-free module generated by {ϕ_{[r,s]}^{NS}}_{[r,s]∈Δ_{p,q}^{NS}} and {ϕ_{[r,s]}^{R}}_{[r,s]∈Δ_{p,q}^{R}}. We introduce the multiplication ∘ on V_{p,q} by

ϕ_{[r_{1},s_{1}]}^{∗_{1}} ∘ ϕ_{[r_{2},s_{2}]}^{∗_{2}} := ∑_{[r_{3},s_{3}]∈Δ_{p,q}^{∗_{3}}} N_{[r_{1},s_{1}],[r_{2},s_{2}]}^{[r_{3},s_{3}]}ϕ_{[r_{3},s_{3}]}^{∗_{3}} for ∗_{1}, ∗_{2}, ∗_{3} ∈ {NS, R},

where ∗_{i} (i = 1, 2, 3) are so chosen that ♯{i|∗_{i} = R} ≡ 0 mod 2 and

N_{[r_{1},s_{1}],[r_{2},s_{2}]}^{[r_{3},s_{3}]} = ∑_{[r′,s′]∈Δ_{p,q}^{NS}} \frac{S_{[r_{1},s_{1}],[r′,s′]}^{∗_{1}, NS}S_{[r_{2},s_{2}],[r′,s′]}^{∗_{2}, NS}S_{[r′,s′],[r_{3},s_{3}]}^{NS, ∗_{3}}}{S_{[1,1],[r′,s′]}^{NS, NS}} .

These formulae have already been discussed in [5] and [19], and their explicit values are computed in a few examples. We call the above formulae modified Verlinde’s formulae and the algebra V_{p,q} equipped with the multiplication ∘ Verlinde’s algebra. In the sequel, we compute the concrete value of N_{[r_{1},s_{1}],[r_{2},s_{2}]}^{[r_{3},s_{3}]}. For (r_{i}, s_{i}) ∈ KT_{p,q}^{∗_{i}} (i = 1, 2), set

I_{(r_{1},s_{1}),(r_{2},s_{2})} := \{(r, s) ∈ KT_{p,q}^{∗_{3}} | \begin{matrix}|r_{1} − r_{2}| < r < min\{r_{1} + r_{2}, 2q − (r_{1} + r_{2})\}, \\ r ≢ |r_{1} − r_{2}| mod 2, \\ |s_{1} − s_{2}| < s < min\{s_{1} + s_{2}, 2p − (s_{1} + s_{2})\}, \\ s ≢ |s_{1} − s_{2}| mod 2 \end{matrix} \} .

We have the following expression.

Lemma 4.3. The following formulae hold:

N_{[r_{1},s_{1}],[r_{2},s_{2}]}^{[r_{3},s_{3}]} = ♯{π^{−1}([r_{3}, s_{3}]) ∩ I_{(r_{1},s_{1}),(r_{2},s_{2})}}.

Proof. Note that from the explicit formulae of the matrices (S_{[r,s],[r′,s′]}^{∗, ∗′}) (∗, ∗′ ∈ {NS, R}), one can prove

\frac{S_{[r_{1},s_{1}],[r′,s′]}^{∗_{1}, NS}S_{[r_{2},s_{2}],[r′,s′]}^{∗_{2}, NS}}{S_{[1,1],[r′,s′]}^{NS, NS}} = ∑_{(r,s)∈I_{(r_{1},s_{1}),(r_{2},s_{2})}} S_{[r,s],[r′,s′]}^{∗_{3}, NS}.

Now the results follow from the unitarity of the matrices (S_{[r,s],[r′,s′]}^{∗, ∗′}) (cf. see equations (10)).

Let V_{p,q}^{NS} be the ℤ-submodule of V_{p,q} generated by {ϕ_{[r,s]}^{NS}|(r, s) ∈ Δ_{p,q}^{NS}}. By Lemma 4.3 and Theorem 3.5, it is easy to see that the following theorem holds.

Theorem 4.4. The correspondence ϕ_{[r,s]}^{∗} ↦ [(V_{r−1}, V_{s−1})] gives rise to the isomorphism

V_{p,q}^{NS} ≅ A_{c}^{NS}.

Conjecture 4.5. One can extend the above isomorphism to an isomorphism

V_{p,q} ≅ A_{c}.

5. Proof of Theorem 3.4 and 3.5

In this section, we give a sketch of a proof of Theorem 3.4 and 3.5, since the proof is similar to the case of ôsp(1|2) [13]. Namely, we will calculate the dimension of the coinvariant

H_{0}(L(E), L′_{ϵ_{1}}(c, h_{1})(w_{1}) ⊗ L′_{ϵ_{2}}(c, h_{2})(w_{2}) ⊗ L′_{ϵ_{3}}(c, h_{3})(w_{3})),

where E = {(w_{1}, ϵ_{1}), (w_{2}, ϵ_{2}), (w_{3}, ϵ_{3})}. From now on, let us fix w_{1} = 0, w_{2} = 1 and w_{3} = ∞ for simplicity, since the dimension of this coinvariant does not depend on the choice of points on ℂP^{1}. For ϵ = \frac{1}{2}, 0, we set E_{ϵ} = \{(0, ϵ), (1, \frac{1}{2}), (∞, ϵ)\}. Note that it suffices to consider the Lie superalgebras L(E_{ϵ}) in our proof.

To prove our main results, we need explicit forms of the action of singular vectors on the loop modules. If c = c(t) and h = ha,b^{\frac{1}{2}}(t) for some a, b ∈ ℤ_{>0} such that a − b ∈ 2ℤ, then there exists S_{a,b} ∈ U(V ir−_{\frac{1}{2}}) \ \{0\} (unique up to scalar) such

that S_{a,b}|c, h⟩_{\frac{1}{2}} is a singular vector of level \frac{1}{2}ab. Set

δ_{i,j}^{(2)} = \{ \begin{matrix}1 & if i ≡ j (mod 2), \\ 0 & if i ≢ j (mod 2). \end{matrix}

S_{a,b} acts on the element F_{0}θ^{γ} ∈ F_{λ,μ}^{\frac{1}{2} ,σ} as follows:

Lemma 5.1.

S_{a,b}.F_{0}θ^{γ} = P_{a,b}^{γ}(λ, μ, t)_{\frac{1}{2}} F_{− \frac{1}{2} ab}θ^{δ_{ab,1−γ}^{(2)}},

where

\{P_{a,b}^{γ}(λ, μ, t)_{\frac{1}{2}} \}^{2} = ∏_{\begin{matrix}0 ≤ k ≤ a − 1 \\ 0 ≤ l ≤ b − 1 \\ k − l ≡ γ (mod 2) \end{matrix}} Q_{a,b}^{k,l}(λ, μ, t)_{\frac{1}{2}}

and

\begin{matrix}Q_{a,b}^{k,l}(λ, μ, t)_{\frac{1}{2}} \\ = [(μ − 2λ) − \frac{1}{2}(kt^{\frac{1}{2}} − lt^{− \frac{1}{2}} )\{(a − k)t^{\frac{1}{2}} − (b − l)t^{− \frac{1}{2}} \}] \\ × [(μ − 2λ) − \frac{1}{2}\{(k + 1)t^{\frac{1}{2}} − (l + 1)t^{− \frac{1}{2}} \}\{(a − k − 1)t^{\frac{1}{2}} − (b − l − 1)t^{− \frac{1}{2}} \}] \\ +\frac{1}{2} \{(a − 1 − 2k)t^{\frac{1}{2}} − (b − 1 − 2l)t^{− \frac{1}{2}} \}^{2} λ \end{matrix} (11)

Proof. We can show this lemma in the same way as in [7] by using embedding diagrams of Verma modules (see Proposition 1.3).

From now on, we will prove Theorem 3.5. We reduce our calculation of coinvariants to computation of cohomology of a nilpotent subalgebra. Let 𝔞_{\frac{1}{2}} be the following nilpotent subalgebra of L(E_{\frac{1}{2}} ):

𝔞_{\frac{1}{2}} = \{f(z) ∈ L(E_{\frac{1}{2}} )|f(z) ∈ [L̂_{w; \frac{1}{2}}^{≥} , L̂_{w; \frac{1}{2}}^{≥}] for w = 0, 1\}.

It is easy to see that

𝔞_{\frac{1}{2}} = z^{2}(z − 1)^{2}ℂ[z]∂_{z} ⊕ z(z − 1)ℂ[z]∂z^{\frac{1}{2}} .

We set

ℓ_{0}^{(0)} = −z(z − 1)^{2}∂_{z} and ℓ_{0}^{(1)} = −z^{2}(z − 1)∂_{z}.

First we show a preliminary lemma, since the idea of the proof is useful in the remainder of this paper.

Lemma 5.2. Suppose that c = c(t) satisfies the condition in Case 1 or Case 2. For any h_{0}, h_{1}, h_{∞} ∈ ℂ, we have

dim H_{0}(L(E_{\frac{1}{2}} ), M_{\frac{1}{2}} (c, h_{0})(0) ⊗ M_{\frac{1}{2}} (c, h_{1})(1) ⊗ M_{\frac{1}{2}} (c, h_{∞})(∞))_{σ} = 1,

where the subscript σ = 0, 1 specifies the even i.e. σ = 0 (resp. odd i.e. σ = 1) subspaces.

