Semi-infinite Weil complex and superconformal algebras II.

Elena Poletaeva

Contents.

0. Introduction.

1. Relative semi-infinite Weil complex.

2. Relative semi-infinite Weil complex of 𝔰𝔩(2) ⊗ ℂ[t, t^{−1}].

2.1. Relative semi-infinite cohomology. 2.2 An action of the Witt algebra. 2.3 An 𝔰𝔩(2)-module structure. 2.4. Another 𝔰𝔩(2)-module structure.

3. The superconformal algebra S(2, 0).

4. An action of Ŝ(2, 0) on the relative semi-infinite cohomology of 𝔰𝔩(2) ⊗ ℂ[t, t^{−1}].

5. The case of the loop algebra of a Lie algebra with a non-degenerate invariant symmetric form.

0. Introduction.

This work is a continuation of [P2]. Recall that in [P2] we considered the semi-infinite Weil complex W^{\frac{∞}{2} +∗}(𝔤) of a graded Lie algebra 𝔤. We proved that in the case when 𝔤 is the loop algebra of a complex finite-dimensional Lie algebra, W^{\frac{∞}{2} +∗}(𝔤) is a module over the N = 2 superconformal algebra. The N = 2 superconformal algebra also acts on the semi-infinite cohomology of 𝔤 with coefficients in its “adjoint semi-infinite symmetric powers” H^{\frac{∞}{2} +∗}(𝔤, S^{\frac{∞}{2} +∗}(𝔤)).

In this work we show that there is a superconformal algebra from the classification given in [K-VDL], which contains the N = 2 superconformal algebra as a subalgebra, and acts on the relative semi-infinite Weil complex and relative semi-infinite cohomology.

In section 1 we review a definition of the relative semi-infinite Weil complex and the relative semi-infinite cohomology H^{\frac{∞}{2} +∗}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)) of a graded Lie algebra 𝔤.

In section 2 we consider the relative semi-infinite Weil complex of the loop algebra of 𝔰𝔩(2). We describe the relative semi-infinite cohomology in terms of the Lie algebra (co)homology and give the proofs of the results announced in [P1]. We observe some 𝔰𝔩(2)-module structures and an action of the W itt algebra on this complex.

In section 3 we recall a definition of the superconformal algebra S(2, 0). The algebra S(2, 0) has a simple ideal S′(2, 0) = [S(2, 0), S(2, 0)]. We prove that the exterior derivations of S′(2, 0) form an 𝔰𝔩(2).

In section 4, using the structures described in section 2, we prove that the relative semiinfinite Weil complex of the loop algebra of 𝔰𝔩(2) (and the relative semi-infinite cohomology) is a module over a central extension Ŝ(2, 0) of S(2, 0). We construct S(2, 0) using the quadratic expansions on the generators of Heisenberg and Clifford algebras related to 𝔤.

In section 5 we consider a complex finite-dimensional Lie algebra 𝔤 with a nondegenerate invariant symmetric form. We obtain a representation of Ŝ(2, 0) with central charge 3dim𝔤 on the relative semi-infinite Weil complex of the loop algebra 𝔤̃ = 𝔤⊗ℂ[t, t^{−1}], and on the relative semi-infinite cohomology H^{\frac{∞}{2} +∗}(𝔤̃, 𝔤̃_{0}, S^{\frac{∞}{2} +∗}(𝔤̃)). Moreover, there is an action of a central extension of the Lie superalgebra DerS′(2, 0) of derivations of S′(2, 0) on these spaces. In particular, the exterior derivations of S′(2, 0) define the 𝔰𝔩(2)-module structure on the relative semi-infinite cohomology, which was observed in [FGZ] as an analogue of the classical structure on the de Rham cohomology in Kähler geometry.

Through this paper we use the notations and definitions given in section 1 of [P2].

I would like to thank Max-Planck-Institut für Mathematik for the hospitality and support.

1. Relative semi-infinite Weil complex.

Let 𝔤 = ⊕_{n∈ℤ}𝔤_{n} be a graded Lie algebra over ℂ. In [P2] we reviewed a construction of the semi-infinite Weil complex

\{S^{\frac{∞}{2} +∗}(𝔤) ⊗ Λ^{\frac{∞}{2} +∗}(𝔤), d + h\}.  (1.1)

Recall that the semi-infinite symmetric power space S^{\frac{∞}{2} +∗}(𝔤) is a module over the Heisenberg algebra H(𝔤), and the semi-infinite exterior power space Λ^{\frac{∞}{2} +∗}(𝔤) is a module over the Clifford algebra C(𝔤). Denote the actions of H(𝔤) and C(𝔤) on these modules by β(x), γ(x′) and τ(x), ε(x′), respectively, for x ∈ 𝔤, x′ ∈ 𝔤′.

Recall the definition of the relative semi-infinite Weil complex (cf. [FGZ]). Let ρ(x) and π(x) be the projective representations of 𝔤 in Λ^{\frac{∞}{2} +∗}(𝔤) and S^{\frac{∞}{2} +∗}(𝔤), respectively. Let

θ(x) = ρ(x) + π(x)  (1.2)

for x ∈ 𝔤. Since the cocycles corresponding to ρ(x) and π(x) cancel, the representation θ of 𝔤 in S^{\frac{∞}{2} +∗}(𝔤) ⊗ Λ^{\frac{∞}{2} +∗}(𝔤) is well-defined. Let

C^{\frac{∞}{2} +∗}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)) = \{c ∈ S^{\frac{∞}{2} +∗}(𝔤) ⊗ Λ^{\frac{∞}{2} +∗}(𝔤)|τ(x)c = 0, θ(x)c = 0 for all x ∈ 𝔤_{0}\},

(1.3)

The differential d preserves this subcomplex since

{d, τ(x)} = dτ(x) + τ(x)d = θ(x),  (1.4)

and

[d, θ(x)] = 0  (1.5)

for any x ∈ 𝔤. The complex (1.3) is called the relative semi-infinite Weil complex. Its cohomology (with respect to the differential d) is called the relative semi-infinite cohomology H^{\frac{∞}{2} +∗}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)). Let Deg ε(x′) = Deg γ(x′) = 1, Deg τ(x) = Deg β(x) = −1 for x ∈ 𝔤, x′ ∈ 𝔤′. Then

Λ^{\frac{∞}{2} +∗}(𝔤) = ⊕_{j∈ℤ}Λ^{\frac{∞}{2} +j}(𝔤), S^{\frac{∞}{2} +∗}(𝔤) = ⊕_{l∈ℤ}S^{\frac{∞}{2} +l}(𝔤).  (1.6)

Accordingly, for any j, l ∈ ℤ we obtain a group H^{\frac{∞}{2} +j}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +l}(𝔤)).

Let deg τ(x) = deg β(x) = n, deg ε(x′) = deg γ(x′) = −n for x ∈ 𝔤_{n}, x′ ∈ 𝔤′_{n}. Since d preserves the graded structure deg of the Weil complex, the cohomology groups also have a graded structure. Notice that they are finite-dimensional in each level:

H^{\frac{∞}{2} +j}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +l}(𝔤)) = ⊕_{n∈ℤ}H^{\frac{∞}{2} +j,n}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +l}(𝔤)).  (1.7)

Recall that given N ∈ ℤ, a polarization of 𝔤 is defined: 𝔫_{+} = ⊕_{n>N}𝔤_{n}, 𝔫 = ⊕_{n≤N}𝔤_{n}. We fix N = 0. Thus 𝔫_{+} = ⊕_{n>0}𝔤_{n}. Let 𝔫_{−} = ⊕_{n<0}𝔤_{n}.

For a homogeneous element in Λ^{a}(𝔫′_{+}) ∧ Λ_{∞}^{b}(𝔫′_{−}), a is the number of added elements, and b is the number of missing elements with respect to the vacuum vector

1_{Λ} = e′_{i_{1}} ∧ e′_{i_{1}−1} ∧ ….  (1.8)

Let

Λ_{rel}^{\frac{∞}{2} +∗}(𝔤) = ⊕_{a≥0,b≥0}Λ^{a}(𝔫′_{+}) ∧ Λ_{∞}^{b}(𝔫′_{−}).  (1.9)

Let

C^{a,b}(𝔤) = [S^{\frac{∞}{2} +∗}(𝔤) ⊗ Λ^{a}(𝔫′_{+}) ∧ Λ_{∞}^{b}(𝔫′_{−})]^{𝔤^{0}}.  (1.10)

Thus

C^{\frac{∞}{2} +j}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)) = ⊕_{a−b=j}C^{a,b}(𝔤).  (1.11)

The first natural example is the Witt algebra:

𝔤 = ⊕_{n∈ℤ}ℂL_{n},  (1.12)
[L_{n}, L_{m}] = (n − m)L_{n+m}.  (1.13)

It was studied by B. Feigin and E. Frenkel in details in [FF]. In particular,

H^{\frac{∞}{2} +∗}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)) = S^{∗}(ℂL_{0}).  (1.14)

Thus the cohomology exists only in deg = 0. The next example is the loop algebra of 𝔰𝔩(2).

