Superconformal algebras and Lie superalgebras of the Hodge theory

We observe a correspondence between the zero modes of superconformal algebras S(2, 1) and W (4) ([8]) and the Lie superalgebras formed by classical operators appearing in the Kähler and hyper-Kähler geometry. 1 Lie superalgebras of the Hodge theory 1.1 Kähler manifolds Let M = (M2n, g, I, ω) be a compact Kähler manifold of real dimension 2n, where g is a Riemannian (Kähler) metric, I is a complex structure on M , and ω is the corresponding closed 2-form defined by ω(x, y) = g(x, I(y)) for any vector fields x and y [10]. A number of classical operators on the Dolbeault algebra A(M) of complex differential forms on M is well-known [5]: the exterior differential d and its holomorphic and antiholomorphic parts, ∂ and ∂̄, and dc = i(∂− ∂̄), their dual operators, and the associated Laplacians. Recall that ∂ : A(M) → A(M), ∂̄ : A(M) → A(M), d = ∂ + ∂̄. (1.1) The Hodge operator ⋆ : A(M) −→ A(M), satisfies ⋆2 = (−1)p+q on A(M). Accordingly, the Hodge inner product is defined on each of A(M): (φ,ψ) = ∫ M φ ∧ ⋆ψ̄. Recall that △ = dd + dd = 2△∂ = 2△∂̄ . In addition, A(M) admits an sl(2)-module structure, where sl(2) = 〈E,H,F 〉 and the generators satisfy [E,F ] = H, [H,E] = 2E, [H,F ] = −2F. (1.2) The operator E : A(M) → Ap+1,q+1(M) is defined by E(φ) = φ ∧ ω. (Clearly, ω is a (1, 1)-form). Let F = E : A(M) → Ap−1,q−1(M) be its dual operator, and H|Ap,q(M) = p+ q−n. According to the Lefschetz theorem, there exists the corresponding action of sl(2) on H(M) [5]. These operators satisfy a series of identities, known as the Hodge identities [5]. Let K be the Lie superalgebra, whose even part is spanned by sl(2) = 〈E,H,F 〉 and the Laplace operator △, and the odd part is spanned by the Copyright c © 2002 by E Poletaeva 2 E Poletaeva differentials d, d, dc, d ∗ c . The non-vanishing commutation relations in K are (1.2) and the following relations (see [5]): [d, d] = [dc, d ∗ c ] = △, [H, d] = d, [H, d] = −d, [H, dc] = dc, [H, d∗c ] = −dc , (1.3) [E, d] = −dc, [E, d∗c ] = d, [F, d] = d∗c , [F, dc] = −d. Thus K = sl(2)+⊃ hei(0|4), where hei(0|4) is the Heisenberg Lie superalgebra: hei(0|4)1̄ = 〈d, dc〉 ⊕ 〈dc, d∗〉 is a direct sum of two isotropic subspaces with respect to the nondegenerate symmetric form given by (d, d) = (dc, d ∗ c) = 1, and hei(0|4)0̄ = 〈△〉 is the center. The isotropic subspaces are standard sl(2)-modules. Since sl(2) ≃ sp(2), the following is a natural generalization. 1.2 Hyper-Kähler manifolds Let M be a compact hyper-Kähler manifold. By definition, M is a Riemannian manifold endowed with three complex structures I, J , and K, such that I ◦J = −J ◦ I = K and M is Kähler with respect to each of the complex structures I, J and K. Let ωI , ωJ and ωK be the corresponding closed 2-forms on M . Having fixed one of the complex structures, for example, I, we obtain the Hodge theory as in the Kähler case. For each of the complex structures I, J and K, the operators Ei, Ej and Ek and their dual operators Fi, Fj and Fk with respect to the Hodge inner product are naturally defined. One can also