Representation Theory of Affine Superalgebras

at the Critical Level

Dedicated to the memory of my father Shoji Wakimoto

Minoru Wakimoto

1991 Mathematics Subject Classification: Primary 17B65; Secondary 17B67, 81R10. Keywords and Phrases: Superalgebra, character, atypical representation, free field construction, mock theta function, modular transformation

0. Introduction.

As is known well, the representation theory of affine Lie algebras has a number of important connection with various area of mathematics and physics, while the representation of superalgebras still remains mysterious, and only a few has been known about it because of serious technical and intrinsic difficulties arising in its analysis.

The representation theory of superalgebras, however, seems quite interesting and fascinating, since the associated Macdonald identities are of different kinds from those of the usual affine Lie algebras, and also the Ramanujan’s famous mock theta functions take place closely related to the denominator identities for certain affine superalgebras.

The effective theory for superalgebras are not well established, and some of the results exposed in this note are obtained with the help of a computer and Reduce 3.6.

1. Characters of affine superalgebras.

Our interest in this note is a finite-dimensional or affine simple Lie superalgebra, with a non-degenerate even super-invariant super-symmetric bilinear form ( | ). The finite-dimensional ones other than Lie algebras are listed in the following table:

[figure: A(m, n); (m, n ≥ 0); (m + n ≥ 1); :; ⋯; ×; ⋯; α_{1}; α_{m}; α_{m+1}; α_{m+2}; α_{m+n+1}]

[figure: B(m, n); (m, n ≥ 1); :; ⋯; ×; ⋯; α_{1}; α_{n−1}; α_{n}; α_{n+1}; α_{m+n−1}; α_{m+n}]

[figure: B(0, n); (n ≥ 1); :; ⋯; α_{1}; α_{2}; α_{n−2}; α_{n−1}; α_{n}]

[figure: C(n); (n ≥ 3); :; ×; ⋯; α_{1}; α_{2}; α_{n−2}; α_{n−1}; α_{n}]

[figure: D(m, n); (m ≥ 2); (n ≥ 1); :; ⋯; α_{n}; ×; ⋯; α_{1}; α_{n−1}; α_{n}; α_{n+1}; α_{m+n−2}; α_{m+n−1}; α_{m+n}]

[figure: D(2, 1; a); (−1, 0 ≠ a ∈ ℂ); :; ×; α_{1}; α_{2}; α_{3}; −1; −a]

[figure: F (4); :; ×; α_{1}; α_{2}; α_{3}; α_{4}]

[figure: G(3); :; ×; α_{1}; α_{2}; α_{3}]

We note that, for a superalgebra with isotropic odd roots, its Dynkin diagram is not determined uniquely, and the diagrams in the above list are standard ones. But non-standard ones are never less important, and a suitable choice of Dynkin diagrams has sometimes a crucial importance in the representation theory. For example, Dynkin diagrams in Fig 1.1 ∼ 1.3 are useful diagrams of A(1, 0) = 𝔰𝔩(2, 1), A(1, 1) = 𝔰𝔩(2, 2)/𝔷 and A(2, 2) = 𝔰𝔩(3, 3)/𝔷 respectively (here and henceforth 𝔷 stands for the center of the superalgebra in question) with their affinizations shown in Fig 1.1′ ∼ 1.3′ :

[figure: ×; ×; α_{1}; α_{2}; Fig.1.1; ×; ×; ×; α_{1}; α_{2}; α_{3}; Fig.1.2; ×; ×; ×; ×; ×; α_{1}; α_{2}; α_{3}; α_{4}; α_{5}; Fig.1.3]

[figure: ×; ×; α_{1}; α_{2}; α_{0}; Fig.1.1′; ×; ×; ×; α_{1}; α_{2}; α_{3}; ×; α_{0}; Fig.1.2′; ×; ×; ×; ×; ×; α_{1}; α_{2}; α_{3}; α_{4}; α_{5}; ×; α_{0}; Fig.1.3′]

Let 𝔤 be a finite-dimensional or affine Lie superalgebra and 𝔥 its Cartan subalgebra. The inner product ( | ) is normalized so that the dual Coxeter number h^{∨} is a non-negative rational number and the square length of the longest roots is equal to 2 or −2. For usual notations and terminologies of superalgebras, we refer to [5] and [9]. In particular the concept of integrable weight is never obvious for superalgebras and given in Chapter 6 of [9].

