ar X iv : h ep - t h / 99 11 19 3 v 1 2 4 N ov 1 99 9 INJE - TP - 99 - 8 Greybody factor for D 3 - branes in B field

We calculate the effect of noncommutative spacetime on the greybody factor on the supergravity side. For this purpose we introduce a system of D3-branes with a constant NS B-field along their world volume directions (x 2 , x 3). In the B → ∞ limit, one has the D=7 black hole solution including the noncommutative effect. Considering the propagation of minimally coupled scalar in this background, we derive an exact form of the greybody factor in B field. It turns out that, although the two expressions of σ B =0 l and σ B=0 l are the same form, one finds σ B =0 l > σ B=0 l. This means that the presence of B-field (the noncommutativity) suppresses the curvature effect surrounding the black hole. 1 Recently noncommutative geometry has attracted much interest in studying on string and M-theory in the B-field [1–7]. For simplicity, we consider supergravity solutions which are related to D3 branes with NS B fields. According to the AdS/CFT correspondence [8], the near horizon geometry of D=7 black hole solution can describe the large N limit of noncommutative super Yang-Mills theory (NCSYM). We take a decoupling limit to isolate the near horizon geometry from the remaining one. It turns out that the noncommutativity affects the ultra violet(UV) but not the infra red(IR) of the Yang-Mills dynamics. The NCSYM is thus not useful for studying the theory at short distances. It is well known that an NCSYM with the noncommutativity scale ∆ on a torus of size Σ is equivalent to an ordinary supersymmetric Yang-Mills theory (OSYM) with a magnetic flux if Θ = ∆ 2 /Σ 2 is a rational number [9]. The equivalence between the NCSYM and the OSYM can be understood from the T-duality of the corresponding string theory. Hence the OSYM with B-field is the proper description in the UV region, while the NCSYM takes over in the IR region. Actually, the noncommutativity comes from B → ∞ limit of the ordinary theories [5,6,10]. On the other hand, it turns out that the total number of physical degrees of freedom for the noncommutative case at any given scale coincides with the commutative case [5]. All thermodynamic quantities of the NCSYM including the entropy are the same as those of the OSYM. However, in the next order correction of the α ′-expansion, the entropy decreases in the NCSYM [11]. We remind the reader …