Proof. Shapiro’s lemma and Poincaré duality imply that

\begin{matrix}dim H_{0}(L(E_{\frac{1}{2}} ), M_{\frac{1}{2}} (c, h_{0})(0) ⊗ M_{\frac{1}{2}} (c, h_{1})(1) ⊗ M_{\frac{1}{2}} (c, h_{∞})(∞)) \\ = dim H^{0}(𝔞_{\frac{1}{2}} , M_{\frac{1}{2}} (c, h_{∞})(∞)^{∗})^{(h_{0},h_{1})}. \end{matrix}

Here V^{∗} stands for the full dual of V and the superscripts (h_{0}, h_{1}) signify the simultaneous eigenspace of the operators ℓ_{0}^{(0)} and ℓ_{0}^{(1)} with the eigenvalues h_{0} and h_{1} respectively. Let us calculate the dimension of the space

H^{0}(𝔞_{\frac{1}{2}} , M_{\frac{1}{2}} (c, h_{∞})(∞)^{∗})^{(h_{0},h_{1})}

in the following way: For Φ ∈ M_{\frac{1}{2}} (c, h_{∞})(∞)^{∗} and j ∈ \frac{1}{2}ℤ_{≥0}, we denote by Φ_{j} the restriction Φ|_{M_{\frac{1}{2}} (c,h_{∞})(∞)_{j}}. Then we have

Fact 5.1. For Φ ∈ M_{\frac{1}{2}} (c, h_{∞})(∞)^{∗}, Φ ∈ H^{0}(𝔞_{\frac{1}{2}} , M_{\frac{1}{2}} (c, h_{∞})(∞)^{∗})^{(h_{0},h_{1})} if and only if

\begin{matrix}L_{−n}^{(∞)}.Φ_{j} = \{−h_{∞} + h_{0} − n(h_{1} − 1) − j\}Φ_{j−n}, \\ G_{−m}^{(∞)}.Φ_{j} = G_{− \frac{1}{2}}^{(∞)}.Φ_{j−m+ \frac{1}{2}} \end{matrix}

for any n ∈ ℤ_{>0} and m ∈ ℤ_{>0} − \frac{1}{2} .

Using this fact, we can show

Lemma 5.3. Set V = ⨁_{j=0, \frac{1}{2}} M_{\frac{1}{2}} (c, h_{∞})(∞)_{j}. Then the map

\begin{matrix}H^{0}(𝔞_{\frac{1}{2}} , M_{\frac{1}{2}} (c, h_{∞})(∞)^{∗})^{(h_{0},h_{1})} & → & V^{∗} \\ Φ & ↦ & Φ|_{V} \end{matrix}  (12)

gives an isomorphism of a ℤ_{2}-graded vector space.

Lemma 5.2 is an immediate consequence of the last lemma.

Till the end of the proof of Theorem 3.5, we assume that c = c(\frac{p}{q}) satisfies the condition in Case 2.

Theorem 5.4. Suppose that c = c(\frac{p}{q}) satisfies the condition in Case 2 and h_{0}, h_{1}, h_{∞} ∈ Λc_{\frac{1}{2}} . Then we have

\begin{matrix}dim H_{0}(L(E_{\frac{1}{2}} ), L_{\frac{1}{2}} (c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, h_{1})(1) ⊗ L_{\frac{1}{2}} (c, h_{∞})(∞)) \\ = dim H_{0}(𝔞_{\frac{1}{2}} , L_{\frac{1}{2}} (c, h_{∞})(∞))^{(−h_{0},−h_{1})}, \end{matrix}

where H_{0}(𝔞_{\frac{1}{2}} , L_{\frac{1}{2}} (c, h_{∞})(∞))^{(−h_{0},−h_{1})} is the simultaneous eigenspace of ℓ_{0}^{(0)} and

ℓ_{0}^{(1)}.

From now on, let us examine

H_{0}(𝔞_{\frac{1}{2}} , L_{\frac{1}{2}} (c, h_{∞})(∞))^{(−h_{0},−h_{1})}.

For h_{0}, h_{1}, h_{∞} ∈ Λc_{\frac{1}{2}} , let [(V_{i_{0}}, V_{j_{0}})], [(V_{i_{1}}, V_{j_{1}})] and [(V_{i_{∞}}, V_{j_{∞}})] be the equivalence

classes corresponding to lh_{0}^{\frac{1}{2}} , lh_{1}^{\frac{1}{2}} and lh_{∞}^{\frac{1}{2}} respectively.

Theorem 5.5. Suppose that c = c(\frac{p}{q}) satisfies the condition in Case 2.

(i) dim H_{0}(𝔞_{\frac{1}{2}} , L_{\frac{1}{2}} (c, h_{∞})(∞)) < ∞, if and only if h_{∞} ∈ Λc_{\frac{1}{2}} .

(ii) If h_{∞} ∈ Λc_{\frac{1}{2}} , then we have

dim H_{0}(𝔞_{\frac{1}{2}} , L_{\frac{1}{2}} (c, h_{∞})(∞))^{(−h_{0},−h_{1})}
= \{ \begin{matrix}2 & if h_{0}, h_{1} ∈ Λc_{\frac{1}{2}} and both C^{0} and C^{1} hold, \\ 1 & if h_{0}, h_{1} ∈ Λc_{\frac{1}{2}} and only one of C^{0} and C^{1} holds, \\ 0 & otherwise, \end{matrix}

where the conditions C^{0} and C^{1} are defined as follows:

P = \{(i, j) ∈ (ℤ)^{2} | \begin{matrix}|i_{0} − i_{1}| ≤ i ≤ min(i_{0} + i_{1}, 2q − i_{0} − i_{1} − 4), \\ |j_{0} − j_{1}| ≤ j ≤ min(j_{0} + j_{1}, 2p − j_{0} − j_{1} − 4), \\ i_{0} + i_{1} + i ≡ 0, j_{0} + j_{1} + j ≡ 0 (mod 2), \end{matrix} \}

and

C^{0} ⇔ [(i_{∞}, j_{∞}) ∈ P], C^{1} ⇔ [(−i_{∞} + q − 2, −j_{∞} + p − 2) ∈ P].

Proof. Since the proof of Theorem 5.5 is very similar to that of Theorem 4.4 in [13], here we only give an outline of the proof. To prove the theorem, we identify the invariant

H^{0}(𝔞_{\frac{1}{2}} , M_{\frac{1}{2}} (c, h_{∞})(∞)^{∗})^{(h_{0},h_{1})}  (13)

with the quotient module G_{λ,μ}^{\frac{1}{2} ,σ}(∞) and further, by using the formulae of the action of singular vectors on the loop modules given in Lemma 5.7, we determine when a form Φ ∈ H^{0}(𝔞_{\frac{1}{2}} , M_{\frac{1}{2}} (c, h_{∞})(∞)^{∗})^{(h_{0},h_{1})} induces a non-trivial element of H^{0}(𝔞_{\frac{1}{2}} , L_{\frac{1}{2}} (c, h_{∞})(∞)^{∗})^{(h_{0},h_{1})}. First we state how one can identify the invariant (13) with the quotient modules. It follows from Lemma 5.2 that

dim H^{0}(𝔞_{\frac{1}{2}} , M_{\frac{1}{2}} (c, h_{∞})(∞)^{∗})_{σ}^{(h_{0},h_{1})} = 1,

for each σ = 0, 1. Let Φ^{σ} ∈ H^{0}(𝔞_{\frac{1}{2}} , M_{\frac{1}{2}} (c, h_{∞})(∞)^{∗})_{σ}^{(h_{0},h_{1})} such that Φ^{σ} ≠ 0. For j ∈ \frac{1}{2}ℤ_{≥0}, we denote by Φ_{j}^{σ} the restriction Φ^{σ}|_{M_{\frac{1}{2}} (c,h_{∞})(∞)_{j}}. If we set

φ_{j}^{σ} = \{ \begin{matrix}Φ_{j}^{σ} & if j ∈ \frac{1}{2}σ + ℤ, \\ G_{− \frac{1}{2}}^{(∞)}.Φ_{j+ \frac{1}{2}}^{σ} & if j ∈ \frac{1}{2}(1 − σ) + ℤ, \end{matrix}

then from the formula (12) we obtain the following lemma.

Lemma 5.6. Let μ = −h_{∞} + h_{0} + h_{1} − \frac{1}{2} and λ = h_{1} − \frac{1}{2} . For each σ = 0, 1,

⨁_{j∈ \frac{1}{2} ℤ_{≥0}} ℂφ_{j}^{σ} ≃ G_{λ,μ}^{\frac{1}{2} ,1−σ}(∞)

as a L̂_{∞; \frac{1}{2}}^{<} -module, via

φ_{j}^{σ} ↦ F_{j}θ^{δ_{2j,σ}^{(2)}} for j ∈ \frac{1}{2}ℤ_{≥0}.

For the action of the singular vector S_{a,b}^{(w)}|c, h_{∞}⟩(w)_{\frac{1}{2}}, we have the following lemma that is an analog of Lemma 5.1:

Lemma 5.7. For F_{\frac{1}{2} ab}θ^{γ} ∈ F_{λ,μ}^{\frac{1}{2} ,σ}(w), we have

a(S_{a,b}^{(w)}).F_{\frac{1}{2} ab}θ^{γ} = ∏_{\begin{matrix}0 ≤ k ≤ a − 1 \\ 0 ≤ l ≤ b − 1 \\ k − l ≡ γ (mod 2) \end{matrix}} Q_{a,b}^{k,l}(λ, μ − \frac{1}{2}ab, −t)_{\frac{1}{2}} F_{0}θ^{δ_{ab,1−γ}^{(2)}},

where a denotes the antipode and the polynomial Q_{a,b}^{k,l}(λ, μ, t)_{\frac{1}{2}} is defined as (11). Using the isomorphism in Lemma 5.6 and the formulae in Lemma 5.7, we obtain the condition that the form Φ^{σ} vanishes on the singular vectors of M_{\frac{1}{2}} (c, h_{∞})(∞). Therefore, we have proved Theorem 5.5.

To complete the proof of Theorem 3.5, we need to show Theorem 5.4. The following lemma implies Theorem 5.4.

Lemma 5.8. Let c = c(\frac{p}{q}) satisfy the condition in Case 2, and h_{0}, h_{1}, h_{∞} ∈ Λc_{\frac{1}{2}} . Then we have

(i)

\begin{matrix}H_{0}(L(E_{\frac{1}{2}} ), M_{\frac{1}{2}} (c, h_{0})(0) ⊗ M_{\frac{1}{2}} (c, h_{1})(1) ⊗ L_{\frac{1}{2}} (c, h_{∞})(∞)) \\ = H_{0}(L(E_{\frac{1}{2}} ), L_{\frac{1}{2}} (c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, h_{1})(1) ⊗ L_{\frac{1}{2}} (c, h_{∞})(∞)), \end{matrix}

(ii)

\begin{matrix}H_{0}(L(E_{\frac{1}{2}} ), M_{\frac{1}{2}} (c, h_{0})(0) ⊗ M_{\frac{1}{2}} (c, h_{1})(1) ⊗ L_{\frac{1}{2}} (c, h_{∞})(∞)) \\ = H_{0}(𝔞_{\frac{1}{2}} , L_{\frac{1}{2}} (c, h_{∞})(∞))^{(−h_{0},−h_{1})}. \end{matrix}

Proof. We can prove this lemma in the same way as in [13] (Lemma 4.5).

