Semi-infinite Weil complex and N = 2 superconformal algebra I.

Elena Poletaeva

Contents.

0. Introduction.

1. Semi-infinite Weil complex.

2. An action of the N = 2 superconformal algebra on the semi-infinite Weil complex of a loop algebra.

3. Semi-infinite Weil complex of a graded vector space.

4. Semi-infinite Weil complex of the standard modules over the Witt algebra.

0. Introduction.

Recently there was a significant progress in the understanding of infinite-dimensional graded Lie algebras. It turns out that the most appropriate generalization of the classical Lie algebra cohomology is the semi-infinite cohomology introduced by B.Feigin. He considered the semi-infinite wedge products of the elements of Lie algebra and defined a differential on such products. The cohomology of the corresponding chain complex (unbounded in both directions) is called the semi-infinite cohomology [F].

In [FF] B.Feigin and E.Frenkel constructed a semi-infinite analogue of the famous Weil complex based on the space

W^{∗}(𝔤) = S^{∗}(𝔤′) ⊗ Λ^{∗}(𝔤′),

where 𝔤 is a Lie algebra.

B.Feigin and E.Frenkel studied this complex in the case when 𝔤 is the Virasoro algebra, and computed the semi-infinite cohomology of 𝔤 with coefficients in its “adjoint semiinfinite symmetric powers” H^{\frac{∞}{2} +∗}(𝔤, S^{\frac{∞}{2} +∗}(𝔤)). The semi-infinite Weil complex of the Virasoro algebra has also been studied by physicists in relation to the string theory.

E. Getzler has shown that the semi-infinite Weil complex of the Virasoro algebra is a module over the N = 2 superconformal algebra [G].

In this work we consider the semi-infinite Weil complex W^{\frac{∞}{2} +∗}(𝔤̃) of the loop algebra 𝔤̃ = 𝔤 ⊗ ℂ[t, t^{−1}] of a complex finite-dimensional Lie algebra 𝔤.

In section 1 we review a construction of the semi-infinite Weil complex of a complex graded Lie algebra. In section 2 we show that the semi-infinite Weil complex of the loop algebra 𝔤̃ of a complex finite-dimensional Lie algebra 𝔤 is a module over the N = 2 superconformal algebra with central charge 3dim𝔤. We also obtain an action of the N = 2 superconformal algebra on the semi-infinite cohomology H^{\frac{∞}{2} +∗}(𝔤̃, S^{\frac{∞}{2} +∗}(𝔤̃)).

In section 3 we observe that given a complex graded vector space, one can repeat the construction of the semi-infinite Weil complex. In section 4 we describe an action of the N = 2 superconformal algebra on the semi-infinite Weil complex related to a standard module of type F_{λ,μ} over the Witt algebra.

This work was done during my staying in Bonn. I would like to thank Max-Planck-Institut für Mathematik for hospitality and for excellent working conditions.

1. Semi-infinite Weil complex.

The semi-infinite Weil complex was introduced by B. Feigin and E. Frenkel in [FF]. Recall the necessary definitions. Let 𝔤 = ⊕_{n∈ℤ}𝔤_{n} be a graded Lie algebra over ℂ, so that dim𝔤_{n} < ∞. Let 𝔤′ = ⊕_{n∈ℤ}𝔤′_{n} be the restricted dual of 𝔤. Linear space 𝔤 ⊕ 𝔤′ carries non-degenerate exterior and symmetric forms: (,) and {, }. Let H(𝔤) and C(𝔤) be the quotients of the tensor algebra T^{∗}(𝔤 ⊕ 𝔤′) by the ideals generated by the elements of the form xy − yx − (x, y) and xy + yx − {x, y}, respectively, where x, y ∈ 𝔤 ⊕ 𝔤′. We fix N ∈ ℤ. Let 𝔫_{+} = ⊕_{n>N}𝔤_{n}, 𝔫 = ⊕_{n≤N}𝔤_{n}.