The character of a suitable irreducible integrable highest weight 𝔤-module L(Λ) is given in [9]:

chL(Λ) = \frac{c_{Λ}}{e^{ρ}R} ∑_{w∈W^{♯}} ε(w)w (\frac{e^{Λ+ρ}}{∏_{i=1}^{k}(1 + e^{−β_{i}})})  (1)

with a rational number c_{Λ}, where {β_{1}, ⋯, β_{k}} is a maximal set of mutually orthogonal isotropic positive odd roots satisfying (Λ + ρ|β_{i}) = 0 for all i, and R is the denominator:

R := \frac{∏_{α∈Δ_{0+}}(1 − e^{−α})^{mult(α)}}{∏_{α∈Δ_{1+}}(1 + e^{−α})^{mult(α)}}  (2)

In the above, W^{♯} is not always the full Weyl group W but its subgroup (cf. [9]), and Δ_{+} stands, as usual, for the set of all positive roots, and the suffix “0” or “1” implies the set of “even roots” or “odd roots” respectively.

The highest weight Λ is called typical when k = 0, and k-atypical or simply atypical when k > 0. The number k is the atypicality of a highest weight Λ.

A particular feature and difficulty of the representation theory of superalgebras is that the formula (1) is not necessarily valid for all representations, but fails to hold in some cases. The formula (1) is true in case when (i) Λ is typical ([6]) or (ii) β_{1}, ⋯, β_{k} are chosen from simple roots with a suitable choice of the Dynkin diagram ([9]).

The trivial representation L(0) for an affine superalgebra with non-zero dual Coxeter number satisfies the condition (ii) above, and the formula (1) applied to this, called the denominator identity, provides a Lie superalgebraic interpretation to some of number theoretic formulas as is discussed in [9].

For affine superalgebras with h^{∨} = 0, whose complete list is Â(n, n)(=the affinization of 𝔰𝔩(n + 1|n + 1)/𝔷) and D̂(n + 1, n)(= 𝔬̂𝔰𝔭(2n + 2, 2n)) and D̂(2, 1; a), the trivial representation is a representation of critical level. The simplest among them is Â(1, 1) with the Dynkin diagram Fig 1.2′. In this case, we claim the following denominator identity, which is obtained from the analysis of the super Boson-Fermion correspondence ([10]) :

R =  ∑_{w∈⟨r_{α_{1}+α_{2}} , t_{α_{1}+α_{2}} ⟩} ε(w)w (\frac{1}{(1+e^{−α_{0}}) ∏_{n=1}^{∞}(1+e^{α_{2}−nδ})(1+e^{−α_{2}−(n−1)δ})}) , (3)

where r_{α} is the reflection with respect to α, and t_{α} is the linear transformation on 𝔥 defined by

t_{α}(λ) := λ + (λ|δ)α − (\frac{|α|^{2}}{2} (λ|δ) + (λ|α)) δ.

We note the following. For Â(1, 1) with the Dynkin diagram Fig 1.2′, one can choose

{α_{0} + nδ, α_{2} + nδ}_{n≥0} ∪ {−α_{0} + nδ, −α_{2} + nδ}_{n>0}  (4)

as a maximal system of mutually orthogonal isotropic positive odd roots which are orthogonal to Λ + ρ since Λ = ρ = 0, where δ := ∑_{i=0}^{3} α_{i} is the canonical imaginary root as usual. So Λ = 0 is “infinitely atypical” in this case. A very remarkable fact is that not all of β_{i}’s in this maximal system (4) does take a part in the denominator identity (3).

