D-branes in pp-wave spacetime with nonconstant NS-NS flux

We find classical solutions of D-branes in pp-wave spacetime with nonconstant NS-NS flux. We also present Dp-Dp' bound state solutions in this background. We further analyze the supersymmetric properties of these brane solutions by solving the type IIB killing spinor equations explicitly.

In this section we present classical solutions of Dp as well as Dp − Dp ′ -branes with nonconstant NS − NS three form flux transverse to brane worldvolume. We start by writing down the supergravity solutions of D-string in pp-wave background with nonconstant NS-NS three form flux. The metric, dilaton and the field strengths are given by: with b(x i ) and K(x i ) satisfying the equations ✷b(x i ) = 0 and ✷K(x i ) = −(∂ i b j ) 2 respectively and f 1 = 1+ Q 1 r 6 is the harmonic function in the transverse space. We have checked that the above solution satisfies type IIB field equations. For constant three form flux this solution reduces to that of ref [20]. All other Dp-brane (p = 2, ..., 5) solutions can be found out by applying T -duality along x 5 , ..., x 8 directions. For example: the classical solution for a system of D3-brane in such a background is given by: with b(x i ) and K(x i ) satisfying the equations ✷b(x i ) = 0 and ✷K(x i ) = −(∂ i b j ) 2 respectively and f 3 = 1 + Q 3 r 4 is the harmonic function satisfying the Green function equation in the transverse space. Now we present classical solution of D1−D5 system as an example of p−p ′ bound state in these background. The supergravity solution for a such system is given by: respectively and f 1 = 1 + Q 1 r 2 and f 5 = 1 + Q 5 r 2 are the harmonic functions of D1 and D5-branes in common transverse space. One can check that the above ansatz do satisfy type IIB field equations.

Supersymmetry Analysis
In this section we present the supersymmetry of the solutions described earlier in section (2).
The supersymmetry variation of dilatino and gravitino fields of type IIB supergravity in ten dimension, in string frame, is given by [25,26]: where we have used (µ, ν, ρ) to describe the ten dimensional space-time indices, and hat's represent the corresponding tangent space indices. Solving the above two equations for D-string solution as given in (2.1), we get several conditions on the spinors. First, the dilatino variation gives: Gravitino variation gives the following conditions on the spinors: In writing the above equations we have used the brane supersymmetry condition: (1 − Γ1234)ǫ 0 ± = 0. (3.12) Therefore in this case, the D-string solution (2.1), preserves 1/8 supersymmetry. Similarly, one can show that the D3-brane solution (2.2) also preserves 1/8 supersymmetry.
Next, we will analyze the supersymmetry properties of (D1 − D5) system that is described in eqn. (2.3) of the previous section.
The dilatino variation gives the following conditions on the spinors: On the other hand, the gravitino variation gives: In writing down the above gravitino variations we have once again made use of the brane conditions: and Taking derivative of the eqn. (3.16) with respect to ∂k and subtracting the derivative of ∂k equation with respect to ∂ˆi, we get (∂k∂ˆibĵ)Γ+ĵǫ ± = 0, (3.19) which can be satisfied for nonconstant ∂ˆibĵ only if Γ+ǫ ± = 0.
Using Γ+ǫ ± = 0 and brane supersymmetry conditions (3.17) and (3.18) , the dilatino condition (3.13) is satisfied. Using Γ+ǫ ± = 0, the supersymmetry condition (3.16) is solved by spinors: ǫ ± = exp(− 1 8 ln(f 1 f 5 ))ǫ 0 ± , with ǫ 0 ± being a function of x + only. Since ǫ 0 ± is independent of x i and (∂ i b j ) is a function of x i only, the gravitino variation gives the following conditions to have nontrivial solutions: and Once again, the number of supersymmetries depend on the existence of solutions of equation (3.20). For the particular case when H +12 = H +34 , the D1 − D5 bound state solution (2.3) also preserves 1/8 supersymmetry.
In this paper we have constructed the supergravity solutions of Dp as well as Dp−Dp ′ branes in pp-wave background with nonconstant NS − NS flux. The supersymmetric properties of these solutions are also verified by analyzing the type IIB killing spinor equations explicitly. All the solutions presented here are shown to preserve 1/8 supersymmetry and the supernumarary supersymmetry is absent for the background presented in this paper. D-brane solutions with nonconstant RR flux can be found out by applying S-duality transformation on the solutions presented here, which will be generalization of those given in [20]. The D-brane solutions presented here have the interpretation of D-branes in nonsupersymmetric sigma model of [10]. It is also desirable to analyze them from the worldvolume point of view following the procedure of [11].