The symmetric algebra S^{∗}(𝔫_{+} ⊕ 𝔫′) is a subalgebra of H(𝔤) and the exterior algebra Λ^{∗}(𝔫_{+} ⊕ 𝔫′) is a subalgebra of C(𝔤). Let S^{\frac{∞}{2} +∗}(𝔤), Λ^{\frac{∞}{2} +∗}(𝔤) be the representations of H(𝔤) and C(𝔤) induced from the trivial representations < 1_{S} > et < 1_{Λ} > of S^{∗}(𝔫_{+} ⊕ 𝔫′) and Λ^{∗}(𝔫_{+} ⊕ 𝔫′), respectively. Thus we obtain some semi-infinite analogues of symmetric and exterior power modules. 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′. Notice that each element of S^{\frac{∞}{2} +∗}(𝔤) and of Λ^{\frac{∞}{2} +∗}(𝔤) is a finite linear combination of the monoms of the type γ(x′_{1}) … γ(x′_{k})β(y_{1}) … β(y_{m})1_{S} and of the type ε(x′_{1}) … ε(x′_{k})τ(y_{1}) … τ(y_{m})1_{Λ}, respectively, where x′_{1}, …, x′_{k} ∈ 𝔫′_{+}, y_{1}, … y_{m} ∈ 𝔫.

Let {e_{i}}_{i∈ℤ} be a homogeneous basis of 𝔤. In other words, for any i ∈ ℤ, e_{i} ∈ 𝔤_{n} for some n ∈ ℤ, and if e_{i} ∈ 𝔤_{n}, then e_{i+1} ∈ 𝔤_{n} or e_{i+1} ∈ 𝔤_{n+1}. Let {e′_{i}}_{i∈ℤ} be the dual basis. Let i_{0} ∈ ℤ be such that e_{i_{0}} ∈ 𝔤_{N} and e_{i_{0}+1} ∈ 𝔤_{N+1}.

Notice that one can think of Λ^{\frac{∞}{2} +∗}(𝔤) as the vector space spanned by the elements w = e′_{i_{1}} ∧ e′_{i_{2}} ∧ … such that there exists K(w) ∈ ℤ such that i_{k+1} = i_{k} − 1 for k > K(w). Then 1_{Λ} = e′_{i_{0}} ∧ e′_{i_{0}−1} ∧ … is a vacuum vector in this space. The actions of ε(x′), τ(x) are, respectively, the exterior multiplication and contraction in the space of semi-infinite exterior products.

Let ρ and π be the projective representations of 𝔤 in Λ^{\frac{∞}{2} +∗}(𝔤) and S^{\frac{∞}{2} +∗}(𝔤), respectively, defined as follows :

ρ(x) = Σ_{i∈ℤ} : τ([x, e_{i}])ε(e′_{i}) :,  (1.1)
π(x) = Σ_{i∈ℤ} : β([x, e_{i}])γ(e′_{i}) :,  (1.2)

where x ∈ 𝔤, and where the double colons : : denote a normal ordering operation :

: τ(e_{j})ε(e′_{i}) := \{ \begin{matrix}τ(e_{j})ε(e′_{i}) if i ≤ i_{0} \\ −ε(e′_{i})τ(e_{j}) if i > i_{0} \end{matrix} \}  (1.3)

: β(e_{j})γ(e′_{i}) := \{ \begin{matrix}β(e_{j})γ(e′_{i}) if i ≤ i_{0} \\ γ(e′_{i})β(e_{j}) if i > i_{0} \end{matrix} \}  (1.4)

Thus

ρ(x)1_{Λ} = π(x)1_{S} = 0 for x ∈ 𝔤_{0}  (1.5)

and

[ρ(x), ρ(y)] = ρ([x, y]) + c_{Λ}(x, y),  (1.6)
[π(x), π(y)] = π([x, y]) + c_{S}(x, y),  (1.7)

where x, y ∈ 𝔤 and c_{Λ}, c_{S} are 2-cocycles. Notice that c_{Λ} = −c_{S}. Let

W^{\frac{∞}{2} +∗}(𝔤) = S^{\frac{∞}{2} +∗}(𝔤) ⊗ Λ^{\frac{∞}{2} +∗}(𝔤).  (1.8)

Since the cocycles corresponding to the projective representations cancel, the representation θ(x) = ρ(x) + π(x) of 𝔤 in W^{\frac{∞}{2} +∗}(𝔤) is well-defined. We obtain the semi-infinite Weil complex

\{W^{\frac{∞}{2} +∗}(𝔤), d + h\},  (1.9)

where

d = Σ_{i<j} : τ([e_{i}, e_{j}])ε(e′_{j})ε(e′_{i}) : +Σ_{i,j} : β([e_{j}, e_{i}])γ(e′_{i})ε(e′_{j}) :  (1.10)
h = Σ_{i}γ(e′_{i})τ(e_{i}).  (1.11)