2. Relative semi-infinite Weil complex of 𝔰𝔩(2) ⊗ ℂ[t, t^{−1}].

2.1. Relative semi-infinite cohomology. Let 𝔤 = 𝔰𝔩(2) ⊗ ℂ[t, t^{−1}], 𝔤_{n} = 𝔰𝔩(2) ⊗ t^{n}, let {e, h, f} be the standard basis of 𝔰𝔩(2), {e_{n}, h_{n}, f_{n}} be a basis of 𝔤_{n}, {e′_{n}, h′_{n}, f′_{n}} be the dual basis.

Let V^{a} ⊂ Λ^{a}(𝔫′_{+}) be an irreducible 𝔤_{0}-module with highest vector f′_{1} ∧ f′_{2} ∧ … ∧ f′_{a}. Let V^{b} be an irreducible 𝔤_{0}-module with highest vector e_{−1} ∧ e_{−2} ∧ … ∧ e_{−b}. We identify V^{b} with a subspace of Λ_{∞}^{b}(𝔫′_{−}). Let

φ : S^{\frac{∞}{2} +∗}(𝔤) ⊗ V^{a} ⊗ V^{b} → S^{\frac{∞}{2} +∗}(𝔤) ⊗ V^{b} ⊗ V^{a}  (2.1.1)

be the natural isomorfism. The following statement was announced in [P1]. Theorem 2.1.1. The representatives of the elements of H^{\frac{∞}{2} +j}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)) in C^{\frac{∞}{2} +j}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)) are the arbitrary linear combinations of the elements

x ∈ [S^{\frac{∞}{2} +∗}(𝔤) ⊗ V^{a} ⊗ V^{b}]^{𝔤^{0}}, where a − b = j,  (2.1.2)

such that x ∈ H^{a}(𝔫_{+}, S^{\frac{∞}{2} +∗}(𝔤)) ⊗ V^{b}, and φx ∈ H_{b}(𝔫_{−}, S^{\frac{∞}{2} +∗}) ⊗ V^{a}.

To prove this theorem we will use some Hochschild-Serre spectral sequences. We will use the notation C^{\frac{∞}{2} +∗}(𝔤) = C^{\frac{∞}{2} +∗}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)). Spectral sequence for H^{\frac{∞}{2} +∗}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)). We define a filtration

… ⊃ F^{p}C^{\frac{∞}{2} +∗}(𝔤) ⊃ F^{p+1}C^{\frac{∞}{2} +∗}(𝔤) ⊃ ….  (2.1.3)

corresponding to the number of missing elements:

F^{p}C^{\frac{∞}{2} +∗}(𝔤) = \{c ∈ ⊕_{a≥0,b≤−p}C^{a,b}(𝔤)\}.  (2.1.4)

We follow the usual construction of the Hochschild-Serre spectral sequence (see [Fu], [GM]) and set

Z_{r}^{p,q} = \{c ∈ F^{p}C^{\frac{∞}{2} +p+q}(𝔤)| dc ∈ F^{p+r}C^{\frac{∞}{2} +p+q+1}(𝔤)\}.  (2.1.5)

Then set

E_{r}^{p,q} = Z_{r}^{p,q}/(Z_{r−1}^{p+1,q−1} + dZ_{r−1}^{p−r+1,q+r−2}).  (2.1.6)

The differential d induces the differentials

d_{r}^{p,q} : E_{r}^{p,q} ⟶ E_{r}^{p+r,q−r+1}  (2.1.7)

and E_{r+1}^{p,q} = H^{p,q}(E_{r}). Let

E_{r} = ⊕_{p,q}E_{r}^{p,q}.  (2.1.8)

Denote by d_{r} the corresponding differential. Thus we have

E_{0} = ⊕_{a,b≥0}[S^{\frac{∞}{2} +∗}(𝔤) ⊗ Λ^{a}(𝔫′_{+}) ∧ Λ_{∞}^{b}(𝔫′_{−})]^{𝔤^{0}},  (2.1.9)

and

E_{1} = ⊕_{a,b≥0}[H^{a}(𝔫_{+}, S^{\frac{∞}{2} +∗}(𝔤)) ⊗ H_{∞}^{b}(𝔫_{−}, ℂ)]^{𝔤^{0}}.  (2.1.10)

Notice that H_{∞}^{b}(𝔫_{−}, ℂ) coincides with the usual homology H_{b}(𝔫_{−}) of the Lie algebra 𝔫_{−} with coefficients in the trivial representation. Lemma 2.1.1. H_{b}(𝔫_{−}) is an irreducible 𝔰𝔩(2)-module with highest weight 2b, and highest vector

e_{−1} ∧ e_{−2} ∧ … ∧ e_{−b}.  (2.1.11)

Proof. We will prove an equivalent statement for H^{i}(𝔫_{−}) using the generalized Bott-Kostant theorem ([GL]). Consider the affine Kac-Moody Lie algebra A_{1}^{(1)}:

A_{1}^{(1)} = ⊕_{n}𝔤̃_{n}, where  (2.1.12)
𝔤̃_{n} = 𝔰𝔩(2) ⊗ t^{n} for n ≠ 0,  (2.1.13)
𝔤̃_{0} = 𝔰𝔩(2) ⊕ ℂc ⊕ ℂ𝔡.  (2.1.14)

Recall that

𝔤_{0} = 𝔰𝔩(2) =< e, h, f >,  (2.1.15)

where {e, h, f} is the standard basis. Let the corresponding roots be ±α. The abelian subalgebra

h̃_{0} = ℂh ⊕ ℂc ⊕ ℂ𝔡  (2.1.16)

is identified with its dual by means of the nonsingular symmetric bilinear form on A_{1}^{(1)}. Let

h_{0} = c − α, h_{1} = α.  (2.1.17)

Let W be the Weyl group of A_{1}^{(1)}. Recall that W is generated by the linear automorphisms r_{k} of h̃_{0}, where k = 0, 1, defined by the condition

r_{k}φ = φ− < φ, h_{k} > h_{k} for all φ ∈ h̃_{0}.  (2.1.18)

Let

Δ_{+} = {α + nc for n ≥ 0, −α + nc for n > 0, nc for n > 0}  (2.1.19)
Δ_{−} = {−α + nc for n ≤ 0, α + nc for n < 0, nc for n < 0}  (2.1.20)
Δ(𝔫_{−}) = {−α + nc for n < 0, α + nc for n < 0, nc for n < 0}. (2.1.21)

According to the generalized Bott-Kostant theorem, the irreducible 𝔤_{0}-submodules of H^{i}(𝔫_{−}) correspond exactly to the elements w of W which satisfy the following conditions:

w(Δ_{+}) ∩ Δ_{−} ⊂ Δ(𝔫_{−}),  (2.1.22)
#(w(Δ_{+}) ∩ Δ_{−}) = i,  (2.1.23)

where # denotes the number of the elements in the set. Define ρ ∈ h̃_{0} by

< ρ, h_{k} >= 1 for k = 0, 1.  (2.1.24)

Then the corresponding highest weights are given by

w(ρ) − ρ.  (2.1.25)

One can check that in our case

ρ = 1/2α + 2𝔡,  (2.1.26)

and the only w ∈ W which satisfies (2.1.22) and (2.1.23) is (of length i):

w = (r_{0}r_{1})^{i/2}  if i is even,  (2.1.27)
(r_{0}r_{1})^{[i/2]}r_{0}  if i is odd.  (2.1.28)

Then

w(ρ) − ρ = −i(i + 1)/2c + iα.  (2.1.29)

Thus H^{i}(𝔫_{−}) is an irreducible 𝔤_{0}-module with highest weight 2i.