The formula becomes more complicated in higher rank cases. For example, the denominator identity of Â(2, 2) is given as follows with respect to the Dynkin diagram Fig 1.3′ :

R · ∏_{n=1}^{∞} \frac{(1 − e^{α_{1}+α_{3}+α_{5}−2nδ})(1 − e^{−α_{1}−α_{3}−α_{5}−2(n−1)δ})}{(1 − e^{−nδ})(1 − e^{−(2n−1)δ})}

= ∑_{w∈W^{♯}} ε(w)w (\frac{1}{(1 + e^{−α_{0}}) ∏_{i=2,4} ∏_{n=1}^{∞}(1 + e^{α_{i}−nδ})(1 + e^{−α_{i}−(n−1)δ})}) (5)

where W^{♯} := ⟨r_{α_{1}+α_{2}}, r_{α_{3}+α_{4}}, t_{α_{1}+α_{2}}, t_{α_{3}+α_{4}}⟩.

We now look at other simplest examples of representations at the critical level, namely integrable 𝔰̂𝔩(2, 1)-modules of level −1. For a linear form Λ := kΛ_{0} − (k + 1)Λ_{1} (k ∈ ℤ) which is atypical with respect to α_{2}, we claim the following:

chL(Λ) = \frac{∏_{n=1}^{∞}(1 − e^{−nδ})}{e^{ρ}R} ×

\begin{cases}∑_{w∈⟨r_{α_{0}} ⟩} ∑_{j=0}^{∞} ε(w)wt_{jα_{0}} (_{\frac{e^{Λ+ρ}}{∏_{n=1}^{∞}(1+e^{α_{2}−nδ})(1+e^{−α_{2}−(n−1)δ})}} ) & (if k ≥ 0) \\ ∑_{w∈⟨r_{α_{1}+α_{2}} ⟩} ∑_{j=0}^{∞} ε(w)wt_{j(α_{1}+α_{2})} (_{\frac{e^{Λ+ρ}}{∏_{n=1}^{∞}(1+e^{α_{2}−nδ})(1+e^{−α_{2}−(n−1)δ})}} ) & (if k < 0). \end{cases} (6)

Note that the sum in the right-hand side is not taken over a subgroup of the affine Weyl group. From this formula, chL(−ρ) is obtained as follows:

chL(−ρ)

= \frac{∏_{n=1}^{∞}(1 − e^{−nδ})}{e^{ρ}R} \{ ∑_{k=1,2} \frac{∑_{j=0}^{∞} e^{jα_{k}− \frac{j(j+1)}{2} δ}}{∏_{n=1}^{∞}(1+e^{α_{k}−nδ})(1+e^{−α_{k}−(n−1)δ})} − ∏_{n=1}^{∞}(1 − e^{−nδ})\}

= \frac{∏_{n=1}^{∞}(1 − e^{−nδ})}{e^{ρ}R}×

\{\frac{∑_{j≥0} e^{jα_{1}− \frac{j(j+1)}{2} δ}}{∏_{n=1}^{∞}(1+e^{α_{1}−nδ})(1+e^{−α_{1}−(n−1)δ})} − \frac{∑_{j<0} e^{jα_{2}− \frac{j(j+1)}{2} δ}}{∏_{n=1}^{∞}(1+e^{α_{2}−nδ})(1+e^{−α_{2}−(n−1)δ})}\} .

Documenta Mathematica · Extra Volume ICM 1998 · II · 605–614

The second equality in the above is due to the Jacobi triple product identity

∏_{n=1}^{∞}(1 − e^{−nδ}) = \frac{∑_{j∈ℤ} e^{jα}e^{− \frac{j(j+1)}{2} δ}}{∏_{n=1}^{∞}(1 + e^{α−nδ})(1 + e^{−α−(n−1)δ})}.  (7)

2. Free field construction of atypical 𝔰̂𝔩(2, 1)-modules at the critical level.

It is known in [2], [3] and [7] that, in case of affine algebras, irreducible highest weight representations and their characters display a remarkable behavior at the critical level. This may also be expected for superalgebras. To get an information on the structure of these representations, we give an explicit construction of atypical highest weight modules at the critical level for the simplest affine superalgebra 𝔰̂𝔩(2, 1).

