Thermal Partition Function of Superstring on Compactified PP-Wave

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We study the thermal partition function of superstring on the pp-wave background with the circle compactification along a transverse direction. We calculate it in the two ways: the operator formalism and the path-integral calculation. The former gives the finite result with no subtlety of the Wick rotation, which only contains the contributions of physical states. On the other hand, the latter yields the manifestly modular invariant expression, even though we only have the winding modes along the transverse circle (no Kaluza-Klein excitations). We also check the equivalence of these two analyses. The DLCQ approach makes the path-integration quite easy. Remarkably, we find that the contributions from the transverse winding sectors disappear in the non-DLCQ limit, while they indeed contribute in the DLCQ model, depending non-trivially on the longitudinal quantum numbers.