■

Corollary 2.1.1. Let x ∈ H_{b}(𝔫_{−}) . We can choose a representative x such that

degx = −b(b + 1)/2.  (2.1.30)

Corollary 2.1.2. H^{a}(𝔫_{+}) is an irreducible 𝔰𝔩(2)-module with highest weight 2a, and highest vector

f′_{1} ∧ f′_{2} ∧ … ∧ f′_{a}.  (2.1.31)

Spectral sequence for H^{a}(𝔫_{+}, S^{\frac{∞}{2} +∗}(𝔤)). Let

C^{a}(𝔤) = S^{\frac{∞}{2} +∗}(𝔤) ⊗ Λ^{a}(𝔫′_{+}).  (2.1.32)

Let v ∈ S^{\frac{∞}{2} +∗}(𝔤), w_{+} ∈ Λ^{a}(n′_{+}). Define

fdeg(v ⊗ w_{+}) = degv − degw_{+}.  (2.1.33)

Consider a spectral sequence associated with the filtration

… ⊃ F^{p}C^{a}(𝔤) ⊃ F^{p+1}C^{a}(𝔤) ⊃ …,  (2.1.34)

defined as follows:

F^{p}C^{a}(𝔤) = {c ∈ C^{a}(𝔤)|fdegc ≥ p}.  (2.1.35)

Then

E_{0} = S^{\frac{∞}{2} +∗}(𝔤) ⊗ Λ^{a}(𝔫′_{+}),  (2.1.36)

and

E_{1} = S^{\frac{∞}{2} +∗}(𝔤) ⊗ H^{a}(𝔫_{+}, ℂ).  (2.1.37)

According to corollary 2.1.2, H^{a}(𝔫_{+}, ℂ) and V^{a} are isomorphic as 𝔤_{0}-modules. Thus the elements of S^{\frac{∞}{2} +∗}(𝔤) ⊗ V^{a} are the representatives of the elements of E_{1} in the space of cochains. Notice that

deg(V^{a}) = −a(a + 1)/2.  (2.1.38)

Recall that the differential preserves the grading deg. Thus

H^{a}(𝔫_{+}, S^{\frac{∞}{2} +∗}(𝔤)) = ⊕_{n∈ℤ}H^{a}(𝔫_{+}, S^{\frac{∞}{2} +∗}(𝔤))_{deg=n}.  (2.1.39)

Consider the filtration defined by (2.1.35) in the space of a-cochains of deg = n. Then the representatives of the term E_{1} of the corresponding spectral sequence constitute

[S^{\frac{∞}{2} +∗}(𝔤) ⊗ V^{a}]_{deg=n}.  (2.1.40)

Notice that

fdeg([S^{\frac{∞}{2} +∗}(𝔤) ⊗ V^{a}]_{deg=n}) = n + a(a + 1).  (2.1.41)

Thus

E_{1}^{p,a−p} = 0, if p ≠ n + a(a + 1).  (2.1.42)

Recall that the differential d_{r} acts as follows :

E_{r}^{p,a−p} ⟶^{d_{r}} E_{r}^{p+r,a−p−r+1}.  (2.1.43)

Thus

d_{r} = 0, if r ≠ 2a + 2.  (2.1.44)

Thus E_{∞} is determined by the complex :

… ⟶^{d_{2a−2}} S^{\frac{∞}{2} +∗}(𝔤) ⊗ V^{a−1} ⟶^{d_{2a}} S^{\frac{∞}{2} +∗}(𝔤) ⊗ V^{a} ⟶^{d_{2a+2}} S^{\frac{∞}{2} +∗}(𝔤) ⊗ V^{a+1} ⟶^{d_{2a+4}} …. (2.1.45)

Spectral sequence for H_{b}(𝔫_{−}, S^{\frac{∞}{2} +∗}(𝔤)). Let

C^{b}(𝔤) = S^{\frac{∞}{2} +∗}(𝔤) ⊗ Λ^{b}(𝔫_{−}).  (2.1.46)

Let v ∈ S^{\frac{∞}{2} +∗}(𝔤), w_{−} ∈ Λ^{b}(n_{−}). Define

fdeg′(w_{−} ⊗ v) = degw_{−} − degv.  (2.1.47)

Define the filtration

F^{p}C^{b}(𝔤) = {c ∈ C^{b}(𝔤)|fdeg′c ≥ p}.  (2.1.48)

Then the first term of the corresponding spectral sequence is (cf. (2.1.37))

E_{1} = S^{\frac{∞}{2} +∗}(𝔤) ⊗ H_{b}(𝔫_{−}, ℂ).  (2.1.49)

According to lemma 2.1.1, H_{b}(𝔫_{−}, ℂ) and V^{b} are isomorphic as 𝔤_{0}-modules. Thus the elements of S^{\frac{∞}{2} +∗}(𝔤) ⊗ V^{b} are the representatives of the elements of E_{1} in the space of cochains. By analogy with (2.1.44) we obtain that

d_{r} = 0, if r ≠ 2b.  (2.1.50)

Thus E_{∞} is determined by the complex

… ⟶ S^{\frac{∞}{2} +∗}(𝔤) ⊗ V^{b+1} ⟶^{d_{2b+2}} S^{\frac{∞}{2} +∗}(𝔤) ⊗ V^{b} ⟶^{d_{2b}} S^{\frac{∞}{2} +∗}(𝔤) ⊗ V^{b−1} ⟶ …. (2.1.51)

Proof of theorem 2.1.1.

We use the spectral sequence corresponding to the filtration (2.1.4). Recall that

E_{1}^{−b,a} = [H^{a}(𝔫_{+}, S^{\frac{∞}{2} +∗}(𝔤)) ⊗ H_{∞}^{b}(𝔫_{−}, ℂ)]^{𝔤^{0}}.  (2.1.52)

First we will show that the representatives of the elements of E_{2}^{−b,a} are the elements

x ∈ [S^{\frac{∞}{2} +∗}(𝔤) ⊗ V^{a} ⊗ V^{b}]^{𝔤^{0}},  (2.1.53)

such that x ∈ H^{a}(𝔫_{+}, S^{\frac{∞}{2} +∗}(𝔤)) ⊗ V^{b}, and φx ∈ H_{b}(𝔫_{−}, S^{\frac{∞}{2} +∗}(𝔤)) ⊗ V^{a}.

Recall that the term E_{2}^{−b,a} is determined by the complex

… ⟶^{d_{1}} [H^{a}(𝔫_{+}, S^{\frac{∞}{2} +∗}(𝔤)) ⊗ H_{∞}^{b+1}(𝔫_{−}, ℂ)]^{𝔤^{0}} ⟶^{d_{1}} [H^{a}(𝔫_{+}, S^{\frac{∞}{2} +∗}(𝔤)) ⊗ H_{∞}^{b}(𝔫_{−}, ℂ)]^{𝔤^{0}} ⟶^{d_{1}}

⟶^{d_{1}} [H^{a}(𝔫_{+}, S^{\frac{∞}{2} +∗}(𝔤)) ⊗ H_{∞}^{b−1}(𝔫_{−}, ℂ)]^{𝔤^{0}} ⟶^{d_{1}} ….  (2.1.54)

According to (2.1.45), the elements of H^{a}(𝔫_{+}, S^{\frac{∞}{2} +∗}(𝔤)) have representatives in S^{\frac{∞}{2} +∗}(𝔤)⊗ V^{a}. Hence the elements of E_{1}^{−b,a} have representatives in [S^{\frac{∞}{2} +∗}(𝔤) ⊗ V^{a} ⊗ V^{b}]𝔤^{0}. Let

x = Σ_{k}Σ_{i}v^{k,i} ⊗ w_{+}^{k} ⊗ w_{−}^{k,i}, where v^{k,i} ∈ S^{\frac{∞}{2} +∗}(𝔤), w_{+}^{k} ∈ V^{a}, w_{−}^{k,i} ∈ V^{b},  (2.1.55)

be a representative of an element of E_{1}^{−b,a}. Thus x ∈ H^{a}(𝔫_{+}, S^{\frac{∞}{2} +∗}(𝔤)) ⊗ V^{b}. Notice that

d_{1}x = φ^{−1}(Σ_{k}d_{2b}(Σ_{i}v^{k,i} ⊗ w_{−}^{k,i}) ⊗ w_{+}^{k}).  (2.1.56)

Thus x is a representative of an element of E_{2}^{−b,a} if and only if

φx ∈ H_{b}(𝔫_{−}, S^{\frac{∞}{2} +∗}(𝔤)) ⊗ V^{a}.  (2.1.57)

Next we will show that

E_{2}^{−b,a} = E_{∞}^{−b,a}.  (2.1.58)

Let x be a representative of an element of E_{r}^{−b,a}, where r ≥ 2, in Z_{r}^{−b,a}. Then

dx ∈ F^{−b+r}C^{\frac{∞}{2} +a−b+1}(𝔤).  (2.1.59)

Hence

dx ∈ ⊕_{a_{1}≥0,b_{1}≤b−2}C^{a_{1},b_{1}}(𝔤).  (2.1.60)

On the other hand,

d = d_{+} + d_{−},  (2.1.61)

where

d_{+} : C^{a,b}(𝔤) → C^{a+1,b}(𝔤),  (2.1.62)
d_{−} : C^{a,b}(𝔤) → C^{a,b−1}(𝔤).  (2.1.63)

Thus if x ∈ [S^{\frac{∞}{2} +∗}(𝔤) ⊗ V^{a} ⊗ V^{b}]𝔤^{0}, then

dx ∈ C^{a+1,b}(𝔤) ⊕ C^{a,b−1}(𝔤).  (2.1.64)

Then according to (2.1.60) and (2.1.64), we have

dx = 0.  (2.1.65)
■

Remark 2.1.1. Notice that to compute the relative semi-infinite cohomology one can use as well the filtration defined by the number of added elements :

F̃^{p}C^{\frac{∞}{2} +∗}(𝔤) = \{c ∈ ⊕_{a≥p,b≥0}C^{a,b}(𝔤)\}.  (2.1.66)

The first term of the corresponding spectral sequence is

Ẽ_{1} = ⊕_{a,b≥0}[H^{a}(𝔫_{+}, ℂ) ⊗ H_{∞}^{b}(𝔫_{−}, S^{\frac{∞}{2} +∗}(𝔤))]^{𝔤^{0}}.  (2.1.67)

Both spectral sequences corresponding to the filtrations (2.1.4) and (2.1.66) collapse in the second term.

