Light-cone gauge in string theory

Light-cone gauge uses residual conformal transformations to make the target-space coordinate $X^{+}$ linear in worldsheet time. The Virasoro constraints then determine $X^{-}$ from the transverse fields.

This is the string-worldsheet realization of light cone gauge.

On a patch with ${\alpha }_0^{+}\ne 0$, use worldsheet diffeomorphisms and worldsheet supersymmetry to fix ${\alpha }_{n\ne 0}^{+}=0$ and ${\psi }_r^{+}=0$. The Virasoro constraints and superconformal current constraints then determine the longitudinal modes by the displayed transverse bilinears. In the Ramond sector, reduction of the fermionic zero modes includes the zero-mode Dirac constraint; fixing only nonzero modes does not complete the reduction. Eight transverse bosons and eight transverse fermions remain in the critical ten-dimensional RNS string.

Fixing nonzero longitudinal oscillators leaves the center-of-mass symplectic term, transverse oscillator terms, and a zero-mode multiplier imposing the mass-shell condition. On a patch with $p^{+}\ne 0$, further choosing $x^{+}$ as time solves that constraint for $p^{-}=[p_{\perp }^2+(N-a)/{\alpha }^{\prime }]/(2p^{+})$. This distinguishes partial oscillator gauge fixing from the subsequent time choice.