Let {h_{i}, e_{i}, f_{i}}_{i=1,2} be a system of Chevalley generators of 𝔰𝔩(2, 1) with respect to its Dynkin diagram Fig 1.1. Then these elements together with e_{12} := −[e_{1}, e_{2}] and f_{12} := [f_{1}, f_{2}], satisfying [e_{12}, f_{12}] = h_{1}+h_{2}, form a basis of 𝔰𝔩(2, 1). Choosing the super-invariant super-symmetric bilinear form ( | ) such that (e_{i}|f_{i}) = 1 (i = 1, 2), we consider its affinization

𝔰̂𝔩(2, 1) := 𝔰𝔩(2, 1) ⊗ ℂ[t, t^{−1}] ⊕ ℂ · K ⊕ ℂ · t\frac{∂}{∂t}

with bracket

[X ⊗ t^{m}, Y ⊗ t^{n}] := [X, Y] ⊗ t^{m+n} + m(X|Y)Kδ_{m+n,0},

which is written as follows

X(z)Y (w) = \frac{[X, Y](w)}{z − w} + \frac{(X|Y)K}{(z − w)^{2}} ,

in terms of operator products of fields X(z) := ∑_{n∈ℤ} X ⊗ t^{n} · z^{−n−1}.

Let a_{j}, a_{j}^{∗}, b_{j}, ψ_{j}, ψ_{j}^{∗} (j ∈ ℤ) be linear operators on a linear space
V := ℂ[x_{n}; n ∈ ℤ] ⊗ ℂ[y_{n}; n ∈ ℤ_{>0}] ⊗ ∧[ξ_{n}; n ∈ ℤ],

defined by

a_{j} := \begin{cases}\frac{∂}{∂x_{j}} & if j ≥ 0 \\ x_{j} & if j < 0, \end{cases}  a_{j}^{∗} := \begin{cases}−x_{j} & if j ≥ 0 \\ \frac{∂}{∂x_{j}} & if j < 0, \end{cases}

b_{j} := \begin{cases}\frac{∂}{∂y_{j}} & if j > 0 \\ −jy_{−j} & if j < 0, \end{cases}

ψ_{j} := \begin{cases}\frac{∂}{∂ξ_{j}} & if j ≥ 0 \\ ξ_{j}∧ & if j < 0, \end{cases}  ψ_{j}^{∗} := \begin{cases}ξ_{j}∧ & if j ≥ 0 \\ \frac{∂}{∂ξ_{j}} & if j < 0, \end{cases}

b_{0} being the scalar operator with a complex number b_{0}. As usual, we introduce the following fields :

a(z) := ∑_{j∈ℤ} a_{j}z^{−j−1},  a^{∗}(z) := ∑_{j∈ℤ} a_{j}^{∗}z^{j},  b(z) := ∑_{j∈ℤ} b_{j}z^{−j−1},
ψ(z) := ∑_{j∈ℤ} ψ_{j}z^{−j−1},  ψ^{∗}(z) := ∑_{j∈ℤ} ψ_{j}^{∗}z^{j}.

Proposition 2.1. The space V is an 𝔰̂𝔩(2, 1)-module by the following action:

h_{1}(z) := − : a(z)a^{∗}(z) : + : ψ(z)ψ^{∗}(z) :
h_{2}(z) := − : a(z)a^{∗}(z) : −b(z)
e_{1}(z) := a(z)ψ^{∗}(z)
e_{2}(z) := −ψ(z)
e_{12}(z) := a(z)
f_{1}(z) := −a^{∗}(z)ψ(z)
f_{2}(z) := ∂ψ^{∗}(z)+ : a(z)a^{∗}(z)ψ^{∗}(z) : +b(z)ψ^{∗}(z)
f_{12}(z) := −∂a^{∗}(z)− : a(z)a^{∗}(z)a^{∗}(z) : + : a^{∗}(z)ψ(z)ψ^{∗}(z) : −a^{∗}(z)b(z).

And the constant function 1 is a highest weight vector in V of weight (b_{0} − 1)Λ_{0} − b_{0}Λ_{2}.