2.2. An action of the Witt algebra.

We will show that the relative semi-infinite cohomology admits an action of the Witt algebra.

Let u run through the standard basis {e, h, f} of 𝔰𝔩(2), let {u_{n}} be a basis of 𝔤_{n}, {u′_{n}} be the dual basis. Let

L_{n} = −Σ_{u}Σ_{m}m : τ(u_{m+n})ε(u′_{m}) : −Σ_{u}Σ_{m}m : β(u_{m})γ(u′_{m−n}) :,  (2.2.1)

We obtain the Witt algebra L = ⊕_{n∈ℤ}ℂL_{n}, [L_{n}, L_{m}] = (n − m)L_{n+m}. Recall that 1_{S} is the vacuum vector in S^{\frac{∞}{2} +∗}(𝔤), and 1_{Λ} is the vacuum vector in Λ_{rel}^{\frac{∞}{2} +∗}(𝔤). Thus

1 = 1_{S} ⊗ 1_{Λ}  (2.2.2)

is the vacuum vector in C^{\frac{∞}{2} +∗}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)). Notice that L_{n}1 = 0 if and only if n ≥ −1. Lemma 2.2.1. For any j, l ∈ ℤ, C^{\frac{∞}{2} +j}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +l}(𝔤)) is a module over L. Proof. Notice that

[L_{n}, τ(u_{m})] = −mτ(u_{m+n}),  (2.2.3)
[L_{n}, ε(u′_{m})] = (m − n)ε(u′_{m−n});  (2.2.4)
[L_{n}, β(u_{m})] = −(m + n)β(u_{m+n}),  (2.2.5)
[L_{n}, γ(u′_{m})] = mγ(u′_{m−n}).  (2.2.6)

It follows from (2.2.3)-(2.2.6) that S^{\frac{∞}{2} +l}(𝔤) ⊗ Λ^{\frac{∞}{2} +j}(𝔤) is a module over L. Notice that according to (2.2.3)

[L_{n}, τ(𝔤_{0})] = 0,  (2.2.7)

Let us show that

[L_{n}, θ(u_{m})] = −mθ(u_{m+n}).  (2.2.8)

Recall that

θ(u_{m}) = ρ(u_{m}) + π(u_{m}).  (2.2.9)

We will show that

[L_{n}, ρ(u_{m})] = −mρ(u_{m+n}),  (2.2.10)
[L_{n}, π(u_{m})] = −mπ(u_{m+n}).  (2.2.11)

In fact, according to (2.2.3) and (2.2.4)

[L_{n}, ρ(u_{m})] = [L_{n}, Σ_{i} : τ([u_{m}, v_{i}])ε(v′_{i}) :] = [L_{n}, Σ_{i} : τ([u, v]_{m+i})ε(v′_{i}) :] =
Σ_{i}(: −(m + i)τ([u, v]_{m+i+n})ε(v′_{i}) : + : τ([u, v]_{m+i})(i − n)ε(v′_{i−n}) :) =
Σ_{i}(: −(m + i)τ([u, v]_{m+i+n})ε(v′_{i}) : + : τ([u, v]_{m+i+n})iε(v′_{i}) :) =
− mΣ_{i} : τ([u, v]_{m+i+n})ε(v′_{i}) := −mρ(u_{m+n}).  (2.2.12)

According to (2.2.5) and (2.2.6)

[L_{n}, π(u_{m})] = [L_{n}, Σ_{i} : β([u_{m}, v_{i}])γ(v′_{i}) :] = [L_{n}, Σ_{i} : β([u, v]_{m+i})γ(v′_{i}) :] = Σ_{i}(: −(m + i + n)β([u, v]_{m+i+n})γ(v′_{i}) : + : β([u, v]_{m+i})iγ(v′_{i−n}) :) = Σ_{i}(: −(m + i + n)β([u, v]_{m+i+n})γ(v′_{i}) : + : β([u, v]_{m+i+n})(i + n)γ(v′_{i}) :) =

− mΣ_{i} : β([u, v]_{m+i+n})γ(v′_{i}) := −mπ(u_{m+n}).  (2.2.13)

Thus (2.2.8) follows. From (2.2.8) we obtain

[L_{n}, θ(𝔤_{0})] = 0.  (2.2.14)

According to (2.2.7) and (2.2.14) C^{\frac{∞}{2} +j}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +l}(𝔤)) is a module over L.

■

Corollary 2.2.1. For any j, l ∈ ℤ, H^{\frac{∞}{2} +j}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +l}(𝔤)) is a module over L. Proof. Notice that

[L_{n}, d] = 0,  (2.2.15)

see corollary 2.1 in [P2].

2.3 An 𝔰𝔩(2)-module structure.

We will show that the relative semi-infinite Weil complex is a module over an 𝔰𝔩(2). Define SL(2) =< E, H, F > as follows:

E = −\frac{1}{2}iΣ_{m}[γ(e′_{m})γ(f′_{1−m}) + γ(f′_{m})γ(e′_{1−m}) + 2γ(h′_{m})γ(h′_{1−m})],  (2.3.1)
H = −Σ_{m}[: γ(e′_{m})β(e_{m}) : + : γ(f′_{m})β(f_{m}) : + : γ(h′_{m})β(h_{m}) :],  (2.3.2)
F = −\frac{1}{2}iΣ_{m}[β(e_{m})β(f_{1−m}) + β(f_{m})β(e_{1−m}) + (1/2)β(h_{m})β(h_{1−m})]. (2.3.3)

Thus

[H, E] = 2E, [H, F] = −2F, [E, F] = H,

E(F) : S^{\frac{∞}{2} +l}(𝔤) → S^{\frac{∞}{2} +l±2}(𝔤),  (2.3.4)

H is the multiplication by l.

Clearly, the action of SL(2) on Λ^{\frac{∞}{2} +∗}(𝔤) is trivial. Lemma 2.3.1. For any j ∈ ℤ, C^{\frac{∞}{2} +j}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)) is a module over SL(2).

Proof. Notice that

[E, β(e_{m})] = −iγ(f′_{1−m}),
[E, β(f_{m})] = −iγ(e′_{1−m}),  (2.3.5)
[E, β(h_{m})] = −2iγ(h′_{1−m}),
[E, γ(u′_{m})] = 0;

[H, γ(u′_{m})] = γ(u′_{m}),  (2.3.6)
[H, β(u_{m})] = −β(u_{m});

[F, γ(e′_{m})] = iβ(f_{1−m}),
[F, γ(f′_{m})] = iβ(e_{1−m}),
[F, γ(h′_{m})] = \frac{1}{2}iβ(h_{1−m}),  (2.3.7)
[F, β(u_{m})] = 0.

Obviously,

[SL(2), τ(𝔤)] = 0,  (2.3.8)
[SL(2), ρ(𝔤)] = 0.  (2.3.9)

One can check that

[SL(2), π(𝔤)] = 0.  (2.3.10)

Hence

[SL(2), θ(𝔤)] = 0.  (2.3.11)

It follows from (2.3.8) and (2.3.11) that C^{\frac{∞}{2} +j}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)) is a module over SL(2).

■

Corollary 2.3.1. For any j ∈ ℤ, H^{\frac{∞}{2} +j}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)) is a module over SL(2). Proof. It follows from (2.3.10) that

[SL(2), d] = 0.  (2.3.12)

2.4. Another 𝔰𝔩(2)-module structure.

There exists another 𝔰𝔩(2)-module structure on the relative semi-infinite Weil complex and relative semi-infinite cohomology. Define 𝒮ℒ(2) =< E, H, F > as follows :

E = iΣ_{m>0}m[ε(e′_{−m})ε(f′_{m}) + ε(f′_{−m})ε(e′_{m}) + 2ε(h′_{−m})ε(h′_{m})],  (2.4.1)
H = −Σ_{u,m≠0} : τ(u_{m})ε(u′_{m}) :,  (2.4.2)
F = −iΣ_{m>0} \frac{1}{m}[τ(f_{m})τ(e_{−m}) + τ(e_{m})τ(f_{−m}) + \frac{1}{2}τ(h_{m})τ(h_{−m})].  (2.4.3)

Thus

[H, E] = 2E, [H, F] = −2F, [E, F] = H,

E(F) : Λ_{rel}^{\frac{∞}{2} +j}(𝔤) → Λ_{rel}^{\frac{∞}{2} +j±2}(𝔤),  (2.4.4)

H is the multiplication by j.