From now on, we will consider Case 1 i.e. Theorem 3.4. Here we assume that c = c(t) for t ∈ ℝ_{>0} \ ℚ_{>0}. Similarly to Case 2, we introduce a nilpotent subalgebra, which plays the same role as 𝔞_{\frac{1}{2}} in this case.

\begin{aligned}𝔠_{\frac{1}{2}} & = & \{f(z) ∈ L(E_{\frac{1}{2}} )|f(z) ∈ [L̂_{w; \frac{1}{2}}^{≥} , L̂_{w; \frac{1}{2}}^{≥}] at w = ∞\} \\ & = & ℂ[z^{−1}, (1 − z)^{−1}]∂_{z} ⊕ ℂ[z^{−1}, (1 − z)^{−1}]∂z^{\frac{1}{2}} . \end{aligned}

We set

ℓ_{0}^{(∞)} = z∂_{z}.

Theorem 5.9. Suppose that c = c(t) satisfies the condition in Case 1 and h_{0}, h_{1}, h_{∞} ∈ Λc_{\frac{1}{2}} , then we have

\begin{matrix}dim H_{0}(L(E_{\frac{1}{2}} ), L_{\frac{1}{2}} (c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, h_{1})(1) ⊗ L_{\frac{1}{2}} (c, h_{∞})(∞)) \\ = dim H_{0}(𝔠_{\frac{1}{2}} , L_{\frac{1}{2}} (c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, h_{1})(1))^{(−h_{∞})}, \end{matrix}

where H_{0}(𝔠_{\frac{1}{2}} , L_{\frac{1}{2}} (c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, h_{1})(1))^{(−h_{∞})} denotes the eigenspace of the oper-

ator ℓ_{0}^{(∞)} with the eigenvalue −h_{∞}.

To prove Theorem 3.4, we calculate the right hand side of Theorem 5.9. For h_{0}, h_{1}, h_{∞} ∈ Λc_{\frac{1}{2}} , let (V_{i_{0}}, V_{j_{0}}), (V_{i_{1}}, V_{j_{1}}) and (V_{i_{∞}}, V_{j_{∞}}) be the pairs of two

irreducible representations of sl_{2} which correspond to lh_{0}^{\frac{1}{2}} , lh_{1}^{\frac{1}{2}} and lh_{∞}^{\frac{1}{2}} respectively. We say that a triple (h_{0}, h_{1}, h_{∞}) is proper if (V_{i_{∞}}, V_{j_{∞}}) is a component of (V_{i_{0}} ⊗ V_{i_{1}}, V_{j_{0}} ⊗ V_{j_{1}}).

Theorem 5.10. We assume that c = c(t) satisfies the condition in Case 1 and h_{0}, h_{1} ∈ Λc_{\frac{1}{2}} , then we have

dim H_{0}(𝔠_{\frac{1}{2}} , L_{\frac{1}{2}} (c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, h_{1})(1))^{(−h_{∞})}

= \{ \begin{matrix}1 & if h_{∞} ∈ Λc_{\frac{1}{2}} and (h_{0}, h_{1}, h_{∞}):proper, \\ 0 & otherwise. \end{matrix}

Proof. Here we only identify (co-)invariants with the loop modules. It follows from Lemma 5.2 that

dim H^{0}(𝔠_{\frac{1}{2}} , (M_{\frac{1}{2}} (c, h_{0})(0) ⊗ M_{\frac{1}{2}} (c, h_{1})(1))^{∗})_{σ}^{(h_{∞})} = 1,

for any σ = 0, 1. For a non-zero element

Ψ^{σ} ∈ H^{0}(𝔠_{\frac{1}{2}} , (M_{\frac{1}{2}} (c, h_{0})(0) ⊗ M_{\frac{1}{2}} (c, h_{1})(1))^{∗})_{σ}^{(h_{∞})},

we introduce functionals Ψ^{σ,w} (w = 0, 1) on M_{\frac{1}{2}} (c, h_{w})(w) by

Ψ^{σ,0}(u_{0}) = Ψ^{σ}(u_{0} ⊗ |c, h_{1}⟩(1)_{\frac{1}{2}}) and Ψ^{σ,1}(u_{1}) = Ψ^{σ}(|c, h_{0}⟩(0)_{\frac{1}{2}} ⊗ u_{1}),

where u_{w} ∈ M_{\frac{1}{2}} (c, h_{w})(w). Let Ψ_{j}^{σ,w} be the restriction Ψ^{σ,w}|_{M_{\frac{1}{2}} (c,h_{w})(w)_{j}}. We set

ψ_{j}^{σ,w} = \{ \begin{matrix}Ψ_{j}^{σ,w} & if j ∈ \frac{1}{2}σ + ℤ, \\ G_{− \frac{1}{2}}^{(w)}.Ψ_{j+ \frac{1}{2}}^{σ,w} & if j ∈ \frac{1}{2}(1 − σ) + ℤ, \end{matrix}

then we have the following lemma.

Lemma 5.11.

(i) Let μ = h_{∞} − h_{0} + h_{1} − \frac{1}{2} and λ = h_{1} − \frac{1}{2} . For each σ = 0, 1,

⨁_{j∈ \frac{1}{2} ℤ_{≥0}} ℂψ_{j}^{σ,0} ≃ G_{λ,μ}^{\frac{1}{2} ,1−σ}(0)

as a L̂_{0; \frac{1}{2}}^{<} -module, via ψ_{j}^{σ,0} ↦ F_{j}θ^{δ_{2j,σ}^{(2)}} ( j ∈ \frac{1}{2}ℤ_{≥0} ).

(ii) Let μ = h_{∞} − h_{1} + h_{0} − \frac{1}{2} and λ = h_{0} − \frac{1}{2} . For each σ = 0, 1,

⨁_{j∈ \frac{1}{2} ℤ_{≥0}} ℂψ_{j}^{σ,1} ≃ G_{λ,μ}^{\frac{1}{2} ,1−σ}(1)

as a L̂_{1; \frac{1}{2}}^{<} -module, via ψ_{j}^{σ,1} ↦ F_{j}θ^{δ_{2j,σ}^{(2)}} ( j ∈ \frac{1}{2}ℤ_{≥0} ).

Since the other parts of the proof of Theorem 5.10 are the very analogous to those of Theorem 4.4 in [13], we omit them.

From the following lemma, we obtain Theorem 5.9 and thus we complete the proof of Theorem 3.4.

Lemma 5.12. Assume that c = c(t) satisfies the condition in Case 1 and h_{0}, h_{1}, h_{∞} ∈ Λc_{\frac{1}{2}} . Then we have

(i)

\begin{matrix}H_{0}(L(E_{\frac{1}{2}} ), L_{\frac{1}{2}} (c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, h_{1})(1) ⊗ M_{\frac{1}{2}} (c, h_{∞})(∞)) \\ = H_{0}(L(E_{\frac{1}{2}} ), L_{\frac{1}{2}} (c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, h_{1})(1) ⊗ L_{\frac{1}{2}} (c, h_{∞})(∞)), \end{matrix}

(ii)

\begin{matrix}H_{0}(L(E_{\frac{1}{2}} ), L_{\frac{1}{2}} (c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, h_{1})(1) ⊗ M_{\frac{1}{2}} (c, h_{∞})(∞)) \\ = H_{0}(𝔠_{\frac{1}{2}} , L_{\frac{1}{2}} (c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, h_{1})(1))^{(−h_{∞})}. \end{matrix}

6. On Conjecture 3.6

Here we state two propositions, which support Conjecture 3.6. To prove the first proposition, we identify coinvariants with the quotient module G_{λ,μ}^{0,0} and calculate the action of singular vectors. We need a formulae of the action of singular vectors on the loop module F_{λ,μ}^{0,0}. For τ = ±, let { \atop |c, h : τ⟩_{0}} be a highest weight vector of the half Verma module M̃_{0}^{τ}(c, h). Suppose that c = c(t) for t ∈ ℝ_{>0} and h = h_{a,b}^{0}(t) for a, b ∈ ℤ_{>0} such that a − b ≡ 1 (mod 2). Then there exists an even element S_{a,b}^{0} (resp. odd element S_{a,b}^{1}) of U(V ir_{0}^{−})\{0} such that a singular vector of the half Verma module M_{0}^{τ}(c, h) can be written as

(S_{a,b}^{0} + τS_{a,b}^{1}){ \atop |c, h : τ⟩_{0}}.

For α, β ∈ {0, 1}, let P_{a,b}^{α,β}(λ, μ, t) be polynomials of t defined by

S_{a,b}^{0}.F_{0} = P_{a,b}^{0,0}(λ, μ, t)_{0}F_{− \frac{1}{2} ab},
S_{a,b}^{0}.F_{0}θ = P_{a,b}^{1,1}(λ, μ, t)_{0}F_{− \frac{1}{2} ab}θ,
S_{a,b}^{1}.F_{0} = P_{a,b}^{1,0}(λ, μ, t)_{0}F_{− \frac{1}{2} ab}θ,
S_{a,b}^{1}.F_{0}θ = P_{a,b}^{0,1}(λ, μ, t)_{0}F_{− \frac{1}{2} ab},

where F_{i}, F_{i}θ ∈ F_{λ,μ}^{0,0}. Set

P_{a,b}(λ, μ, t)_{0} = ( \begin{matrix}P_{a,b}^{0,0}(λ, μ, t)_{0} & P_{a,b}^{0,1}(λ, μ, t)_{0} \\ P_{a,b}^{1,0}(λ, μ, t)_{0} & P_{a,b}^{1,1}(λ, μ, t)_{0} \end{matrix} ) .  (14)

Lemma 6.1.

