Closed Bosonic String Partition Function in Time Independent Exact PP-Wave Background

The modular invariance of the one-loop partition function of the closed bosonic string in four dimensions in the presence of general time independent exact pp-wave background is studied. In the absence of an axion field the partition function is found to be modular invariant. In the presence of an axion field modular invariace is broken. This can be attributed to the light-cone gauge which breaks the symmetry in the $\sigma$-, $t$-directions