Clearly, the action of 𝒮ℒ(2) on S^{\frac{∞}{2} +∗}(𝔤) is trivial. Lemma 2.4.1. For any l ∈ ℤ, C^{\frac{∞}{2} +∗}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +l}(𝔤)) is a module over 𝒮ℒ(2). Proof. Notice that

[E, τ(e_{m})] = imε(f′_{−m}),

[E, τ(f_{m})] = imε(e′_{−m}),  (2.4.5)

[E, τ(h_{m})] = 2imε(h′_{−m}),

[E, ε(u′_{m})] = 0;

[H, ε(u′_{m})] = ε(u′_{m}),  (2.4.6)

[H, τ(u_{m})] = −τ(u_{m});

[F, ε(e′_{m})] = \frac{i}{m}τ(f_{−m}),

[F, ε(f′_{m})] = \frac{i}{m}τ(e_{−m}),  (2.4.7)

[F, ε(h′_{m})] = \frac{i}{2m}τ(h_{−m}),

[F, τ(u_{m})] = 0.

It follows from (2.4.5) that

[𝒮ℒ(2), τ(𝔤_{0})] = 0.  (2.4.8)

Clearly,

[𝒮ℒ(2), π(𝔤)] = 0.  (2.4.9)

One can check that

[𝒮ℒ(2), ρ(𝔤_{0})] = 0.  (2.4.10)

Hence

[𝒮ℒ(2), θ(𝔤_{0})] = 0.  (2.4.11)

It follows from (2.4.8) and (2.4.11) that C^{\frac{∞}{2} +∗}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +l}(𝔤)) is a module over 𝒮ℒ(2).

■

Corollary 2.4.1. For any l ∈ ℤ, H^{\frac{∞}{2} +∗}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +l}(𝔤)) is a module over 𝒮ℒ(2). Proof. Notice that 𝒮ℒ(2) does not commute with d. In fact,

[F, d] ≠ 0.  (2.4.12)

Nevertheless, an 𝒮ℒ(2)-module structure on the relative semi-infinite cohomology does exist, as it was pointed out in [FGZ]. We will explain this in lemma 5.1.

3. The superconformal algebra S(2, 0).

Recall the definition of superconformal algebras ([K-VDL]). Let

W (N) = derℂ[t^{−1}, t, θ_{1}, …, θ_{N}],  (3.1)

where θ_{1}, …, θ_{N} are odd variables. Let

S(N, α) = {D ∈ W (N)|Div(t^{α}D) = 0} for α ∈ ℂ,  (3.2)

where

Div(f∂/∂t + Σ_{i}f_{i}∂/∂θ_{i}) = ∂f/∂t + Σ_{i}(−1)^{p(f_{i})}∂f_{i}/∂θ_{i}.  (3.3)

Assume that N > 1. Notice that S(N, α) is simple, if α ∉ ℤ; if α ∈ ℤ, then S(N, α) contains the simple ideal S′(N, α) = [S(N, α), S(N, α)]:

0 → S′(N, α) → S(N, α) → ℂt^{−α}θ_{1} … θ_{N}∂/∂t → 0.  (3.4)

Notice that S(N, α) has a nontrivial central extension if and only if N = 2 (see [K-VDL]), which is important for our consideration. In what follows N = 2 and α = 0. Let

{ℒ_{n}, E_{n}, H_{n}, F_{n}, h_{n}, p_{n}, x_{n}, y_{n}}_{n∈ℤ}  (3.5)

be the basis of S′(2, 0) defined as follows:

ℒ_{n} = −t^{n}(t∂/∂t + (1/2)(n + 1)(Σ_{i}θ_{i}∂/∂θ_{i})),  (3.6)
E_{n} = t^{n}θ_{2}∂/∂θ_{1},  (3.7)
H_{n} = t^{n}(θ_{2}∂/∂θ_{2} − θ_{1}∂/∂θ_{1}),  (3.8)
F_{n} = t^{n}θ_{1}∂/∂θ_{2},  (3.9)
h_{n} = t^{n}θ_{2}∂/∂t − nt^{n−1}θ_{1}θ_{2}∂/∂θ_{1},  (3.10)
p_{n} = −t^{n+1}∂/∂θ_{2},  (3.11)
x_{n} = t^{n+1}∂/∂θ_{1},  (3.12)
y_{n} = t^{n}θ_{1}∂/∂t + nt^{n−1}θ_{1}θ_{2}∂/∂θ_{2}.  (3.13)

The nonvanishing commutation relations are

[ℒ_{n}, ℒ_{m}] = (n − m)ℒ_{n+m},  (3.14)
[E_{n}, F_{k}] = H_{n+k}, [H_{n}, E_{k}] = 2E_{n+k}, [H_{n}, F_{k}] = −2F_{n+k},  (3.15)
[ℒ_{n}, E_{k}] = −kE_{n+k}, [ℒ_{n}, H_{k}] = −kH_{n+k}, [ℒ_{n}, F_{k}] = −kF_{n+k},  (3.16)
[ℒ_{n}, h_{k}] = (1/2)(n − 2k + 1)h_{n+k}, [ℒ_{n}, p_{k}] = (1/2)(n − 2k − 1)p_{n+k},  (3.17)
[ℒ_{n}, x_{k}] = (1/2)(n − 2k − 1)x_{n+k}, [ℒ_{n}, y_{k}] = (1/2)(n − 2k + 1)y_{n+k},  (3.18)
[E_{n}, y_{k}] = h_{n+k}, [F_{n}, h_{k}] = y_{n+k}, [H_{n}, h_{k}] = h_{n+k}, [H_{n}, y_{k}] = −y_{n+k},  (3.19)
[E_{n}, p_{k}] = x_{n+k}, [F_{n}, x_{k}] = p_{n+k}, [H_{n}, x_{k}] = x_{n+k}, [H_{n}, p_{k}] = −p_{n+k},  (3.20)
[h_{n}, x_{k}] = (k + 1 − n)E_{n+k}, [p_{n}, y_{k}] = (k − n − 1)F_{n+k}.  (3.21)

[h_{n}, p_{k}] = ℒ_{n+k} − (1/2)(k − n + 1)H_{n+k}, [x_{n}, y_{k}] = −ℒ_{n+k} + (1/2)(k − n − 1)H_{n+k},

(3.22)

Let

F = −θ_{1}θ_{2}∂/∂t.  (3.23)

Then

ℂF⋉S′(2, 0) = S(2, 0),  (3.24)

and the additional nonvanishing commutation relations are

[F, x_{k}] = h_{k+1}, [F, p_{k}] = y_{k+1}.  (3.25)

There exists only one (up to equivalence) cocycle ψ on S(2, 0) ([K-VDL]):

ψ(ℒ_{m}, ℒ_{n}) = (1/12)(m^{3} − m)δ_{m,−n},  (3.26)
ψ(h_{m}, p_{n}) = (1/6)m(m − 1)δ_{m,−n},  (3.27)
ψ(x_{m}, y_{n}) = −(1/6)m(m + 1)δ_{m,−n},  (3.28)
ψ(H_{m}, H_{n}) = (1/3)mδ_{m,−n},  (3.29)
ψ(E_{m}, F_{n}) = (1/6)mδ_{m,−n},  (3.30)

and it is zero otherwise. Denote the corresponding central extension of S(2, 0) by Ŝ(2, 0). Thus

Ŝ(2, 0) = S(2, 0)⊕ < C >,  (3.31)
[s_{1}, s_{2}]_{|Ŝ(2,0)} = [s_{1}, s_{2}]_{|S(2,0)} + ψ(s_{1}, s_{2})C, for s_{1}, s_{2} ∈ S(2, 0),  (3.32)

The cocycle ψ also defines the central extension Ŝ′(2, 0) of S′(2, 0) Remark 3.1. The superalgebra S′(2, 0) can be described as follows:

S′(2, 0) = S′(2, 0)¯0 ⊕ S′(2, 0)¯1;  (3.33)

where

S′(2, 0)¯0 = W itt⋉ŜL(2),  (3.34)
W itt =< ℒ_{n} >_{n∈ℤ}, ŜL(2) =< E_{n}, H_{n}, F_{n} >_{n∈ℤ},  (3.35)
S′(2, 0)¯1 =< h_{n}, y_{n} >_{n∈ℤ} ⊕ < x_{n}, p_{n} >_{n∈ℤ} .  (3.36)

Thus S′(2, 0)¯1 is a direct sum of two standard (odd) ŜL(2)-modules. Remark 3.2. The N = 2 superconformal algebra is isomorphic to a subalgebra of Ŝ(2, 0).

