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The modular invariance of the one-loop partition function of the closedbosonic string in four dimensions in the presence of certain homogeneous exactpp-wave backgrounds is studied. In the absence of an axion field the partitionfunction is found to be modular invariant. In the presence of an axion fieldmodular invariace is broken. This can be attributed to the light-cone gaugewhich breaks the symmetry in the $\sigma$-, $t$-directions. Recovery of thisbroken modular invariance suggests the introduction of twists in theworld-sheet directions. However, one needs to go beyond the light-cone gauge tointroduce such twists.Comment: 17 pages, added reference

CLOSED BOSONIC STRING PARTITION FUNCTION IN TIME INDEPENDENT EXACT pp-WAVE BACKGROUND