Extended Superconformal Algebras on Ads 3

We study a supersymmetric extension of the Virasoro algebra on the boundary of the anti-de Sitter space-time AdS3. Using the free field realization of the currents, we show that the world-sheet affine Lie superalgebras osp(1|2)(1), sl(1|2)(1) and sl(2|2)(1) provide the boundary N = 1, 2 and 4 extended superconformal algebras, respectively. The duality between the type IIB string theory on AdS3 × S ×M4, where M = K3 or T , and two-dimensional N = 4 superconformal field theory on a symmetric product of M [1, 2, 3], is one of interesting examples of the AdS/CFT correspondence[4]. Conformal symmetry on the boundary of AdS3, firstly introduced by Brown and Henneaux [5], is realized as the chiral algebra of the boundary conformal field theory associated with the SL(2, R) Chern-Simons theory[6]. On the other hand, Giveon et al. [3] constructed the boundary Virasoro algebra from the string theory on AdS3. The generators of the algebra are identified as the global charges associated with the world-sheet sl(2, R) current algebra, which is expressed in terms of the free fields[7]. They also constructed (a part of) the N = 4 superconformal algebra from the superstrings on AdS3 × S × T . Their boundary algebra, however, is not manifestly supersymmetric since the supercurrents are introduced by the bosonization of the world-sheet fermions and the algebra is defined up to the picture changing operator[8]. In the present paper, we study the boundary extended superconformal algebra realized in a manifestly supersymmetric way. By replacing the world-sheet affine Lie algebra sl(2, R) to an affine Lie superalgebra which includes the subalgebra sl(2, R) and employing the free field realization of the currents, we will obtain the extended superconformal algebra which acts on the boundary AdS3 superspace. In particular, we will show that the world-sheet affine Lie superalgebras osp(1|2)(1), sl(1|2)(1) and sl(2|2)(1) provide the boundary N = 1, 2 and 4 extended superconformal algebras, respectively. We begin with reviewing the boundary Virasoro algebra associated with AdS3. The anti-de Sitter space AdS3 is the hypersurface −U2−V +X+Y 2 = −l2 embedded in the flat space R. In the coordinates (ρ, τ, φ) defined by U = l cosh ρ sin τ , V = l cosh ρ cos τ , X = l sinh ρ cosφ, Y = l cosh ρ sinφ, the metric is given by ds l2 = − cosh ρdτ 2 + sinh ρdφ + dρ. (1) In the AdS3 space, there is SL(2, R)L×SL(2, R)R symmetry. The generators of SL(2, R)L read [1] L0 = i∂u, L±1 = ie ±iu ( coth 2ρ∂u − 1 sinh 2ρ ∂v ∓ i 2 ∂ρ )