Recall that the N = 2 superconformal algebra is spanned by the generators ℒ_{n} of Witt algebra, two fermionic fields G_{r}^{±}, a U(1) current H_{n}, and a central element C, where n ∈ ℤ, r ∈ ℤ + 1/2 ([FST]). The commutation relations are

[ℒ_{n}, ℒ_{m}] = (n − m)ℒ_{n+m} + \frac{C}{12}(n^{3} − n)δ_{n,−m},  (3.37)
[ℒ_{n}, H_{m}] = −mH_{n+m},  (3.38)
[ℒ_{n}, G_{r}^{±}] = (\frac{n}{2} − r)G_{n+r}^{±},  (3.39)
[G_{r}^{+}, G_{s}^{−}] = 2ℒ_{r+s} + (r − s)H_{r+s} + \frac{C}{3}(r^{2} − \frac{1}{4})δ_{r,−s},  (3.40)
[H_{n}, H_{m}] = \frac{C}{3}nδ_{n,−m},  (3.41)
[H_{n}, G_{r}^{±}] = ±G_{n+r}^{±},  (3.42)

and they are zero otherwise. Thus the N = 2 superconformal algebra is isomorphic to the subalgebra of Ŝ(2, 0) spanned by ℒ_{n}, H_{n}, h_{n}, p_{n}, and C. The isomorphism is given as follows:

ℒ_{n} → ℒ_{n},  (3.43)
H_{n} → H_{n},  (3.44)
G_{n− \frac{1}{2}}^{+} → \sqrt{2}h_{n},  (3.45)
G_{n+ \frac{1}{2}}^{−} → \sqrt{2}p_{n},  (3.46)
C → C.  (3.47)

Let DerS′(2, 0) be the Lie superalgebra of all derivations of S′(2, 0), and Der_{ext}S′(2, 0) be the exterior derivations of S′(2, 0). In other words,

Der_{ext}S′(2, 0) = DerS′(2, 0)/{ads|s ∈ S′(2, 0)}.  (3.48)

Theorem 3.1. Der_{ext}S′(2, 0) has the structure of the 𝒮ℒ(2) =< E, H, F >, where

[H, E] = 2E, [H, F] = −2F, [E, F] = H.  (3.49)

The action of 𝒮ℒ(2) on S′(2, 0)¯0 is trivial and on S′(2, 0)¯1 is given as follows:

[E, h_{k}] = x_{k−1}, [F, x_{k}] = h_{k+1}, [H, x_{k}] = x_{k}, [H, h_{k}] = −h_{k},  (3.50)
[E, y_{k}] = p_{k−1}, [F, p_{k}] = y_{k+1}, [H, p_{k}] = p_{k}, [H, y_{k}] = −y_{k}.  (3.51)

Proof. Recall that (see [Fu])

Der_{ext}S′(2, 0) ≅ H^{1}(S′(2, 0), S′(2, 0)),  (3.52)

Notice that the superalgebra S′(2, 0) has the following ℤ-grading deg:

degℒ_{n} = −n, degE_{n} = −n − 1, degF_{n} = −n + 1, degH_{n} = −n,  (3.53)
degh_{n} = −n, degp_{n} = −n, degx_{n} = −n − 1, degy_{n} = −n + 1.  (3.54)

Then, accordingly, there is the gradind deg in Der_{ext}S′(2, 0). Let

L_{n} = −ℒ_{−n} + (1/2)(1 + n)H_{−n}.  (3.55)

Notice that for a homogeneous element s ∈ S′(2, 0)

[L_{0}, s] = −(degs)s.  (3.56)

Let us consider the natural action of S′(2, 0) on Der_{ext}S′(2, 0). Then for a homogeneous element D ∈ Der_{ext}S′(2, 0)

L_{0}(D) = −(degD)D.  (3.57)

On the other hand, since the action of a Lie superalgebra on its cohomology is trivial, then for any D ∈ H^{1}(S′(2, 0), S′(2, 0)) one must have

L_{0}(D) = 0.  (3.58)

Hence the nonzero elements of Der_{ext}S′(2, 0) must have deg = 0, and one can check that these elements are the arbitrary linear combinations of E, H, F defined in (3.50) and (3.51).

■

4. An action of Ŝ(2, 0) on the relative semi-infinite cohomology of 𝔰𝔩(2) ⊗ ℂ[t, t^{−1}].

Let 𝔤 = 𝔰𝔩(2) ⊗ ℂ[t, t^{−1}]. Theorem 4.1. There is a representation of Ŝ(2, 0) in C^{\frac{∞}{2} +∗}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)) with central charge 9. Proof. We will construct a representation of Ŝ′(2, 0) in C^{\frac{∞}{2} +∗}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)) using the quadratic expansions on the generators of the Heisenberg algebra H(𝔤) and Clifford algebra C(𝔤). Then we will extend it to a representation of Ŝ(2, 0) using the 𝒮ℒ(2) introduced in section 2.4.

Let us write down the elements of the basis (3.5) of S′(2, 0) in terms of the quadratic expansions as follows:

L_{n} = −Σ_{u,m}m : τ(u_{m+n})ε(u′_{m}) : −Σ_{u,m}m : β(u_{m})γ(u′_{m−n}) :,  (4.1)
E_{n} = −(1/2)iΣ_{m}[γ(e′_{m})γ(f′_{1−m+n}) + γ(f′_{m})γ(e′_{1−m+n}) + 2γ(h′_{m})γ(h′_{1−m+n})],  (4.2)
H_{n} = −Σ_{u,m} : γ(u′_{m+n})β(u_{m}) :,  (4.3)
F_{n} = −(1/2)iΣ_{m}[β(e_{m})β(f_{1−m−n}) + β(f_{m})β(e_{1−m−n}) + (1/2)β(h_{m})β(h_{1−m−n})], (4.4)
h_{n} = Σ_{u,m}γ(u′_{m+n})τ(u_{m}),  (4.5)
p_{n} = Σ_{u,m}mβ(u_{m−n})ε(u′_{m}),  (4.6)
y_{n} = iΣ_{m}[β(e_{m})τ(f_{1−m−n}) + β(f_{m})τ(e_{1−m−n}) + (1/2)β(h_{m})τ(h_{1−m−n})],  (4.7)

x_{n} = −iΣ_{m}[(1 − m + n)(ε(e′_{1−m+n})γ(f′_{m}) + ε(f′_{1−m+n})γ(e′_{m}) + 2ε(h′_{1−m+n})γ(h′_{m}))],

(4.8)

Set

ℒ_{n} = −L_{−n} + (1/2)(1 − n)H_{n}.  (4.9)

One can check that the commutation relations (3.14)-(3.22) are satisfied.

Notice that L =< L_{n} >_{n∈ℤ} is the Witt algebra described in section 2.2, and ŜL(2) =< E_{n}, H_{n}, F_{n} >_{n∈ℤ} is the loop algebra of SL(2) =< E, H, F > defined in section 2.3:

E_{0} = E, H_{0} = H, F_{0} = F.  (4.10)

Let us show that the elements (4.1)-(4.8) act on C^{\frac{∞}{2} +∗}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)).

Notice that these elements commute with τ(𝔤_{0}). Recall that

[L_{n}, θ(𝔤_{0})] = [SL(2), θ(𝔤_{0})] = 0,  (4.11)

see (2.2.14) and (2.3.11). Then it follows from the commutation relations

[L_{n}, E_{k}] = (1 + n + k)E_{k−n}, [L_{n}, F_{k}] = (k − n − 1)F_{k−n}, [E_{n}, F_{k}] = H_{n+k}  (4.12)

that

[ŜL(2), θ(𝔤_{0})] = 0.  (4.13)

Let us show that

[h_{n}, θ(𝔤_{0})] = 0.  (4.14)

Recall that

θ(u_{i}) = ρ(u_{i}) + π(u_{i}) = Σ_{v,j}(: τ([u_{i}, v_{j}])ε(v′_{j}) : + : β([u_{i}, v_{j}])γ(v′_{j}) :),  (4.15)

where u, v run through the standard basis {e, h, f} of 𝔰𝔩(2), and i, j ∈ ℤ. Notice that

[h_{n}, ε(u′_{m})] = γ(u′_{m+n}),  (4.16)
[h_{n}, β(u_{m})] = τ(u_{m−n}).  (4.17)

Thus for any i ∈ ℤ

[h_{n}, θ(u_{i})] = −Σ_{v,j} : τ([u, v]_{i+j})γ(v′_{j+n}) : +Σ_{v,j} : τ([u, v]_{i+j−n})γ(v′_{j}) := 0.  (4.18)

In particular, we have (4.14). Let us show that

[p_{n}, θ(𝔤_{0})] = 0.  (4.19)

Notice that

[p_{n}, τ(u_{m})] = mβ(u_{m−n}),  (4.20)
[p_{n}, γ(u_{m})] = −(m + n)ε(u′_{m+n}).  (4.21)

Thus for any i ∈ ℤ

[p_{n}, θ(u_{i})] = Σ_{v,j} : (i + j)β([u, v]_{i+j−n})ε(v′_{j}) : +Σ_{v,j} : β([u, v]_{i+j})(−(j + n))ε(v′_{j+n}) :=

Σ_{v,j} : iβ([u, v]_{i+j−n})ε(v′_{j}) : .  (4.22)

In particular, we have (4.19).

