Time-dependent AdS/CFT duality and null singularity

We consider AdS/CFT correspondence for time-dependent IIB backgrounds in this paper. The supergravity solutions we construct are supersymmetric pp-waves on AdS and may have null singularity in the bulk. The dual gauge theory is also constructed explicitly and is given by a time-dependent supersymmetric Yang-Mills theory living on the boundary. Apart from the usual terms that are dictated by the geometry, our gauge theory action features also a time-dependent axion coupling and a time-dependent gauge coupling. Both of which are necessary due to the presence of a nontrivial dilaton and axion profile in the supergravity solution. The proposal is supported by a precise matching in the symmetries and functional dependence on the null coordinate of the two theories. As applications, we show how the bulk Einstein equation may be reproduced from the gauge theory. We also study and compare the behaviour of the field theory two-point functions. We find that the two-point function computed by using duality is different from that by doing a direct field theory computation. In particular the spacetime singularity is not seen in our gauge theory result, suggesting that the spacetime singularity may be resolved in the gauge theory.