\{det P_{a,b}(λ, μ, t)_{0}\}^{2} = ∏_{\begin{matrix}0 ≤ k ≤ a − 1 \\ 0 ≤ l ≤ b − 1 \end{matrix}} Q_{a,b}^{k,l}(λ, μ, t)_{0},

where

Q_{a,b}^{k,l}(λ, μ, t)_{0}

= [(μ − 2λ) + \frac{1}{2}ab − \frac{1}{2}(kt^{\frac{1}{2}} + lt^{− \frac{1}{2}} )\{(a − k)t^{\frac{1}{2}} + (b − l)t^{− \frac{1}{2}} \}]

× [(μ − 2λ) + \frac{1}{2}ab − \frac{1}{2}\{(k + 1)t^{\frac{1}{2}} + (l + 1)t^{− \frac{1}{2}} \}\{(a − k − 1)t^{\frac{1}{2}} + (b − l − 1)t^{− \frac{1}{2}} \}]

+\frac{1}{2} \{(a − 1 − 2k)t^{\frac{1}{2}} + (b − 1 − 2l)t^{− \frac{1}{2}} \}^{2} λ.

Remark 6.2. At present, only the determinant of the matrix P_{a,b}(λ, μ, t)_{0} can be calculated and explicit forms of the matrix elements P_{a,b}^{α,β}(λ, μ, t)_{0} are not obtained except for some special cases.

From now on, we give two supporting evidences of Conjecture 3.6. First one is a necessary condition that the coinvariants

H_{0}(L(E_{0}), L_{0}(c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, h_{1})(1) ⊗ L_{0}(c, h_{∞})(∞))  (15)

do not vanish. In the sequel we suppose that c = c(t) satisfies the condition in Case 1 or Case 2 and h_{0}, h_{∞} ∈ Λ_{0}^{c}, h_{1} ∈ Λc_{\frac{1}{2}} . In Case 1, let (V_{i_{0}}, V_{j_{0}}), (V_{i_{1}}, V_{j_{1}}) and (V_{i_{∞}}, V_{j_{∞}}) be the pairs of two irreducible representations of sl_{2} which correspond to l_{h_{0}}^{0} , lh_{1}^{\frac{1}{2}} and l_{h_{∞}}^{0} respectively. In Case 2, let [(V_{i_{0}}, V_{j_{0}})], [(V_{i_{1}}, V_{j_{1}})] and [(V_{i_{∞}}, V_{j_{∞}})]

be the equivalence classes which correspond to l_{h_{0}}^{0} , lh_{1}^{\frac{1}{2}} and l_{h_{∞}}^{0} respectively. We say that a triple (h_{0}, h_{1}, h_{∞}) is proper if (V_{i_{∞}}, V_{j_{∞}}) (resp. [(V_{i_{∞}}, V_{j_{∞}})]) is a component of (V_{i_{0}} ⊗ V_{i_{1}}, V_{j_{0}} ⊗ V_{j_{1}}) (resp. [(V_{i_{0}} ˙⊗_{q−2}V_{i_{1}}, V_{j_{0}} ˙⊗_{p−2}V_{j_{1}})]).

Proposition 6.3. Suppose that c = c(t) satisfies one of the conditions in Case 1 and Case 2. If at least two of (h_{0}, 0), (h_{1}, \frac{1}{2}) and (h_{∞}, 0) are elements of S_{c} and

dim H_{0}(L(E_{0}), L_{0}(c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, h_{1})(1) ⊗ L_{0}(c, h_{∞})(∞)) ≠ 0,  (16)

then (h_{0}, 0), (h_{1}, \frac{1}{2}), (h_{∞}, 0) ∈ S_{c} and the triple (h_{0}, h_{1}, h_{∞}) is proper.

Proof. Similarly to NS × NS → NS case, by using Shapiro’s lemma and Poincaré duality, we can reduce calculation of coinvariants (16) to computation of coinvariants of nilpotent subalgebras. Furthermore in the case of h_{0}, h_{∞} ≠ \frac{1}{24}c, these coinvariants of nilpotent subalgebras can be identified with the quotient module G_{λ,μ}^{0,0}. By using Lemma 6.1 and its variants, we can show this proposition by an argument similar to the proof of Theorem 5.4 and 5.5. In particular, we will calculate the coinvariants (16) explicitly for h_{0} = \frac{1}{24}c or h_{∞} = \frac{1}{24}c in Proposition 6.7.

Remark 6.4. The above coinvariant

H_{0}(L(E_{0}), L_{0}(c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, h_{1})(1) ⊗ L_{0}(c, h_{∞})(∞))

has even dimension. In fact, we see that

\begin{matrix}H_{0}(L(E_{0}), L_{0}(c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, h_{1})(1) ⊗ L_{0}(c, h_{∞})(∞)) \\ = H_{0}(𝔞̃_{0}, L_{0}(c, h_{0})(0) ⊗ L_{0}(c, h_{∞})(∞))^{(−h_{1})}, \end{matrix}

where

\begin{aligned}𝔞̃_{0} & = & \{f(z) ∈ L(E_{0})|f(z) ∈ [L_{1: \frac{1}{2}}^{≥} , L_{1: \frac{1}{2}}^{≥}]\} \\ & = & ℂ[z, z^{−1}](z − 1)^{2}∂_{z} ⊕ ℂ[z, z^{−1}](z − 1)z^{\frac{1}{2}} ∂z^{\frac{1}{2}} \end{aligned}

and H_{0}(𝔞̃_{0}, L_{0}(c, h_{0})(0)⊗L_{0}(c, h_{∞})(∞))^{(−h_{1})} denotes the eigenspace of the operator −(z − 1)∂_{z} with the eigenvalue −h_{1}. The parity shift Π acts on the coinvariant

H_{0}(𝔞̃_{0}, L_{0}(c, h_{0})(0) ⊗ L_{0}(c, h_{∞})(∞))^{(−h_{1})}

as an isomorphism, since

Π(L_{0}(c, h_{0})(0) ⊗ L_{0}(c, h_{∞})(∞)) = L_{0}(c, h_{0})(0) ⊗ L_{0}(c, h_{∞})(∞).

This implies the desired result.

Next we give the second supporting evidence (Proposition 6.5). Consider the following subcase of Case 2.

Case 2’ c = c(\frac{p}{q}) for p, q ∈ 2ℤ_{>0}, p ≠ q and (\frac{1}{2}(p − q), q) = 1.

We prove

Proposition 6.5. If c = c(\frac{p}{q}) satisfies the condition in Case 2’ and h_{0} = \frac{1}{24}c or h_{∞} = \frac{1}{24}c, then Conjecture 3.6 holds. For the proof, we have to calculate

dim H_{0}(L(E_{0}), L_{0}(c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, h_{1})(1) ⊗ L_{0}(c, h_{∞})(∞))  (17)

for h_{1} ∈ Λc_{\frac{1}{2}} and h_{0}, h_{∞} ∈ Λ_{0}^{c} such that h_{0} = \frac{1}{24}c or h_{∞} = \frac{1}{24}c. To examine the above coinvariants, we introduce the following nilpotent subalgebra 𝔞_{0}:

\begin{aligned}𝔞_{0} & = \{f(z) ∈ L(E_{0}) |f(z) ∈ [L̂_{w,ϵ}^{≥}, L̂_{w,ϵ}^{≥}] for (w, ϵ) = (0, 0), (∞, 0)\} \\ & = z^{2}(z − 1)^{−2}ℂ[(z − 1)^{−1}]∂_{z} ⊕ z^{\frac{3}{2}} (z − 1)^{−2}ℂ[(z − 1)^{−1}]∂z^{\frac{1}{2}} . \end{aligned}  (18)

We set

𝔞̄_{0} = 𝔞_{0} ⨁_{w=0,∞}(ℂℓ_{0}^{(w)} ⊕ ℂg_{0}^{(w)}),  (19)

where

\begin{matrix}ℓ_{0}^{(0)} = −z(z − 1)^{−2}∂_{z}, & ℓ_{0}^{(∞)} = z^{2}(z − 1)^{−1}∂_{z}, \\ g_{0}^{(0)} = −\sqrt{−1}z^{\frac{1}{2}} (z − 1)^{−1}∂z^{\frac{1}{2}} , & g_{0}^{(∞)} = z^{\frac{3}{2}} (z − 1)^{−1}∂z^{\frac{1}{2}} . \end{matrix}

To reduce our calculation by using Sapiro’s lemma, we need the following representations of 𝔞̄_{0}: For w = 0, ∞, let W_{c,h_{w}}^{w} = ⨁_{α=0,1} ℂ1_{c,h_{w}:α}^{w} be the 2-dimensional representation of 𝔞̄_{0} defined by

x.1_{c,h_{w}:α}^{w} = 0  if x ∈ 𝔞_{0}
ℓ_{0}^{(w′)}.1_{c,h_{w}:α}^{w} = δ_{w′,w}h_{w}1_{c,h_{w}:α}^{w}  for w′ = 0, ∞,
g_{0}^{(w′)}.1_{c,h_{w}:α}^{w} = (h_{w} − \frac{1}{24}c)^{\frac{1}{2}} 1_{c,h_{w}:1−α}^{w} for w′ = 0, ∞,
deg 1_{c,h_{w}:α}^{w} = α.

(20)

Now we have

Proposition 6.6. Suppose that c = c(\frac{p}{q}) satisfies the condition in Case 2’ and

h_{0}, h_{∞} ∈ Λ_{0}^{c}, h_{1} ∈ Λc_{\frac{1}{2}} . If h_{0} = \frac{1}{24}c or h_{∞} = \frac{1}{24}c then

\begin{matrix}dim H_{0}(L(E_{0}), L_{0}(c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, h_{1})(1) ⊗ L_{0}(c, h_{∞})(∞)) \\ = dim H_{0}(𝔞̄_{0}, W_{c,h_{0}}^{0} ⊗ L_{\frac{1}{2}} (c, h_{1})(1) ⊗ W_{c,h_{∞}}^{∞}). \end{matrix}

Proof. Using Shapiro’s lemma, the resolution for L_{ϵ}(c, h_{w})(w) and Theorem 6.6, we can prove this theorem by the same argument as in the proof of Lemma 4.5 in [13].