The differential d is the analogues of the classical differential for the Lie algebra cohomology and h is the Koszul differential. Notice that

d^{2} = 0, h^{2} = 0, {d, h} = 0, (d + h)^{2} = 0.  (1.12)

Notice that if 𝔤 is a finite-dimensional Lie algebra, 𝔫_{+} = 𝔤, 𝔫 = 0, then (1.9) is the classical Weil complex. 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),  (1.13)

and we obtain a ℤ-grading on W^{\frac{∞}{2} +∗}(𝔤):

W^{\frac{∞}{2} +i}(𝔤) = ⊕_{2l+j=i}S^{\frac{∞}{2} +l}(𝔤) ⊗ Λ^{\frac{∞}{2} +j}(𝔤).  (1.14)

As in the case of the classical Weil complex, on can construct two filtrations, for one of them (F_{1}^{p}, l + m ≥ p) the complex is acyclic, the second term of the spectral sequence associated to the other filtration (F_{2}^{p}, 2l ≥ p) is the semi-infinite cohomology of Lie algebra 𝔤 with coefficients in its “adjoint semi-infinite symmetric powers” H^{\frac{∞}{2} +∗}(𝔤, S^{\frac{∞}{2} +∗}(𝔤)). B. Feigin and E. Frenkel calculated this cohomology in the case when 𝔤 is the Virasoro algebra ([FF]). The next natural example is the loop algebra of a complex finite-dimensional Lie algebra.

2. An action of the N = 2 superconformal algebra on the semi-infinite Weil complex of a loop algebra.

In this section we fix N = 0. Thus 𝔫_{+} = ⊕_{n>0}𝔤_{n}, 𝔫 = ⊕_{n≤0}𝔤_{n}.

Recall that the N = 2 superconformal algebra is spanned by the Virasoro generators ℒ_{n}, Heisenberg generators H_{n}, two fermionic fields G_{r}^{±}, and a central element C, where n ∈ ℤ, r ∈ ℤ + 1/2, and where the nonvanishing commutation relations are as follows [FST]:

[ℒ_{n}, ℒ_{m}] = (n − m)ℒ_{n+m} + \frac{C}{12}(n^{3} − n)δ_{n,−m},

[ℒ_{n}, H_{m}] = −mH_{n+m},

[ℒ_{n}, G_{r}^{±}] = (\frac{n}{2} − r)G_{n+r}^{±},

[G_{r}^{+}, G_{s}^{−}] = 2ℒ_{r+s} + (r − s)H_{r+s} + \frac{C}{3}(r^{2} − \frac{1}{4})δ_{r,−s},  (2.1)

[H_{n}, H_{m}] = \frac{C}{3}nδ_{n,−m},

[H_{n}, G_{r}^{±}] = ±G_{n+r}^{±}.

Theorem 2.1. Let 𝔤̃ = 𝔤 ⊗ ℂ[t, t^{−1}], where 𝔤 is a complex finite-dimensional Lie algebra. Then W^{\frac{∞}{2} +∗}(𝔤̃) is a module over the N = 2 superconformal algebra with central charge 3dim𝔤. Proof. We fix a basis in 𝔤. Let element u run through this basis. Define

L_{n} = −Σ_{u,m∈ℤ}[m : τ(u_{m+n})ε(u′_{m}) : +m : β(u_{m+n})γ(u′_{m}) :]
H_{n} = −Σ_{u,m∈ℤ} : γ(u′_{m+n})β(u_{m}) :
h_{n} = Σ_{u,m∈ℤ}γ(u′_{m+n})τ(u_{m}),  (2.2)
p_{n} = Σ_{u,m∈ℤ}mβ(u_{m−n})ε(u′_{m}).

Notice that h_{0} is the Koszul differential and p_{0} is an anologue of the homotopy operator [Fu]. Let

ℒ_{n} = −L_{−n} + \frac{n + 1}{2}H_{n}.  (2.3)

The nonvanishing commutation relations are:

[ℒ_{n}, ℒ_{k}] = (n − k)ℒ_{n+k},
[ℒ_{n}, H_{k}] = −kH_{n+k},
[ℒ_{n}, h_{k}] = (1/2)(n − 2k + 1)h_{n+k},
[ℒ_{n}, p_{k}] = (1/2)(n − 2k − 1)p_{n+k},  (2.4)
[H_{n}, h_{k}] = h_{n+k},
[H_{n}, p_{k}] = −p_{n+k},
[h_{n}, p_{k}] = ℒ_{n+k} − (1/2)(k − n + 1)H_{n+k}.