In this case, the weights of variables are given by

wt(x_{j}) = \begin{cases}−jδ − (α_{1} + α_{2}) & if j ≥ 0 \\ jδ + (α_{1} + α_{2}) & if j < 0, \end{cases}
wt(y_{j}) = −jδ,
wt(ξ_{j}) = \begin{cases}−jδ − α_{2} & if j ≥ 0 \\ jδ + α_{2} & if j < 0, \end{cases}

and so, by counting the character of V, one obtains the following:

Corollary 2.1. Let 𝔤 = 𝔰̂𝔩(2, 1) with Dynkin diagram Fig 1.1′, and Λ ∈ 𝔥^{∗} be a linear form of level −1, atypical with respect to α_{i} (i = 1 or 2). Then

chL(Λ) ≤ \frac{e^{Λ}}{R} ∏_{n=1}^{∞} \frac{(1 − e^{−nδ})}{(1 + e^{−nδ+α_{i}})(1 + e^{−(n−1)δ−α_{i}})}

≤ \frac{e^{Λ}}{R} ∏_{n=1}^{∞}(1 − e^{−nδ})^{2}.

The second inequality in the above is due to (7).

Documenta Mathematica · Extra Volume ICM 1998 · II · 605–614

3. Mock theta identities and the associated modular functions.

The formula (1) applied to the trivial representation gives rise to some kinds of identities of Lambert series. The simplest and most remarkable ones among them are mock theta identities obtained from the denominator identities of affine superalgebras 𝔰̂𝔩(2, 1) and B̂(1, 1) (cf. [9]). The mock theta function associated to 𝔰̂𝔩(2, 1), already appearing in the classical book [12], has a crucial importance in conformal field theory since it gives rise to the formula of the modular transformation of the characters of the minimal series representations of N=2 superconformal algebras (cf. [9]). In this section, we give an exposition of the modular transformation of another simplest mock theta functions associated to B̂(1, 1).

First we look at the formula (3). Putting u := e^{−α_{1}}, w := e^{−α_{2}}, v := e^{−α_{3}} and q := e^{−δ}, this is written as follows:

∏_{n=1}^{∞} \frac{(1 − q^{n})^{2}(1 − uwq^{n−1})(1 − (uw)^{−1}q^{n})(1 − vwq^{n−1})(1 − (vw)^{−1}q^{n})}{(1 + uq^{n−1})(1 + u^{−1}q^{n})(1 + vq^{n−1})(1 + v^{−1}q^{n})}

× ∏_{n=1}^{∞} \frac{1}{(1 + wq^{n−1})(1 + w^{−1}q^{n})(1 + uvwq^{n−1})(1 + (uvw)^{−1}q^{n})}

=  \frac{1}{∏_{n=1}^{∞}(1 + wq^{n−1})(1 + w^{−1}q^{n})} · ∑_{k∈ℤ} \frac{w^{−k}q^{k(k+1)/2}}{1 + (uvw)^{−1}q^{k+1}}

− \frac{1}{∏_{n=1}^{∞}(1 + uq^{n−1})(1 + u^{−1}q^{n})} · ∑_{k∈ℤ} \frac{u^{k+1}q^{k(k+1)/2}}{1 + v^{−1}q^{k+1}} .  (8)

Letting w=u, we obtain the following formula, which coincides with the mock theta identity associated to B̂(1, 1) (cf. [1] and [4]):

∏_{n=1}^{∞} \frac{(1 − q^{n})^{2}(1 − u^{2}q^{n−1})(1 − u^{−2}q^{n})(1 − uvq^{n−1})(1 − u^{−1}v^{−1}q^{n})}{(1+uq^{n−1})(1+u^{−1}q^{n})(1+vq^{n−1})(1+v^{−1}q^{n})(1+u^{2}vq^{n−1})(1+u^{−2}v^{−1}q^{n})}

= ∑_{k∈ℤ} \frac{u^{−k}q^{k(k+1)/2}}{1 + u^{−2}v^{−1}q^{k+1}} − ∑_{k∈ℤ} \frac{u^{k+1}q^{k(k+1)/2}}{1 + v^{−1}q^{k+1}}  (9)

= { ∑_{j,k≥0} − ∑_{j,k<0} }(−1)^{j} (u^{−2j−k}v^{−j} − u^{1+k}v^{−j}) q^{(k+1)(k+2j)/2}

= \{ ∑_{\begin{matrix}m,n≥0 \\ s.t. m≡n mod 2 \end{matrix}} − ∑_{\begin{matrix}m,n<0 \\ s.t. m≡n mod 2 \end{matrix}} \}(−1)^{\frac{m−n}{2}} v^{\frac{m}{2}} (u^{−2}v^{−1})^{\frac{n}{2}} q^{\frac{(m+1)n}{2}} .  (10)