To show Conjecture 3.6 in Case 2’, we calculate

dim H_{0}(𝔞̄_{0}, W_{c,h_{0}}^{0} ⊗ L_{\frac{1}{2}} (c, h_{1})(1) ⊗ W_{c,h_{∞}}^{∞}).

For the sake of the commutativity of the tensor products in the coinvariant (17), we can assume that h_{∞} = \frac{1}{24}c without loss of generality. We show the following proposition.

Proposition 6.7. Suppose that c = c(\frac{p}{q}) satisfies the condition in Case 2’ and

h_{1} = hr_{1},s_{1}^{\frac{1}{2}}(\frac{p}{q}) ∈ Λc_{\frac{1}{2}} . If h_{∞} = \frac{1}{24}c then

H_{0}(𝔞̄_{0}, W_{c,h_{0}}^{0} ⊗ L_{\frac{1}{2}} (c, h_{1})(1) ⊗ W_{c,h_{∞}}^{∞})

= \{ \begin{matrix}4 & if h_{0} = h_{r,s}^{0}(\frac{p}{q}) for some (r, s) ∈ E_{min(r_{1},q−r_{1}),min(s_{1},p−s_{1})}, \\ 0 & otherwise, \end{matrix}

where, for a, b ∈ ℤ_{>0} such that a ≡ b (mod 2), the set E_{a,b} is defined by

E′_{a,b} = \{(r, s) ∈ (ℤ_{>0} × ℤ) ⊔ (\{0\} × ℤ_{>0}) | \begin{matrix}r ≤ a − 1 ∧ |s| ≤ b − 1 \\ r ≡ a − 1, s ≡ b − 1 (mod 2) \end{matrix} \} , (21)

and

E_{a,b} = \{ \begin{matrix}E′_{a,b} ⊔ \{(0, 0)\} & if a ≡ b ≡ 1 (mod 2), \\ E′_{a,b} & if a ≡ b ≡ 0 (mod 2). \end{matrix}  (22)

Proof. By Poincaré duality we have

\begin{matrix}dim H_{0}(𝔞̄_{0}, W_{c,h_{0}}^{0} ⊗ L_{\frac{1}{2}} (c, h_{1})(1) ⊗ W_{c,h_{∞}}^{∞}) \\ = dim H^{0}(𝔞̄_{0}, (W_{c,h_{0}}^{0} ⊗ L_{\frac{1}{2}} (c, h_{1})(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}). \end{matrix}  (23)

In the sequel, we calculate the invariant in the right hand side of (23). By using the following part of BGG type resolution:

N_{\frac{1}{2}} (c, h_{1})(1) → M_{\frac{1}{2}} (c, h_{1})(1) → L_{\frac{1}{2}} (c, h_{1})(1) → 0,

we obtain

H^{0}(𝔞̄_{0}, (W_{c,h_{0}}^{0} ⊗ N_{\frac{1}{2}} (c, h_{1})(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}) ←^{ξ}
H^{0}(𝔞̄_{0}, (W_{c,h_{0}}^{0} ⊗ M_{\frac{1}{2}} (c, h_{1})(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}) ←
H^{0}(𝔞̄_{0}, (W_{c,h_{0}}^{0} ⊗ L_{\frac{1}{2}} (c, h_{1})(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}) ← 0.

Therefore, we see that

H^{0}(𝔞̄_{0}, (W_{c,h_{0}}^{0} ⊗ L_{\frac{1}{2}} (c, h_{1})(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}) ≃ ker ξ.

To calculate dim ker ξ, we first determine how the nilpotent subalgebra L̂_{1, \frac{1}{2}}^{<} acts on the invariant:

H^{0}(𝔞̄_{0}, (W_{c,h_{0}}^{0} ⊗ M_{\frac{1}{2}} (c, h_{1})(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}).  (24)

Let us consider the following functionals on the Verma module M_{\frac{1}{2}} (c, h_{1})(1). For Ψ ∈ H^{0}(𝔞̄_{0}, (W_{c,h_{0}}^{0} ⊗ M_{\frac{1}{2}} (c, h_{1})(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}) and α, β ∈ \{0, 1\}, we define Ψ^{α,β} ∈ M_{\frac{1}{2}} (c, h_{1})(1)^{∗} by

Ψ^{α,β}(v_{1}) = Ψ(1_{c,h_{0}:α}^{0} ⊗ v_{1} ⊗ 1_{c,h_{∞}:β}^{∞}) for v_{1} ∈ M_{\frac{1}{2}} (c, h_{1})(1),  (25)

and set

ψ_{j}^{α,β} = Ψ^{α,β}|_{M_{\frac{1}{2}} (c,h_{1})(1)_{j}} for j ∈ \frac{1}{2}ℤ_{≥0}.  (26)

By direct calculation, we can show that the subalgebra L_{1, \frac{1}{2}}^{<} ⊕ ℂL_{0}^{(1)} ⊕ ℂC acts on

the functionals ψ_{j}^{α,β} as follows:

Lemma 6.8. For j ∈ \frac{1}{2}ℤ_{≥0} and m, n ∈ ℤ_{>0},

(i) C.φ_{j}^{α,β} = −cφ_{j}^{α,β},
(ii) L_{−n}^{(1)}.φ_{j}^{α,β} = (−1)^{n−1}(n(h_{0} − 1) + h_{1} − h_{∞} + j)φ_{j−n}^{α,β},
L_{0}^{(1)}.φ_{j}^{α,β} = −(h_{1} + j)φ_{j}^{α,β}
(iii) G_{−m+ \frac{1}{2}}^{(1)} .φ_{m− \frac{1}{2} +j}^{α,β} = (−\frac{1}{4})^{m−1} ( \begin{matrix}2m − 2 \\ m − 1 \end{matrix} )

× {(−1)^{2j+α}(2m − 1)K_{0}φ_{j}^{1−α,β} + K_{∞}φ_{j}^{α,1−β}}

+ (−\frac{1}{4})^{m} ( \begin{matrix}2m \\ m \end{matrix} ) ∑_{k=0}^{[j− \frac{1}{2} ]} ( \begin{matrix}−\frac{1}{2} \\ k \end{matrix} ) \frac{m}{m + k} G_{k+ \frac{1}{2}}^{(1)} .φ_{j−k− \frac{1}{2}}^{α,β} ,

where K_{0} = (h_{0} − \frac{1}{24}c)^{\frac{1}{2}} , K_{∞} = \sqrt{−1}(h_{∞} − \frac{1}{24}c)^{\frac{1}{2}} and [∗] stands for the Gaussian symbol.

Using this lemma, we can obtain the dimension of the invariant (24). Indeed, if we set

V_{0} = W_{c,h_{0}}^{0} ⊗ M_{\frac{1}{2}} (c, h_{1})(1)_{0} ⊗ W_{c,h_{∞}}^{∞},  (27)

then we have

Lemma 6.9. The map

\begin{matrix}H^{0}(𝔞̄_{0}, (W_{c,h_{0}}^{0} ⊗ M_{\frac{1}{2}} (c, h_{1})(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}) & → & (V_{0})^{∗} \\ Ψ & ↦ & Ψ|_{V_{0}} \end{matrix}  (28)

gives an isomorphism of a vector space.

In particular, we have

dim H^{0}(𝔞̄_{0}, (W_{c,h_{0}}^{0} ⊗ M_{\frac{1}{2}} (c, h_{1})(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}) = dim V_{0} = 4.

330 K. Iohara and Y. Koga Remark 6.10. In the above case, it follows that

dim H_{0}(L(E_{0}), M_{0}(c, h_{0})(0) ⊗ M_{\frac{1}{2}} (c, h_{1})(1) ⊗ M_{0}(c, h_{∞})(∞)) = 4, since Shapiro’s lemma implies

\begin{matrix}H_{0}(L(E_{0}), M_{0}(c, h_{0})(0) ⊗ M_{\frac{1}{2}} (c, h_{1})(1) ⊗ M_{0}(c, h_{∞})(∞)) \\ ≃ H_{0}(𝔞̄_{0}, W_{c,h_{0}}^{0} ⊗ M_{\frac{1}{2}} (c, h_{1})(1) ⊗ W_{c,h_{∞}}^{∞}). \end{matrix}

Next we calculate the action of singular vectors on functionals {ψ_{j}^{α,β}}. Suppose that c = c(t) for t ∈ ℝ_{>0} (not necessarily rational) and h_{1} = hr_{1},s_{1}^{\frac{1}{2}}(t). Let S_{r_{1},s_{1}}^{(1)}|c, h_{1}⟩(1)_{\frac{1}{2}} be a singular vector of level N_{1} = \frac{1}{2}r_{1}s_{1}, and A(h_{0}, h_{∞}; t) be the 4 × 4 matrix defined by

a(S_{r_{1},s_{1}}^{(1)}) : {ψ_{N_{1}}^{0,0}, ψ_{N_{1}}^{1,1}, ψ_{N_{1}}^{0,1}, ψ_{N_{1}}^{1,0}} ↦ {ψ_{0}^{0,0}, ψ_{0}^{1,1}, ψ_{0}^{0,1}, ψ_{0}^{1,0}}A(h_{0}, h_{∞}; t), By Lemma 6.8, A(h_{0}, h_{∞}; t) become the direct sum of the following 2×2 matrices

\begin{matrix}a(S_{r_{1},s_{1}}^{(1)}) : \{ψ_{N_{1}}^{0,0}, ψ_{N_{1}}^{1,1}\} ↦ \{ψ_{0}^{0,β}, ψ_{0}^{1,1−β}\}A′(h_{0}, h_{∞}; t), \\ a(S_{r_{1},s_{1}}^{(1)}) : \{ψ_{N_{1}}^{0,1}, ψ_{N_{1}}^{1,0}\} ↦ \{ψ_{0}^{0,1−β}, ψ_{0}^{1,β}\}A′′(h_{0}, h_{∞}; t), \end{matrix}

where β is given by

β = \{ \begin{matrix}0 & if N_{1} ∈ ℤ, \\ 1 & if N_{1} ∈ ℤ + \frac{1}{2}. \end{matrix}