Set

G_{n− \frac{1}{2}}^{+} = \sqrt{2}h_{n}, G_{n+ \frac{1}{2}}^{−} = \sqrt{2}p_{n}.  (2.5)

Then the central extension of the superalgebra spanned by ℒ_{n}, H_{n}, G_{n− \frac{1}{2}}^{+} , G_{n+ \frac{1}{2}}^{−} is the N = 2 superconformal algebra. To show that we obtain a representation of this algebra in W^{\frac{∞}{2} +∗}(𝔤̃), one should check the commutation relations on the vacuum vector 1 = 1_{S} ⊗1_{Λ}. For example,

[H_{n}, H_{−n}]1 = (H_{n}H_{−n} − H_{−n}H_{n})1.  (2.6)

Let n > 0. Then H_{−n}1 = 0. Hence

[H_{n}, H_{−n}]1 = −H_{−n}(−Σ_{u}Σ_{m=1−n}^{0}γ(u′_{m+n})β(u_{m}))1 =
(−Σ_{u}Σ_{m=1}^{n}γ(u′_{m−n})β(u_{m}))(Σ_{u}Σ_{m=1−n}^{0}γ(u′_{m+n})β(u_{m}))1 =  (2.7)
− Σ_{u}Σ_{m=1−n}^{0}γ(u′_{m})β(u_{m+n})γ(u′_{m+n})β(u_{m})1 = −dim𝔤(−n)1,

since γ(u′_{i})β(u_{i}) − β(u_{i})γ(u′_{i}) = 1, and u runs through a basis of 𝔤. Notice that

[H_{n}, H_{m}]1 = 0, if m ≠ −n.  (2.8)

Hence

[H_{n}, H_{m}]1 = ndim𝔤δ_{n,−m}1.  (2.9)

Thus the central charge is 3dim𝔤. The other commutation relations on the vacuum vector 1 are calculated in the same way.

■

Corollary 2.1. The space H^{\frac{∞}{2} +∗}(𝔤̃, S^{\frac{∞}{2} +∗}(𝔤̃)) is a module over the N = 2 superconformal algebra with central charge 3dim𝔤. Proof. We will show that the N = 2 superconformal algebra defined by (2.2) commutes with d. Recall that

d = d_{1} + d_{2},  (2.10)

where

d_{1} = (1/2)Σ_{u,v,i,j} : τ([u_{i}, v_{j}])ε(v′_{j})ε(u′_{i}) :,  (2.11)
d_{2} = Σ_{u,v,i,j} : β([u_{i}, v_{j}])γ(v′_{j})ε(u′_{i}) :,  (2.12)

u, v run through a fixed basis of 𝔤, and i, j ∈ ℤ. Notice that

[L_{n}, τ(u_{m})] = −mτ(u_{m+n}),
[L_{n}, ε(u′_{m})] = (m − n)ε(u′_{m−n}),  (2.13)
[L_{n}, β(u_{m})] = −mβ(u_{m+n}),
[L_{n}, γ(u′_{m})] = (m − n)γ(u′_{m−n}),

Thus

[L_{n}, d_{1}] = (1/2)Σ_{u,v,i,j} : −(i + j)τ([u, v]_{i+j+n})ε(v′_{j})ε(u′_{i}) : +
: τ([u_{i}, v_{j}])(j − n)ε(v′_{j−n})ε(u′_{i}) : + : τ([u_{i}, v_{j}])ε(v′_{j})(i − n)ε(u′_{i−n}) := 0. (2.14)

Similarly,

[L_{n}, d_{2}] = Σ_{u,v,i,j} : −(i + j)β([u, v]_{i+j+n})γ(v′_{j})ε(u′_{i}) : +
: β([u_{i}, v_{j}])(j − n)γ(v′_{j−n})ε(u′_{i}) : + : β([u_{i}, v_{j}])γ(v′_{j})(i − n)ε(u′_{i−n}) := 0. (2.15)

Notice that

[H_{n}, γ(u′_{m})] = γ(u′_{m+n}),  (2.16)
[H_{n}, β(u_{m})] = −β(u_{m−n}).

Thus

[H_{n}, d_{1}] = 0,  (2.17)

and

[H_{n}, d_{2}] = Σ_{u,v,i,j} : −β([u, v]_{i+j−n})γ(v′_{j})ε(u′_{i}) : +β([u, v]_{i+j})γ(v′_{j+n})ε(u′_{i}) := 0. (2.18)

Next,

[h_{n}, ε(u′_{m})] = γ(u′_{m+n}),  (2.19)
[h_{n}, β(u_{m})] = τ(u_{m−n}).