Similarly, one can show that

[y_{n}, θ(𝔤_{0})] = 0, [x_{n}, θ(𝔤_{0})] = 0.  (4.23)

To show that we obtain a representation of Ŝ′(2, 0) in C^{\frac{∞}{2} +∗}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)), one should check the commutation relations on the vacuum vector 1 (see theorem 2.1. of [P2]). Then we get the central charge 9.

To extend this representation to a representation of Ŝ(2, 0), recall that

0 → Ŝ′(2, 0) → Ŝ(2, 0) → ℂF → 0,  (4.24)

where F is a derivation of S′(2, 0) defined by

[F, x_{k}] = h_{k+1}, [F, p_{k}] = y_{k+1}.  (4.25)

Let the action of F on C^{\frac{∞}{2} +∗}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)) be given by the quadratic expansion (2.4.3). Notice that the commutators of the element (2.4.3) with x_{k} and p_{k}, differ from h_{k+1} and y_{k+1}, respectively, by some linear combinations of the terms, which involve τ(u_{0}). Since τ(𝔤_{0})c = 0 for any c ∈ C^{\frac{∞}{2} +∗}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)), then a representation of Ŝ(2, 0) in C^{\frac{∞}{2} +∗}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)) is well-defined.

■

Corollary 4.1. There is a representation of Ŝ(2, 0) in H^{\frac{∞}{2} +∗}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)) with central charge 9. Proof. We will prove a more general statement in theorem 5.3. Conjecture. (cf. [P1]). The space H^{\frac{∞}{2} +∗}(𝔤, 𝔤_{0}, S^{\frac{∞}{2} +∗}(𝔤)) is generated by the vacuum vector 1 under the action of Ŝ(2, 0).

5. The case of the loop algebra of a Lie algebra with a non-degenerate invariant symmetric form.

Let us consider a more general situation. Let V be a complex finite-dimensional vector space, Ṽ = V ⊗ ℂ[t, t^{−1}]. Thus Ṽ = ⊕_{n∈ℤ}Ṽ_{n}, where Ṽ_{n} = V ⊗ t^{n}. Using the polarization

Ṽ = Ṽ_{+} + Ṽ_{−},  (5.1)

where

Ṽ_{+} = ⊕_{n>0}Ṽ_{n}, Ṽ_{−} = ⊕_{n≤0}Ṽ_{n}  (5.2)

one can define the semi-infinite symmetric and exterior power spaces S^{\frac{∞}{2} +∗}(Ṽ) and Λ^{\frac{∞}{2} +∗}(Ṽ). Let

W^{\frac{∞}{2} +∗}(Ṽ) = S^{\frac{∞}{2} +∗}(Ṽ) ⊗ Λ^{\frac{∞}{2} +∗}(Ṽ).  (5.3)

The following statement generalizes the theorem 3.1 of [P2]. Theorem 5.1. The space W^{\frac{∞}{2} +∗}(Ṽ) is a module over the superconformal algebra Ŝ′(2, 0) with central charge 3dimV. Proof. Let element u run through a fixed basis of V. Let {u_{n}} be a basis of Ṽ_{n}, {u′_{n}} be the dual basis. Define

L_{n} = −Σ_{u,m}[m : τ(u_{m+n})ε(u′_{m}) : +m : β(u_{m+n})γ(u′_{m}) :]  (5.4)
E_{n} = −(1/2)iΣ_{u,m}γ(u′_{m})γ(u′_{1−m+n}),  (5.5)
H_{n} = −Σ_{u,m} : γ(u′_{m+n})β(u_{m}) :  (5.6)
F_{n} = −(1/2)iΣ_{u,m}β(u_{m})β(u_{1−m−n}),  (5.7)
h_{n} = Σ_{u,m}γ(u′_{m+n})τ(u_{m}),  (5.8)
p_{n} = Σ_{u,m}mβ(u_{m−n})ε(u′_{m}),  (5.9)
y_{n} = iΣ_{u,m}β(u_{m})τ(u_{1−m−n}),  (5.10)
x_{n} = −iΣ_{u,m}(1 − m + n)ε(u′_{1−m+n})γ(u′_{m}).  (5.11)

Set

ℒ_{n} = −L_{−n} + \frac{n + 1}{2}H_{n}.  (5.12)
■

Suppose that 𝔤 is a complex finite-dimensional Lie algebra, ϕ is a representation of 𝔤 in V, and <, > is a non-degenerate 𝔤-invariant symmetric form on V.

Let 𝔤̃ = 𝔤 ⊗ ℂ[t, t^{−1}] = ⊕_{n∈ℤ}𝔤̃_{n}. We extend ϕ to a representation of 𝔤̃ in Ṽ:
ϕ(g ⊗ t^{n})(v ⊗ t^{k}) = (ϕ(g)v) ⊗ t^{n+k}, for g ∈ 𝔤, v ∈ V.  (5.13)

Let u, v run through a fixed basis in V. One can define the projective representations ρ and π of 𝔤̃ in Λ^{\frac{∞}{2} +∗}(Ṽ) and S^{\frac{∞}{2} +∗}(Ṽ), respectively:

ρ(x) = Σ_{u,i∈ℤ} : τ((ϕ(x)u_{i}))ε(u′_{i}) :,  (5.14)
π(x) = Σ_{u,i∈ℤ} : β(ϕ(x)u_{i}))γ(u′_{i}) :,  (5.15)

where x ∈ 𝔤̃. The normal ordering operation is defined by

: τ(u_{i})ε(v′_{j}) := \{ \begin{matrix}τ(u_{i})ε(v′_{j}) if j ≤ 0 \\ −ε(v′_{j})τ(u_{i}) if j > 0 \end{matrix} \}  (5.16)

: β(u_{i})γ(v′_{j}) := \{ \begin{matrix}β(u_{i})γ(v′_{j}) if j ≤ 0 \\ γ(v′_{j})β(u_{i}) if j > 0 \end{matrix} \} .  (5.17)

Since the cocycles corresponding to ρ and π cancel, then θ(x) = ρ(x) + π(x) is a representation of 𝔤̃ in W^{\frac{∞}{2} +∗}(Ṽ). Let

W_{rel}^{\frac{∞}{2} +∗}(Ṽ) = \{c ∈ W^{\frac{∞}{2} +∗}(Ṽ)|τ(x)c = 0 for all x ∈ Ṽ_{0}, θ(x)c = 0 for all x ∈ 𝔤̃_{0}\}. (5.18)

By theorem 3.1,

DerS′(2, 0) =< E, H, F > ⋉S′(2, 0).  (5.19)

Let D̂erS′(2, 0) be a central extension of DerS′(2, 0). Thus

D̂erS′(2, 0) =< E, H, F > ⋉Ŝ′(2, 0).  (5.20)

Theorem 5.2. The space W_{rel}^{\frac{∞}{2} +∗}(Ṽ) is a module over D̂erS′(2, 0) with central charge 3dimV. Proof. Let {v_{i}} be a basis of V so that < v_{i}, v_{j} >= δ_{i,j}. Let u run through this basis. Let {u_{n}} be a basis of Ṽ_{n}, {u′_{n}} be the dual basis. Let us define S′(2, 0) using the quadratic expansions (5.4)-(5.12). By Theorem 5.1, there is a representation of Ŝ′(2, 0) in W^{\frac{∞}{2} +∗}(Ṽ). Since the form <, > is 𝔤-invariant, then there is an action of Ŝ′(2, 0) on W_{rel}^{\frac{∞}{2} +∗}(Ṽ).