Lemma 6.11. Suppose that c = c(\frac{p}{q}) satisfies the condition in Case 2’ and

h_{1} = h_{r_{1},s_{1}}(\frac{p}{q}) ∈ Λc_{\frac{1}{2}} . If h_{∞} = \frac{1}{24}c then

A′(h_{0}, h_{∞}; t) = A′′(h_{0}, h_{∞}; t)

= \{ \begin{matrix}( \begin{matrix}0 & −X_{r_{1},s_{1}} \\ X_{r_{1},s_{1}} & 0 \end{matrix} ) & if r_{1} ≡ s_{1} ≡ 1 (mod 2), \\ ( \begin{matrix}Y_{r_{1},s_{1}} & 0 \\ 0 & Y_{r_{1},s_{1}} \end{matrix} ) & if r_{1} ≡ s_{1} ≡ 0 (mod 2), \end{matrix}

where

\begin{matrix}X_{r_{1},s_{1}} = (−1)^{♯E′_{r_{1},s_{1}}}(h_{0} − h_{0,0}^{0}(t))^{\frac{1}{2}} ∏_{(a,b)∈E′_{r_{1},s_{1}}}(h_{0} − h_{a,b}^{0}(t)), \\ Y_{r_{1},s_{1}} = ∏_{(a,b)∈E′_{r_{1},s_{1}}}(h_{0} − h_{a,b}^{0}(t)), \end{matrix}

Remark 6.12. For c = c(\frac{p}{q}) which satisfies the condition in Case 2’, and

r_{1}, s_{1} ∈ ℤ_{>0} such that hr_{1},s_{1}^{\frac{1}{2}}(\frac{p}{q}) ∈ Λc_{\frac{1}{2}} ,

\{h_{r,s}^{0}(\frac{p}{q}) |(r, s) ∈ E_{min(r_{1},q−r_{1}),min(s_{1},p−s_{1})} \}

= \{h_{r,s}^{0}(\frac{p}{q}) | \begin{matrix}|\frac{q}{2} − r_{1}| + 1 ≤ r ≤ min\{\frac{3q}{2} − r_{1}, \frac{q}{2} + r_{1}\} − 1 \\ |\frac{p}{2} − s_{1}| + 1 ≤ s ≤ min\{\frac{3p}{2} − s_{1}, \frac{q}{2} + s_{1}\} − 1 \\ r ≡ \frac{q}{2} − r_{1} + 1, s ≡ \frac{p}{2} − s_{1} + 1 (mod 2) \end{matrix} \} .

Using this lemma, we can show Proposition 6.7. The maximal proper submodule N_{\frac{1}{2}} (c, h_{1})(1) of the Verma module M_{\frac{1}{2}} (c, h_{1})(1) is generated by the singular vectors

S_{r_{1},s_{1}}^{(1)}|c, h_{1}⟩(1)_{\frac{1}{2}} and S_{q−r_{1},p−s_{1}}^{(1)}|c, h_{1}⟩(1)_{\frac{1}{2}} of level \frac{1}{2}r_{1}s_{1} and \frac{1}{2}(q − r_{1})(p − s_{1}). We set

S_{1} = S_{r_{1},s_{1}}^{(1)} , S_{2} = S_{q−r_{1},p−s_{1}}^{(1)} , N_{1} = \frac{1}{2}r_{1}s_{1} and N_{2} = \frac{1}{2}(q − r_{1})(p − s_{1}). For i = 1, 2, let A′_{i} and A′′_{i} be the above 2×2 matrices A′(h_{0}, h_{∞}; t) and A′′(h_{0}, h_{∞}; t) defined by the singular vector S_{i}|c, h_{1}⟩(1)_{\frac{1}{2}}. Set

β_{i} = \{ \begin{matrix}0 & if N_{i} ∈ ℤ, \\ 1 & if N_{i} ∈ ℤ + \frac{1}{2}. \end{matrix}

Notice that for Ψ ∈ H^{0}(𝔞̄_{0}, (W_{c,h_{0}}^{0} ⊗ M_{\frac{1}{2}} (c, h_{1})(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}),

Ψ ∈ ker ξ ⇔ ψ_{N_{i}}^{α,β}(S_{i}|c, h_{1}⟩(1)_{\frac{1}{2}}) = 0 for any α, β ∈ \{0, 1\} and i = 1, 2,
⇔ {ψ_{0}^{0,β_{i}}, ψ_{0}^{1,1−β_{i}}}A′_{i} = 0 and {ψ_{0}^{0,1−β_{i}}, ψ_{0}^{1,β_{i}}}A′′_{i} = 0 for i = 1, 2.

Therefore, from Lemma 6.9, 6.11 and Remark 6.12, we obtain Proposition 6.7.

Remark 6.13. Combining Theorem 6.6 and 6.7, we have proved Conjecture 3.6 in Case 2’, since we have

\begin{matrix}Φ_{3}(l_{h_{0}}^{0}, lh_{1}^{\frac{1}{2}}, l_{h_{∞}}^{0}) \\ = \frac{1}{2} dim H_{0}(L(E_{0}), L_{0}(c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, h_{1})(1) ⊗ L_{0}(c, h_{∞})(∞)) \end{matrix}

and Φ_{2}(l_{h_{∞}}^{0}, l_{h_{∞}}^{0}) = 2.

Remark 6.14. From the proof of Proposition 6.7, it seems that we have to determine the matrix elements of the matrix (14) to prove Conjecture 3.6 along this line. At the moment, we can do it only in the following two cases:

1. h_{0} = \frac{1}{24}c or h_{∞} = \frac{1}{24}c,

2. h_{0} = h_{∞} (See Lemma 7.4).

7. Rationality of SVOA L_{\frac{1}{2}} (c, 0)

In this section, we prove Theorems 3.2 and 3.3, which are relevant to the rationality of the SVOA L_{\frac{1}{2}} (c, 0).

It follows from Theorem 5.4 and 5.9 that the sufficiency in Theorem 3.2 (i) and Theorem 3.3 for ϵ = \frac{1}{2} is a corollary of Theorem 5.5 and 5.10. Similarly, the sufficiency in Theorem 3.2 (i) and Theorem 3.3 are immediate consequences of Proposition 6.3, that is stated only with a sketch of a proof. The necessity in Theorem 3.2 (i) follows from Theorem 3.2 (ii). So we indicate how one can prove Theorem 3.2 (ii) for both ϵ = \frac{1}{2}, 0. Since Theorem 3.3 is important for its relation with the rationality of the minimal models, we give an alternative proof of Theorem 3.3.

First we comment on the proof of Theorem 3.2 (ii) briefly. Notice that for a generic central charge c, if h ∉ Λ_{ϵ}^{c} then M_{ϵ}(c, h)(w) is irreducible. In the case of h_{1} ∉ Λ_{ϵ_{1}}^{c} and h_{2} ∉ Λ_{ϵ_{2}}^{c},

dim H_{0}(L(E), M_{ϵ_{1}}(c, h_{1})(w_{1}) ⊗ M_{ϵ_{1}}(c, h_{2})(w_{2}) ⊗ M_{ϵ_{3}}(c, h_{3})(w_{3})) ≠ 0,

for any h_{3} ∉ Λ_{ϵ_{3}}^{c} (See Remark 6.10). On the other hand, in the case of h_{1} ∈ Λ_{ϵ_{1}}^{c} and h_{2} ∉ Λ_{ϵ_{2}}^{c}, the Verma module M_{\frac{1}{2}} (c, h)(0) (ϵ = \frac{1}{2} ) or the half Verma module M̃_{0}^{τ}(c, h)(0) (ϵ = 0) has unique (up to scalar) singular vector. As in Theorem 5.5, Theorem 5.10 and Proposition 6.3, this singular vector defines lines where (r_{3}, s_{3}) such that h_{3} = h_{r_{3},s_{3}}^{ϵ_{3}}(t) should lie. Combining these equations with Theorem 5.5, Theorem 5.10 and Proposition 6.3, we obtain the result. To prove Theorem 3.3, we first show the following proposition.

Proposition 7.1. Suppose that c = c(t) for t ∈ ℝ_{>0}. If

H_{0}(L(E_{0}), L′_{0}(c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, 0)(1) ⊗ L′_{0}(c, h_{∞})(∞)) ≠ 0,

then h_{0} = h_{∞}.

Proof. Notice that M_{\frac{1}{2}} (c, 0)(1) has the singular vector G_{− \frac{1}{2}}^{(1)} |c, 0⟩(1)_{\frac{1}{2}}. We set

N ′_{\frac{1}{2}} (c, 0)(1) = U(L̂_{1: \frac{1}{2}}^{<})G_{− \frac{1}{2}}^{(1)} |c, 0⟩(1)_{\frac{1}{2}} and V_{\frac{1}{2}} (c, 0)(1) = M_{\frac{1}{2}} (c, 0)(1)/N′(c, 0)(1). To

prove Proposition 7.1, it is enough to show that

dim H_{0}(L(E_{0}), M′_{0}(c, h_{0})(0) ⊗ V_{\frac{1}{2}} (c, 0)(1) ⊗ M′_{0}(c, h_{∞})(∞)) = 2δ_{h_{0},h_{∞}}2^{δ_{h_{0}, \frac{1}{24} c}}.(29)

Indeed, if we can prove (29), then by using the exact sequences

V_{\frac{1}{2}} (c, 0)(1) → L_{\frac{1}{2}} (c, 0)(1) → 0

and

M′_{0}(c, h_{w})(w) → L′_{0}(c, h_{w})(w) → 0 (w = 0, ∞),

we obtain Proposition 7.1 by an argument similar to Lemma 4.5 in [13]. By Shapiro’s lemma and Poincaré duality, we have

\begin{matrix}dim H_{0}(L(E_{0}), M′_{0}(c, h_{0})(0) ⊗ V_{\frac{1}{2}} (c, 0)(1) ⊗ M′_{0}(c, h_{∞})(∞)) \\ = dim H^{0}(𝔞̄_{0}, (W_{c,h_{0}}^{0} ⊗ V_{\frac{1}{2}} (c, 0)(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}). \end{matrix}  (30)

Then we need to calculate the right hand side of (30). Using the exact sequence

N ′_{\frac{1}{2}} (c, 0)(1) → M_{\frac{1}{2}} (c, 0)(1) → V_{\frac{1}{2}} (c, 0)(1) → 0,

we have

H^{0}(𝔞̄_{0}, (W_{c,h_{0}}^{0} ⊗ N ′_{\frac{1}{2}} (c, 0)(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}) ←^{ξ}
H^{0}(𝔞̄_{0}, (W_{c,h_{0}}^{0} ⊗ M_{\frac{1}{2}} (c, 0)(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}) ←
H^{0}(𝔞̄_{0}, (W_{c,h_{0}}^{0} ⊗ V_{\frac{1}{2}} (c, 0)(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}) ← 0,

and then

H^{0}(𝔞̄_{0}, (W_{c,h_{0}}^{0} ⊗ V_{\frac{1}{2}} (c, 0)(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}) ≃ ker ξ.