Thus

[h_{n}, d_{1}] = (1/2)Σ_{u,v,i,j} : −τ([u, v]_{i+j})γ(v′_{j+n})ε(u′_{i}) : + : τ([u, v]_{i+j})ε(v′_{j})γ(u′_{i=n}) :=
− Σ_{u,v,i,j} : τ([u, v]_{i+j})γ(v′_{j+n})ε(u′_{i}) :,  (2.20)

[h_{n}, d_{2}] = Σ_{u,v,i,j} : τ([u, v]_{i+j−n})γ(v′_{j})ε(u′_{i}) : + : β([u, v]_{i+j})γ(v′_{j})γ(u′_{i+n}) :=
Σ_{u,v,i,j} : τ([u, v]_{i+j})γ(v′_{j+n})ε(u′_{i}) :,  (2.21)

since Σ_{u,v,i,j} : β([u, v]_{i+j})γ(v′_{j})γ(u′_{i+n}) := 0. Hence

[h_{n}, d_{2}] = −[h_{n}, d_{1}].  (2.22)

Finally,

[p_{n}, τ(u_{m})] = mβ(u_{m−n}),
[p_{n}, γ(u′_{m})] = −(m + n)ε(u′_{m+n}).  (2.23)

Thus

[p_{n}, d_{1}] = (1/2)Σ_{u,v,i,j} : (i + j)β([u, v]_{i+j−n})ε(v′_{j})ε(u′_{i}) :,  (2.24)

[p_{n}, d_{2}] = Σ_{u,v,i,j} : −β([u, v]_{i+j})(j + n)ε(v′_{j+n})ε(u′_{i}) :=  (2.25)

Σ_{u,v,i,j} : −β([u, v]_{i+j−n})jε(v′_{j})ε(u′_{i}) := −(1/2)Σ_{u,v,i,j} : (j + i)β([u, v]_{i+j−n})ε(v′_{j})ε(u′_{i}) : .

Hence

[p_{n}, d_{2}] = −[p_{n}, d_{1}].  (2.26)
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3. Semi-infinite Weil complex of a graded vector space.

Let V = ⊕_{n∈ℤ}V_{n} be a graded vector space over ℂ, such that dimV_{n} < ∞. Let V′ = ⊕_{n∈ℤ}V′_{n} be its restricted dual. We fix N ∈ ℤ. Let

V_{+} = ⊕_{n>N}V_{n}, V_{−} = ⊕_{n≤N}V_{n}.  (3.1)

Using the space T^{∗}(V ⊕ V′) one can construct the infinite-dimensional Heisenberg and Clifford algebras H(V) and C(V), and define their representations induced from the trivial representations 1_{S} and 1_{Λ} of S^{∗}(V_{+} ⊕ V′_{−}) and Λ^{∗}(V_{+} ⊕ V′_{−}), respectively. We denote these representations by S^{\frac{∞}{2} +∗}(V) and Λ^{\frac{∞}{2} +∗}(V).

Let 𝔤 = ⊕_{n∈ℤ}𝔤_{n} be a graded Lie algebra over ℂ. Let ϕ be a representation of 𝔤 in V:

ϕ(𝔤_{n})V_{k} ⊂ V_{k+n}.  (3.2)

Let {v_{i}}_{i∈ℤ} be a homogeneous basis in V, {v′_{i}}_{i∈ℤ} be the dual basis. Let i_{0} be such that e_{i_{0}} ∈ 𝔤_{N} and e_{i_{0}+1} ∈ 𝔤_{N+1}.

One can define the projective representations ρ and π of 𝔤 in Λ^{\frac{∞}{2} +∗}(V) and S^{\frac{∞}{2} +∗}(V), respectively:

ρ(x) = Σ_{i∈ℤ} : τ(ϕ(x)v_{i})ε(v′_{i}) :,  (3.3)
π(x) = Σ_{i∈ℤ} : β(ϕ(x)v_{i})γ(v′_{i}) :,  (3.4)

where x ∈ 𝔤, and the normal ordering operation is defined by analogy to (1.3) and (1.4). Let

W^{\frac{∞}{2} +∗}(V) = S^{\frac{∞}{2} +∗}(V) ⊗ Λ^{\frac{∞}{2} +∗}(V).  (3.5)

We obtain the representation θ(x) = ρ(x)+π(x) of 𝔤 in W^{\frac{∞}{2} +∗}(V), since the corresponding cocycles cancel. We fix N = 0. Theorem 3.1. Let Ṽ = V ⊗ ℂ[t, t^{−1}], where V is a complex finite-dimensional vector space. Then W^{\frac{∞}{2} +∗}(Ṽ) is a module over the N = 2 superconformal algebra with central charge 3dimV. Proof. The elements of the N = 2 superconformal algebra are defined by the quadratic expansions (2.2), where now u runs through a fixed basis of the vector space V.

