  Exact construction of superstring field theory in some background fields is
very important. We construct the low energy NS-NS sector of superstring field
action in the pp-wave background with the flux of NS-NS antisymmetric tensor
field (NS-NS pp-wave) without gauge fixing up to the second-order where the
action is world-sheet BRST invariant. Here we use the word "covariant" in a
invariant theory for a symmetric transformation of the pp-wave background which
is not the Lorentz transformation in the flat background. Moreover we prove the
exact correspondence between this low energy action and the second-order
perturbation of supergravity action in the same background. We also prove the
correspondence of the gauge transformation in both the actions. This
construction is based on the BRST first quantization of superstrings in the
pp-wave background in our previous paper.