Note that for Ψ ∈ H^{0}(𝔞̄_{0}, (W_{c,h_{0}}^{0} ⊗ M_{\frac{1}{2}} (c, 0)(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}),

Ψ ∈ ker ξ ⇔ G_{− \frac{1}{2}}^{(1)} .ψα,β_{\frac{1}{2}} = 0 for any α, β ∈ \{0, 1\},

where ψ_{j}^{α,β} is defined as (26). By Lemma 6.8, we have

(G_{− \frac{1}{2}}^{(1)} .ψα,β_{\frac{1}{2}}, G_{− \frac{1}{2}}^{(1)} .ψ1−α,1−β_{\frac{1}{2}}) = (ψ_{0}^{α,1−β}, ψ_{0}^{1−α,β}) ( \begin{matrix}K_{∞} & (−1)^{α+1}K_{0} \\ (−1)^{α}K_{0} & K_{∞} \end{matrix} ) ,

and

det ( \begin{matrix}K_{∞} & (−1)^{α+1}K_{0} \\ (−1)^{α}K_{0} & K_{∞} \end{matrix} ) = h_{0} − h_{∞}.

Therefore,

dim\{ Ψ|_{V_{0}} | Ψ ∈ ker ξ\} = \{ \begin{matrix}0 & if h_{0} ≠ h_{∞}, \\ 2 & if h_{0} = h_{∞} ≠ \frac{1}{24}c, \\ 4 & if h_{0} = h_{∞} = \frac{1}{24}c, \end{matrix}

where V_{0} is given in (27). Combining isomorphism (28), we obtain

dim ker ξ = 2δ_{h_{0},h_{∞}}2^{δ_{h_{0}, \frac{1}{24} c}},

and we have proved Proposition 7.1.

From now on, we will prove Theorem 3.3. First we remark that, in the minimal central charge case, Theorem 3.3 holds in the following strong form:

Theorem 7.2. Let ϵ = \frac{1}{2}, 0. Suppose that c satisfies the condition in Case 2. Then we have

H_{0}(L(E_{0}), L_{ϵ}(c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, 0)(1) ⊗ L_{ϵ}(c, h_{∞})(∞)) ≠ 0

if and only if h_{0} = h_{∞} and h_{0} ∈ Λ_{ϵ}^{c}.

Proof. For ϵ = \frac{1}{2} , this theorem is a corollary of Theorem 5.5. Now let us prove for ϵ = 0. To do so, it is enough to show that

\begin{matrix}H_{0}(L(E_{0}), M_{0}(c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, 0)(1) ⊗ M′_{0}(c, h_{∞})(∞)) ≠ 0 \\ ⇔ h_{0} = h_{∞} and h_{0} ∈ Λ_{0}^{c}. \end{matrix}  (31)

Indeed, by an argument similar to Lemma 4.5 in [13], we see that (31) implies

H_{0}(L(E_{0}), L_{0}(c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, 0)(1) ⊗ L_{0}(c, h_{∞})(∞)) ≠ 0.

For the proof, we use Shapiro’s lemma in the following way: Set

𝔞̄′_{0} = 𝔞_{0} ⊕ ℂℓ_{0}^{(0)} ⊕ ℂℓ_{0}^{(∞)} ⊕ ℂg_{0}^{(∞)}.

Let W′_{c,h_{0}}^{0} = ℂ1_{c,h_{0}}^{0} be the 1-dimensional representation of 𝔞̄′_{0} given by

x.1_{c,h_{0}}^{0} = 0 if x ∈ 𝔞_{0},
l_{0}^{(w′)}.1_{c,h_{0}}^{0} = δ_{w′,0}h_{0}1_{c,h_{0}}^{0} for w′ = 0, ∞,
g_{0}^{(∞)}.1_{c,h_{0}}^{0} = 0,
deg 1_{c,h_{0}}^{0} = 0

and let W_{c,h_{∞}}^{∞} be the 2-dimensional representation of 𝔞̄′_{0} defined in (20). For σ = ±, we set ˜1_{c,h_{∞}:σ}^{∞} = \frac{1}{2}\{1_{c,h_{∞}:0}^{∞} + σ1_{c,h_{∞}:1}^{∞}\}. By Shapiro’s lemma and Poincaré duality, we have

\begin{matrix}dim H_{0}(L(E_{0}), M_{0}(c, h_{0})(0) ⊗ L_{\frac{1}{2}} (c, 0)(1) ⊗ M′_{0}(c, h_{∞})(∞)) \\ = dim H^{0}(𝔞̄′_{0}, (W′_{c,h_{0}}^{0} ⊗ L_{\frac{1}{2}} (c, 0)(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}). \end{matrix}

From the following part of the BGG type resolution for L_{\frac{1}{2}} (c, 0)(1):

N_{\frac{1}{2}} (c, 0)(1) → M_{\frac{1}{2}} (c, 0)(1) → L_{\frac{1}{2}} (c, 0)(1) → 0,

we obtain

H^{0}(𝔞̄′_{0}, (W′_{c,h_{0}}^{0} ⊗ N_{\frac{1}{2}} (c, 0)(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}) ←^{ξ′}
H^{0}(𝔞̄′_{0}, (W′_{c,h_{0}}^{0} ⊗ M_{\frac{1}{2}} (c, 0)(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}) ←
H^{0}(𝔞̄′_{0}, (W′_{c,h_{0}}^{0} ⊗ L_{\frac{1}{2}} (c, 0)(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}) ← 0.

Then we have

H^{0}(𝔞̄′_{0}, (W′_{c,h_{0}}^{0} ⊗ L_{\frac{1}{2}} (c, 0)(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}) ≃ ker ξ′.

Next we determine when Φ ∈ H^{0}(𝔞̄′_{0}, (W′_{c,h_{0}}^{0} ⊗ M_{\frac{1}{2}} (c, 0)(1) ⊗ W_{c,h_{∞}}^{∞})^{∗}) vanishes on the singular vectors of M_{\frac{1}{2}} (c, 0)(1). For σ = ± we set

Φ^{σ}(v_{1}) = Φ(1_{c,h_{0}}^{0} ⊗ v_{1} ⊗ ˜1_{c,h_{∞}:σ}^{∞}) for v_{1} ∈ M_{\frac{1}{2}} (c, 0)(1)

and

φ_{j}^{σ} = Φ^{σ}|_{M_{\frac{1}{2}} (c,0)(1)_{j}} for j ∈ \frac{1}{2}ℤ_{≥0}.

Since the maximal proper submodule N_{\frac{1}{2}} (c, 0)(1) of M_{\frac{1}{2}} (c, 0)(1) is generated by the singular vectors

S_{q−1,p−1}^{(1)}|c, 0⟩(1)_{\frac{1}{2}} and G_{− \frac{1}{2}}^{(1)} |c, 0⟩(1)_{\frac{1}{2}},

we see that

Φ^{σ} ∈ ker ξ′ ⇔ a(S_{q−1,p−1}^{(1)}).φ_{\frac{1}{2} (q−1)(p−1)}^{σ} = 0 and a(G_{− \frac{1}{2}}^{(1)} ).φσ_{\frac{1}{2}} = 0.

Therefore, we have to know how the above singular vectors act on the functionals φ_{j}^{σ}. We need the following two lemmata: The first one is

Lemma 7.3. For j ∈ \frac{1}{2}ℤ_{≥0}, m ∈ ℤ_{>0} and n ∈ ℤ_{≥0},

(i) C.φ_{j}^{σ} = −cφ_{j}^{σ},
(ii) L_{−m}^{(1)}.φ_{j}^{σ} = (−1)^{m−1}(m(h_{0} − 1) + h_{1} − h_{∞} + j)φ_{j−m}^{σ},
L_{0}^{(1)}.φ_{j}^{σ} = −(h_{1} + j)φ_{j}^{σ}
(iii) G_{−n− \frac{3}{2}}^{(1)} .φ_{n+j+ \frac{3}{2}}^{σ} = (−\frac{1}{4})^{n} (2n + 1) ( \begin{matrix}2n \\ n \end{matrix} )

× \{σK_{∞}φ_{j}^{σ} − \frac{2n + 3}{3} ∑_{k=1}^{[j+ \frac{3}{2} ]} ( k^{\frac{3}{2}} ) \frac{n}{n + k} G_{k− \frac{3}{2}} .φ_{j−k+ \frac{3}{2}}^{σ} \} ,

where K_{∞} = \sqrt{−1}(h_{∞} − \frac{1}{24}c)^{\frac{1}{2}} .