define differentials, dc and (d l c) , where l = i, j, k for I, J and K. Set dc = d, (d 1 c) ∗ = d. (1.4) Let Q = {1, i, j, k} be the set of indices satisfying the quaternionic identities. In the hyper-Kähler case there is a natural action of the Lie algebra so(5) ≃ sp(4) on H(M) [13]. The sp(4) is spanned by the operators Ei, Fi,Ki, where i runs through the set Q \ {1}, and H. The non-vanishing commutation relations are [Ei, Fi] = H, [H,Ei] = 2Ei, [H,Fi] = −2Fi; (1.5) [Ei, Fj ] = Kij, [Ki,Kj ] = −2Kij , Kij = −Kji, [Ki, Ej ] = −2Eij , Eij = −Eji, [Ki, Fj ] = −2Fij , Fij = −Fji, i 6= j. Let H be a Lie superalgebra, whose even part is spanned by the sp(4) and the Laplace operator △, and the odd part is spanned by the differentials dc and (dc), where l ∈ Q. Thus dimH = (11|8). The non-vanishing commutation relations in H are (1.5) and the following relations (cf. [2, 14]): [dc, (d l c) ] = △, [H, dc] = dc, [H, (dc)] = −(dc), (1.6) [Ei, (d l c) ] = −d c , d c = −dc, [Fi, d l c] = (d il c ) , (d c ) ∗ = −(dc), [Ki, d l c] = −d c , [Ki, (dc)] = −(d c ). Thus H = sp(4)+⊃ hei(0|8), where hei(0|8) is the Heisenberg Lie superalgebra: hei(0|8)1̄ = 〈dc, (dc)| l ∈ Q〉, hei(0|8)0̄ = 〈△〉, where 〈△〉 is the center. hei(0|8)1̄ = V1 ⊕ V2 is a Superconformal algebras and Lie superalgebras of the Hodge theory 3 direct sum of two isotropic subspaces with respect to the non-degenerate symmetric form: (dc , (d b c) ) = δab for a, b ∈ Q; V1 = 〈dc + √ −2dc − dc , dc − √ −2dc + dc , (dc) ∗ + (dc) + √ −2(dc ), √ −2(dc) + (dc) − (dc)〉, V2 = 〈dc − √ −2dc − dc , dc − √ −2dc + dc , (dc) ∗ − (dc) − √ −2(dc ), √ −2(dc) + (dc) + (dc)〉. (1.7) The subspaces V1 and V2 are irreducible sp(4)-modules. 2 Superconformal algebras A superconformal algebra (SCA [8, 9]) is a complex Z-graded Lie superalgebra g = ⊕igi, such that g is simple, g contains the centerless Virasoro algebra, i.e. the Witt algebra, L = ⊕n∈ZCLn with the commutation relations [Lm, Ln] = (m− n)Lm+n as a subalgebra, and adL0 is diagonalizable with finite-dimensional eigenspaces: gi = {x ∈ g | [L0, x] = ix}, so that dim gi < C, where C is a constant independent of i. (Other definitions of superconformal algebras, embracing central extensions, are also popular, see [4]; for an intrinsic definition see [6]). In general, a SCA is spanned by a number of fields; the Virasoro field is among them. The basic example of a SCA is W (N). Let Λ(N) be the Grassmann algebra in N variables θ1, . . . , θN . Let Λ(1, N) = C[t, t −1] ⊗ Λ(N) be a supercommutative superalgebra with natural multiplication and with the following parity of generators: p(t) = 0̄, p(θi) = 1̄ for i = 1, . . . , N . By definition, W (N) is the Lie superalgebra of all derivations of Λ(1, N). Let ∂t stand for ∂ ∂t and ∂i stand for ∂ ∂θi . The superalgebra W (N) contains a one-parameter family of SCAs S(N,α). By definition, S(N,α) = {D ∈ W (N) | Div(tD) = 0} for α ∈ C, (2.1) where Div(f∂t + ∑N i=1 fi∂i) = ∂tf + ∑N i=1(−1)∂ifi for f, fi ∈ Λ(1, N). The derived superalgebra S(N,α) = [S(N,α), S(N,α)] is simple. Let g = S(2, 1) or W (4), respectively. Let Ln = −t∂t − 1 2 (n+ 2)t N