We consider the theta function

f(τ, z) := e^{\frac{πiτ}{4}} e^{−πiz} ∏_{n=1}^{∞}(1 − q^{n})(1 − e^{2πiz}q^{n−1})(1 − e^{−2πiz}q^{n}),

Documenta Mathematica · Extra Volume ICM 1998 · II · 605–614

defined for τ ∈ ℂ_{+} := {τ ∈ ℂ; Imτ > 0} and z ∈ ℂ, where q = e^{2πiτ} as usual. This function satisfies the modular transformation:

f (−\frac{1}{τ} , \frac{z}{τ} ) = −i(−iτ)^{\frac{1}{2}} e^{\frac{πiz^{2}}{τ}} f(τ, z).  (11)

We put

F (τ, z_{1}, z_{2}) := \frac{η(τ)^{3}f(τ, z_{1} + z_{2})f(τ, \frac{z_{1}−z_{2}}{2} )}{f(τ, z_{1})f(τ, z_{2})f(τ, \frac{z_{1}+z_{2}}{2} )}.  (12)

Then by (11), we have

F (−\frac{1}{τ} , \frac{z_{1}}{τ} , \frac{z_{2}}{τ} ) = τe^{\frac{πiz_{1}z_{2}}{τ}} F (τ, z_{1}, z_{2}).  (13)

We now fix a positive even integer M, and consider the set Ω of all equivalence classes in ℤ × ℤ with respect to the equivalence relation

(j, k) ∼ (j′, k′) ⟺ \begin{cases}j − k, j′ − k′ & ∈ Mℤ, \\ (j + k) − (j′ + k′) & ∈ 2Mℤ. \end{cases}

For (j, k) ∈ Ω, we put

G_{j,k}(τ, z_{1}, z_{2}) := e^{\frac{πi}{Mτ} \{(z_{1}+jτ)(z_{2}+kτ)−z_{1}z_{2}\}}F (Mτ, z_{1} + 1 + jτ, z_{2} + kτ),
H_{j,k}(τ, z_{1}, z_{2}) := G_{j,k}(τ, z_{1} − 1, z_{2}),

whose Lambert series expressions and power series expansions are easily calculated from (9) and (10).

We note the following important lemma which is deduced from (13) and the power series expansion (10): Lemma 3.1. The function F satisfies the following transformation property:

F (−\frac{M}{τ} , \frac{z_{1}}{τ} , \frac{z_{2}}{τ} ) = \frac{τ}{M} e^{\frac{πiz_{1}z_{2}}{τM}}

× ∑_{\begin{matrix}a,b∈ℤ/Mℤ \\ s.t. a≡b+1 mod 2 \end{matrix}}(−1)^{a}\{e^{\frac{πiτab}{M}} e^{\frac{πi(az_{1}+bz_{2})}{M}} F (Mτ, z_{1} + 1 + bτ, z_{2} + aτ)

+ e^{\frac{πiτa(b+M)}{M}} e^{\frac{πi(az_{1}+(b+M)z_{2})}{M}} F (Mτ, z_{1} + 1 + (b + M)τ, z_{2} + aτ)\}.

By this lemma, we obtain the modular transformation of G_{j,k} and H_{j,k} (j, k ∈ Ω) as follows: Theorem 3.1.

G_{j,k} (−\frac{1}{τ} , \frac{z_{1}}{τ} , \frac{z_{2}}{τ} ) = (−1)^{k} \frac{τ}{M} e^{\frac{πiz_{1}z_{2}}{Mτ}} ×

∑_{\begin{matrix}(a,b)∈Ω \\ s.t. a≡b mod 2 \end{matrix}} e^{− \frac{πi(jb+ka)}{M}} G_{a,b}(τ, z_{1}, z_{2})  if j ≡ k mod 2

∑_{\begin{matrix}(a,b)∈Ω \\ s.t. a≡b mod 2 \end{matrix}} e^{− \frac{πi((j−1)b+ka)}{M}} H_{a,b}(τ, z_{1}, z_{2}) if j ≡ k + 1 mod 2,