To state the second lemma that is a key of our proof, we assume that c = c(t) for t ∈ ℝ_{>0} (not necessarily rational) and h_{1} = hr,s^{\frac{1}{2}}(t) for some r, s ∈ ℤ_{>0} such that r − s ∈ 2ℤ. For a singular vector S_{r,s}^{(1)}|c, h_{1}⟩(1)_{\frac{1}{2}} of level N = \frac{1}{2}rs, we define a 2 × 2-matrix M_{r,s}^{σ}(h_{0}, h_{∞}; t) by

a(S_{r,s}^{(1)}) : \{φ_{N}^{σ}, G_{− \frac{1}{2}}^{(1)} .φ_{N+ \frac{1}{2}}^{σ} \} ↦ \{φ_{0}^{σ}, G_{− \frac{1}{2}}^{(1)} .φσ_{\frac{1}{2}} \}M_{r,s}^{σ}(h_{0}, h_{∞}; t),

where a denotes the antipode of L̂_{1, \frac{1}{2}}^{<}.

Lemma 7.4. If h = h_{0} = h_{∞}, then we have

M_{r,s}^{σ}(h, h; t) = ( \begin{matrix}X_{r,s}^{(0)}Y_{r,s}^{(1)} & −2τX_{r,s}^{(0)}Y_{r,s}^{(1)} \\ \frac{1}{2}σK_{∞}^{−1}(X_{r,s}^{(1)}Y_{r,s}^{(0)} − X_{r,s}^{(0)}Y_{r,s}^{(1)}) & X_{r,s}^{(1−τ)}Y_{r,s}^{(τ)} \end{matrix} ) ( \begin{matrix}(−1)^{\frac{1}{2} (r−s)τ} & 0 \\ 0 & 1 \end{matrix} ) ,

where τ = τ_{r,s} such that r ≡ s ≡ τ_{r,s} (mod 2) and X_{r,s}^{(γ)}, Y_{r,s}^{(γ)} are given by

X_{r,s}^{(γ)} = ∏_{(k,l)∈Ẽ_{r,s}^{γ}} 2^{−3}\{(r − 1 − 2k)t^{\frac{1}{2}} − (s − 1 − 2l)t^{− \frac{1}{2}} \},

Y_{r,s}^{(γ)} = ∏_{(k,l)∈Ẽ_{r,s}^{γ}} 4 \{(h_{\frac{1}{2} (r−1−2k), \frac{1}{2} (s−1−2l)}(t) − \frac{1}{24}c(t)) − (h − \frac{1}{24}c(t))\} .

and

Ẽ_{r,s}^{γ} = \{(k, l) ∈ ℤ^{2} | \begin{matrix}0 ≤ k ≤ r − 1 ∧ 0 ≤ l ≤ s − 1 \\ r − 1 − 2k > 0 ∨ (r − 1 − 2k = 0 ∧ s − 1 − 2l ≥ 0) \\ k + l ≡ γ (mod 2) \end{matrix} \} .

Proof. To show this lemma, we notice the following fact:

Fact 7.1. For any x ∈ U(L̂_{1: \frac{1}{2}}^{<}), x.φ_{n}^{σ} can be written as follows:

x.φ_{n}^{σ} = ∑_{y,α,m} c_{(y,α,m),(x,n)}^{σ} \{y (G_{− \frac{1}{2}} )^{α}\}.φ_{m}^{σ},

where y ∈ U(L̂_{1: \frac{1}{2}}^{≥}), α ∈ \{0, 1\}, m ∈ \frac{1}{2}ℤ_{≥0} and c_{(y,α,m),(x,n)}^{σ} ∈ ℂ.

Using this fact and the embedding diagrams of Verma modules, we can prove this lemma by the same argument as in the proof of Lemma 5.1.

Remark 7.5. For the number c = c(\frac{p}{q}) which satisfies the condition in Case 2’, X_{q−1,p−1}^{(0)}|_{t= \frac{p}{q}} ≠ 0 and Y_{q−1,p−1}^{(1)}|_{t= \frac{p}{q}} = 0 if and only if h ∈ Λ_{0}^{c}.

We return to the proof of Theorem 7.2. Combining Proposition 7.1, Lemma 7.4 and Remark 7.5, we obtain (31). Thus we complete the proof of Theorem 7.2.

At the end of this paper, we comment on the following proposition related with the rationality of the minimal models for the N = 1 super Virasoro algebras.

Proposition 7.6. Let ϵ = \frac{1}{2}, 0. Suppose that c = c(\frac{p}{q}) for some p, q ∈ ℤ_{>0} such that p, q ≥ 2, p − q ∈ 2ℤ and (\frac{1}{2}(p − q), q) = 1. For h, h′ ∈ Λ_{ϵ}^{c} we have

Ext_{(V ir_{ϵ},V ir_{ϵ}^{0})}^{j}(L_{ϵ}(c, h), L_{ϵ}(c, h′)) = \{ \begin{matrix}ℂ & if j = 0 and h = h′, \\ ℂ^{2} & if j ∈ 2ℤ_{>0} and h = h′, \\ 0 & otherwise . \end{matrix}

In particular, Ext_{(V ir_{ϵ},V ir_{ϵ}^{0})}^{1}(L_{ϵ}(c, h), L_{ϵ}(c, h′)) = 0 for any h, h′ ∈ Λ_{ϵ}^{c}.

Proof. Combining Theorem 4.5 in [4] and Theorem 2 in [20], we obtain this theorem by a standard argument, since we already have BGG type resolutions for the minimal series representations over the N = 1 super Virasoro algebras.

Finally, let us remark on a relation between our fusion algebras and the super vertex operator algebra associated with L_{\frac{1}{2}} (c(\frac{p}{q}), 0).

Remark 7.7. 1. By [16], L_{\frac{1}{2}} (c(\frac{p}{q}), 0) becomes a vertex operator superalgebra and the set of the irreducible representations of the SVOA L_{\frac{1}{2}} (c(\frac{p}{q}), 0) is given by

\{L_{\frac{1}{2}} (c(\frac{p}{q}), h) |h ∈ Λc( \frac{p}{q})_{\frac{1}{2}} \} .

The fusion rule of this SVOA was computed in [1], and it gives the same result as in Theorem 3.5.

2. The rationality of the SVOA L_{\frac{1}{2}} (c(\frac{p}{q}), 0) was conjectured in [16] and proved in [1]. From Theorem 3.3 and Proposition 7.6 for ϵ = \frac{1}{2} , we obtain another proof of the rationality of the SVOA L_{\frac{1}{2}} (c(\frac{p}{q}), 0).

References

[1] Adamović D., Rationality of Neveu-Schwarz vertex operator superalgebras., Int. Math. Res. Not. 17 (1997), 865–874.

[2] Astashkevich A., On the structure of Verma modules over Virasoro and Neveu-Schwarz algebras, Comm. Math. Phys. 186 (1997), 531–562.

[3] Awata H., and Y. Yamada, Fusion Rules for the fractional level { \atop 𝔰l(2)} algebra, Mod. Phys. Lett. A 7 (1992), 1185–1195.

[4] Deodhar V. V., D. Gabber, and V. Kac, Structure of some categories of representations of infinite dimensional Lie algebras, Adv. Math. 45 (1982), 92–116.

[5] Eholzer W., and R. Hübel, Fusion Algebras of Fermionic Rational Conformal Field Theories via a Generalized Verlinde Formula, Nucl. Phys. B 414 (1994), 348-378.

[6] Feigin B. L., and D. B. Fuchs, Representations of the Virasoro algebra, Gordon and Breach Science Publ. New York, Adv. Stud. Contemp. Math. 7 (1990), 465–554.

[7] —, Cohomology of some nilpotent subalgebras of the Virasoro and Kac-Moody Lie algebras, J. Geom. Phys. 5 (1988), 209–235.

[8] Feigin, B. L., and E. Frenkel, Quantization of the Drinfel’d-Sokolov reduction, Phys. Lett. B 246 (1990), 75–81.

[9] Feigin B., and F. Malikov, Fusion algebra at a rational level and cohomology of nilpotent subalgebras of ŝl_{2}, Lett. Math. Phys. 31 (1994), 315–325.

[10] —, Modular functor and representation theory of ŝl_{2} at a rational level, Operads: Proceedings of Renaissance Conference, 357–405, Contemp. Math. 202, Amer. Math. Soc. Providence, RI, (1997).

[11] Frenkel E., V. Kac, and M. Wakimoto, Characters and fusion rules for W -algebras via quantized Drinfel’d-Sokolov reduction, Comm. Math. Phys. 147 (1992), 295–328.

[12] Iohara K. and Y. Koga, Résolutions de type Bernstein-Gel’fand-Gel’fand pour les super-algèbres de Virasoro N=1, C. R. Acad. Sci. Paris, t. 328 Série I (1999), 381–384.

[13] —, Fusion algebras for N=1 superconformal field theories through coinvariants I:ôsp(1|2)-symmetry, J. Reine Angew. Math., to appear.

[14] Kimura K., Hamiltonian reduction of super OSP (1, 2) and SL(2, 1) Kac-Moody algebras, Internat. J. Mod. Phys. A, 7, Suppl. 1B (1992), 533–544.

[15] Kac V. G., and M. Wakimoto, Modular invariant representations of infinite dimensional Lie algebras and superalgebras, Proc. Natl. Acad. Soc., 35 (1988), 4956–4960.

[16] Kac V. G., and W. Wang, Vertex operator superalgebras and their representations, Mathematical aspects of conformal and topological field theories and quantum groups, 161–191, Contemp. Math. 175, Amer. Math. Soc. Providence, RI, (1994).

[17] Matsuo Y., and S. Yahikozawa, Superconformal field theory with modular invariance on a torus, Phys. Lett. B 178 (1986), 211-218.

[18] Meurman A., and A. Rocha-Caridi, Highest weight representation of the Neveu-Schwarz and Ramond algebras, Comm. Math. Phys. 107 (1986), 263–294.

[19] Minces P., M. A. Namazie, and C. Núñez, Fusion rules in N = 1 superconformal minimal models, Phys. Lett. B 422 (1998), 117-125.

[20] Rocha-Caridi A., and N. R. Wallach, Projective modules over graded Lie algebras. I, Math. Z. 180 (1982), 151–177.

[21] Verlinde E., Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. B 300 (1988), 360-376.

Kenji Iohara Department of Mathematics Faculty of Science Kobe University Kobe 657-8501 Japan Yoshiyuki Koga Department of Mathematics Faculty of Science Osaka University Osaka 560-0043 Japan

Received April 27, 1999 and in final form October 24, 2000