= String sigma model in an isotropic plane wave
{title2=$ds^2=-2,dX^+dX^--m^2|X|^2(dX^+)^2+dX^I dX^I$}
In <conformal gauge> and <light-cone gauge in string theory> $X^+=\kappa\tau$, the transverse part of the <string sigma model in a spacetime metric> reduces to
$$
S_\perp=\frac T2\int d\tau d\sigma\,\left(\dot X^I\dot X^I-X'^IX'^I-m^2\kappa^2X^IX^I\right).
$$
The omitted longitudinal term is a total time derivative. Each transverse coordinate obeys a massive <wave equation>. If the spatial period is $L$, the longitudinal momentum is $p^+=T\kappa L$ in this normalization. Keeping the gauge slope explicit prevents missing a factor of $p^+$ when changing conventions.
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