Let us define 𝒮ℒ(2) =< E, H, F > as follows

E = iΣ_{u,m>0}mε(u′_{−m})ε(u′_{m}),  (5.21)
H = −Σ_{u,m≠0} : τ(u_{m})ε(u′_{m}) :,  (5.22)
F = −iΣ_{u,m>0}(1/m)τ(u_{m})τ(u_{−m}),  (5.23)

Notice that 𝒮ℒ(2) acts on W_{rel}^{\frac{∞}{2} +∗}(Ṽ). One can check that

[H, h_{n}] = −Σ_{u,m≠0}γ(u′_{m+n})τ(u_{m}) = −h_{n} + Σ_{u}γ(u′_{n})τ(u_{0}),  (5.24)
[H, p_{n}] = p_{n},  (5.25)
[H, x_{n}] = x_{n},  (5.26)
[H, y_{n}] = −iΣ_{u,m≠1−n}β(u_{m})τ(u_{1−m−n}) = −y_{n} + iΣ_{u}β(u_{1−n})τ(u_{0}),  (5.27)
[E, h_{n}] = x_{n−1},  (5.28)
[E, y_{n}] = p_{n−1},  (5.29)
[F, x_{n}] = Σ_{u,m≠0}γ(u′_{m+n+1})τ(u_{m}) = h_{n+1} − Σ_{u}γ(u′_{n+1})τ(u_{0}),  (5.30)
[F, p_{n}] = iΣ_{u,m≠−n}β(u_{m})τ(u_{−m−n}) = y_{n+1} − iΣ_{u}β(u_{−n})τ(u_{0}).  (5.31)

Notice that the relations (5.24)-(5.31) coinside with the relations (3.50), (3.51) up to some terms which involve the elements τ(u_{0}). Since the action of any element τ(u_{0}) on W_{rel}^{\frac{∞}{2} +∗}(Ṽ) is trivial, then a representation of D̂erS′(2, 0) in W_{rel}^{\frac{∞}{2} +∗}(Ṽ) is well-defined.

■

Lemma 5.1. Let 𝔤 be a complex finite-dimensional Lie algebra with a non-degenerate invariant symmetric form <, >. Let {v_{i}} be a basis of 𝔤 so that < v_{i}, v_{j} >= δ_{i,j}. Let u run through this basis. Let {u_{n}} be a basis of 𝔤̃_{n}, {u′_{n}} be the dual basis. Let 𝒮ℒ(2) be defined by (5.21)-(5.23). Then H^{\frac{∞}{2} +∗}(𝔤̃, 𝔤̃_{0}, S^{\frac{∞}{2} +∗}(𝔤̃)) is a module over 𝒮ℒ(2). Proof. Let 𝔩 = ⊕_{n∈ℤ}𝔩_{n} be a complex graded Lie algebra, so that there exists a 2-cocycle γ on 𝔩 such that γ|_{𝔩_{n}×𝔩_{−n}} is non-degenerate if n ∈ ℤ\0 and zero otherwise.

It was shown in [FGZ] that in this case there is an 𝒮ℒ(2)-module structure on the relative semi-infinite cohomology of 𝔩. Namely, identifying 𝔩′_{n} with 𝔩_{−n}, one obtains 𝒮ℒ(2) =< E, H, F > where

E = (i/2)Σ_{n}ε(e_{n})ε(e′_{n}),  (5.32)
F = (i/2)Σ_{n}τ(e_{n})τ(e′_{n}),  (5.33)
H = −Σ_{n} : τ(e_{n})ε(e′_{n}) : .  (5.34)

Notice that in our case there exists a 2-cocycle γ on 𝔤̃ with the desired property:

γ(g_{1} ⊗ t^{n}, g_{2} ⊗ t^{m}) = n < g_{1}, g_{2} > δ_{n,−m}, for g_{1}, g_{2} ∈ 𝔤.  (5.35)

Then the formulas (5.32)-(5.34) give exactly the 𝒮ℒ(2) defined by (5.21)-(5.23). Thus there is an action of 𝒮ℒ(2) on H^{\frac{∞}{2} +∗}(𝔤̃, 𝔤̃_{0}, S^{\frac{∞}{2} +∗}(𝔤̃)).

■

Theorem 5.3. Let 𝔤 be a complex finite-dimensional Lie algebra with a non-degenerate invariant symmetric form. Then H^{\frac{∞}{2} +∗}(𝔤̃, 𝔤̃_{0}, S^{\frac{∞}{2} +∗}(𝔤̃)) is a module over D̂erS′(2, 0) with central charge 3dim𝔤. Proof. By definition, W_{rel}^{\frac{∞}{2} +∗}(𝔤̃) = C^{\frac{∞}{2} +∗}(𝔤̃, 𝔤̃_{0}, S^{\frac{∞}{2} +∗}(𝔤̃)). Hence by theorem 5.2, there is a representation of D̂erS′(2, 0) in C^{\frac{∞}{2} +∗}(𝔤̃, 𝔤̃_{0}, S^{\frac{∞}{2} +∗}(𝔤̃)). By theorem 3.1,

DerS′(2, 0) =< E, H, F > ⋉S′(2, 0).

According to Lemma 5.1, the elements E, H, F act on H^{\frac{∞}{2} +∗}(𝔤̃, 𝔤̃_{0}, S^{\frac{∞}{2} +∗}(𝔤̃)). Recall that

S′(2, 0)¯1 =< h_{n}, p_{n}, x_{n}, y_{n} >_{n∈ℤ} .

By corollary 2.1 of [P2],

[h_{n}, d] = [p_{n}, d] = 0.  (5.36)

Thus h_{n} and p_{n} act on H^{\frac{∞}{2} +∗}(𝔤̃, 𝔤̃_{0}, S^{\frac{∞}{2} +∗}(𝔤̃)). Since

[E, h_{n}] = x_{n−1}, [F, p_{n}] = y_{n+1},  (5.37)

then there is an action of S′(2, 0)¯1 on H^{\frac{∞}{2} +∗}(𝔤̃, 𝔤̃_{0}, S^{\frac{∞}{2} +∗}(𝔤̃)). Finally, one obtains an action of S′(2, 0)¯0 on H^{\frac{∞}{2} +∗}(𝔤̃, 𝔤̃_{0}, S^{\frac{∞}{2} +∗}(𝔤̃)), since

S′(2, 0)¯0 = [S′(2, 0)¯1, S′(2, 0)¯1].  (5.38)
■

Corollary 5.1. Let 𝔤 be a complex finite-dimensional Lie algebra with a non-degenerate invariant symmetric form. Then H^{\frac{∞}{2} +∗}(𝔤̃, 𝔤̃_{0}, S^{\frac{∞}{2} +∗}(𝔤̃)) is a module over Ŝ(2, 0) with central charge 3dim𝔤.

In the conclusion we would like to propose a problem. In this work we described a representation of the superconformal algebra Ŝ(2, α) in a relative semi-infinite Weil complex and relative semi-infinite cohomology in the case when α = 0. Problem. Is there an action of (central extension) of S(2, α) on (some) semi-infinite Weil complex for an arbitrary α ∈ ℂ ?

REFERENCES

[F] B. Feigin, Semi-infinite cohomology of Kac-Moody and Virasoro Lie algebras, Russian Math. Surveys, 39 (1984), pp 155-156.

[FF] B. Feigin, E. Frenkel, Semi-infinite Weil Complex and the Virasoro Algebra, Commun. Math. Phys. 137 (1991), pp 617-639. Erratum: Commun. Math. Phys. 147 (1992), pp 647-648.

[FST] B.L. Feigin, A. M. Semikhatov, I. Yu. Tipunin Equivalence between chain categories of representations of affine 𝔰𝔩(2) and N = 2 superconformal algebras., hep-th/9701043

[FGZ] I. Frenkel. H. Garland, G. Zuckerman, Semi-infinite cohomology and string theory,

Proc. Natl. Acad. Sci. U.S.A. 83, (1986), pp 8442-8446.

[Fu] D. Fuks, Cohomology of infinite-dimensional Lie algebras , Consultants Bureau, New York and London, 1986.

[G] E. Getzler, Two-dimensional topological gravity and equivariant cohomology, Comm.

Math. Phys. 163 (1994) no 3, 473-489.

[GM] S. I. Gelfand, Yu. I. Manin, Methods in homological algebra, vol.1, Introduction to the theory of cohomology, and derived categories, Nauka, Moscow, 1988 (Russian).

[GL] H. Garland, J. Lepowsky, Lie algebra homology and the Macdonald-Kac formulas,

Invent. Math. 34 (1976), pp 37-76.

[K-VDL] V. G. Kac, J. W. Van De Leur, On Classification of Superconformal Algebras in

Strings-88, World Scientific 1989, pp 77-106.

[P1] E. Poletaeva, Semi-infinite cohomology of the loop algebra of 𝔰𝔩(2), Comptes Rendus de l’Académie des Sciences, t. 322, Série I, 1996, pp 1019-1023.

[P2] E. Poletaeva, Semi-infinite Weil complex and N = 2 superconformal algebra I, MPIpreprint 97-78.