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4. Semi-infinite Weil complex of the standard modules over the Witt algebra.

Let L = ⊕_{i∈ℤ}ℂL_{i} be the Witt algebra:

[L_{i}, L_{j}] = (i − j)L_{i+j}.  (4.1)

Let λ, μ ∈ ℂ. Let F_{λ,μ} = ⊕_{j∈ℤ}ℂv_{j} be the module over L defined as follows:

ϕ(L_{i})v_{j} = (−j + μ − (i − 1)λ)v_{i+j}.  (4.2)

We fix i_{0} ∈ ℤ and consider the corresponding polarization of F_{λ,μ}:

F_{λ,μ}^{+} = ⊕_{i>i_{0}}ℂv_{i}, F_{λ,μ}^{−} = ⊕_{i≤i_{0}}ℂv_{i}.  (4.3)

Then W^{\frac{∞}{2} +∗}(F_{λ,μ}) is defined. Let i_{0} = 0. Theorem 4.1. The space W^{\frac{∞}{2} +∗}(F_{λ,μ}) is a module over the N = 2 superconformal algebra with central charge 3 − 6λ. Proof. Notice that the action of the element L_{n} on W^{\frac{∞}{2} +∗}(F_{λ,μ}) is given by the quadratic expansion

θ(L_{n}) = Σ_{m}(−m + n + μ − nλ + λ)(: τ(v_{m})ε(v′_{m−n}) : + : β(v_{m})γ(v′_{m−n}) :).  (4.4)

Let us define the action of the N = 2 superconformal algebra on W^{\frac{∞}{2} +∗}(F_{λ,μ}) using the following quadratic expansions:

θ(H_{n}) = λΣ_{m} : τ(v_{m})ε(v′_{m+n}) : +(λ − 1)Σ_{m} : β(v_{m})γ(v′_{m+n}) : +μδ_{n,0},  (4.5)
θ(h_{n}) = Σ_{m}γ(v′_{m+n})τ(v_{m}),  (4.6)
θ(p_{n}) = Σ_{m}(m − μ − (n + 1)λ)β(v_{m−n})ε(v′_{m}),  (4.7)
θ(ℒ_{n}) = −θ(L_{−n}) + \frac{n + 1}{2}θ(H_{n}).  (4.8)

One can check that θ is a representation of the N = 2 superconformal algebra, and calculate the central charge checking the commutation relations on the vacuum vector 1 = 1_{S} ⊗ 1_{Λ}. For example, let n > 0. Then

θ([H_{n}, H_{−n}])1 = −θ(H_{−n})θ(H_{n})1 =
− θ(H_{−n})(λΣ_{m=1−n}^{0}τ(v_{m})ε(v′_{m+n}) + (λ − 1)Σ_{m=1−n}^{0}β(v_{m})γ(v′_{m+n}))1 = (4.9)
− λ^{2}Σ_{m=1−n}^{0}τ(v_{m+n})ε(v′_{m})τ(v_{m})ε(v′_{m+n})1−
(λ − 1)^{2}Σ_{m=1−n}^{0}β(v_{m+n})γ(v′_{m})β(v_{m})γ(v′_{m+n})1 =
(−λ^{2}n − (λ − 1)^{2}(−n))1 = n(1 − 2λ)1,

since ε(v′_{i})τ(v_{i}) + τ(v_{i})ε(v′_{i}) = 1, and γ(v′_{i})β(v_{i}) − β(v_{i})γ(v′_{i}) = 1. Hence,

θ([H_{n}, H_{m}])1 = n(1 − 2λ)δ_{n,−m}1.  (4.10)

Thus the central charge is 3 − 6λ. The other commutation relations on the vacuum vector 1 are calculated in the same way.

■

Corollary 4.1. In the case when λ = −1, μ = 1, the module F_{λ,μ} is the adjoint representation of L. The corresponding central charge is 9 (cf. [G]).

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 [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.