H_{j,k} (−\frac{1}{τ} , \frac{z_{1}}{τ} , \frac{z_{2}}{τ} ) = (−1)^{k} \frac{τ}{M} e^{\frac{πiz_{1}z_{2}}{Mτ}} ×

∑_{\begin{matrix}(a,b)∈Ω \\ s.t. a≡b+1 mod 2 \end{matrix}} e^{− \frac{πi(jb+k(a+1))}{M}} G_{a,b}(τ, z_{1}, z_{2})  if j ≡ k mod 2

∑_{\begin{matrix}(a,b)∈Ω \\ s.t. a≡b+1 mod 2 \end{matrix}} e^{− \frac{πi((j−1)b+k(a+1))}{M}} H_{a,b}(τ, z_{1}, z_{2}) if j ≡ k + 1 mod 2.

The transformation under τ ⟶ τ + 1 is easily obtained: Theorem 3.2.

G_{j,k}(τ + 1, z_{1}, z_{2}) = (−1)^{k}e^{− \frac{πiM}{4}} × \{ \begin{matrix}e^{\frac{πijk}{M}} G_{j,k}(τ, z_{1}, z_{2}) \\ if j ≡ k mod 2 \\ e^{\frac{πi(j+1)k}{M}} H_{j,k}(τ, z_{1}, z_{2}) \\ if j ≡ k + 1 mod 2, \end{matrix}

H_{j,k}(τ + 1, z_{1}, z_{2}) = (−1)^{k}e^{− \frac{πiM}{4}} × \{ \begin{matrix}e^{\frac{πijk}{M}} H_{j,k}(τ, z_{1}, z_{2}) \\ if j ≡ k mod 2 \\ e^{\frac{πi(j−1)k}{M}} G_{j,k}(τ, z_{1}, z_{2}) \\ if j ≡ k + 1 mod 2. \end{matrix}

Furthermore, an interesting family of modular functions is obtained by putting

g_{j,k}(τ, z) := G_{j,k}(τ, z, −z)  and  h_{j,k}(τ, z) := H_{j,k}(τ, z, −z).

Since g_{j,k} = −g_{−k,−j} and h_{j,k} = −e^{− \frac{πi}{M} (j+k)}h_{−k,−j}, the explicit matrices of modular transformation of these functions are written in terms of the trigonometric function “sin”. They may look similar to at a glance but turn to be quite different from those of the characters of the N=2 superconformal algebra.

It is known in [9] that a specialization of the denominator identity of 𝔰̂𝔩(2, 1) gives the formula

∏_{n=1}^{∞} \frac{(1 − q^{n})^{2}}{(1 + zq^{n−1})(1 + z^{−1}q^{n})} = ∑_{n∈ℤ}(−1)^{n} \frac{q^{n(n+1)/2}}{1 + zq^{n}} ,  (14)

which appeared in [11] and was rediscovered by [8] in connection with Hecke indefinite modular forms. A similar identity

∏_{n=1}^{∞} \frac{(1 − q^{\frac{n}{2}} )(1 − q^{2n})(1 − zq^{2n−1})(1 − z^{−1}q^{2n−1})}{(1 + zq^{n− \frac{1}{2}} )(1 + z^{−1}q^{n− \frac{1}{2}} )} = ∑_{n∈ℤ}(−1)^{n} \frac{q^{n(n+1)/4}}{1 + zq^{n+ \frac{1}{2}}} (15)

is deduced from (10) by letting u = q^{\frac{1}{4}} and v = z^{−1}q^{\frac{1}{4}} .

Documenta Mathematica · Extra Volume ICM 1998 · II · 605–614

References

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[9] V. G. Kac and M. Wakimoto : Integrable highest weight modules over affine superalgebras and number theory, in “Lie Theory and Geometry ∼ in honor of Bertram Kostant” ed. by J. L. Brylinski, R. Brylinski, V. G. Guillemin and V. Kac, Progress in Math. Phys. Vol. 123, Birkhäuser, 1994, 415-456.

[10] V. G. Kac and M. Wakimoto : in preparation.

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Minoru Wakimoto

Graduate School of Mathematics Kyushu University Fukuoka, 812-8581, Japan