Conformal gauge

Conformal gauge uses worldsheet diffeomorphisms and Weyl transformations to set $g_{\alpha \beta }=e^{2\varphi }{\eta }_{\alpha \beta }$ locally.

After conformal gauge fixing, independent reparameterizations ${\sigma }^{+}\mapsto f({\sigma }^{+})$ and ${\sigma }^{-}\mapsto \tilde{f}({\sigma }^{-})$ remain, accompanied by a compensating Weyl transformation.

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.

The worldsheet metric equation in conformal gauge sets the stress tensor to zero, $T_{++}=T_{--}=0$. In oscillator language these are the Virasoro constraints.

Critical closed bosonic strings have independent left- and right-moving Virasoro constraints. Positive modes in both sectors annihilate physical kets, while both zero-mode conditions impose the mass shell and closed-string level matching. Quotient by physical null string states, equivalently use the appropriate BRST cohomology. With $L_0={\alpha }^{\prime }{k}^2/4+N$, the result is $N=\tilde{N}$ and $M^2=4(N-1)/{\alpha }^{\prime }$. Negative matter modes are adjoints rather than additional ket-annihilation constraints.

The covariant physical-state conditions impose positive-mode Virasoro constraints on kets and the adjoint conditions on bras, with intercept $a$ in the zero mode. They are a string version of Gupta-Bleuler quantization. Imposing every positive and negative matter mode on the same state would conflict with the Virasoro central extension. In critical bosonic string theory, $a=1$ and the matter central charge is $26$, canceled by the reparameterization ghosts.

With the oscillator symplectic form of an open string, the quadratic generators $L_n={\sum }_k{\alpha }_k\cdot {\alpha }_{n-k}/2$ obey $\{{\alpha }_k,L_n\}=-ik{\alpha }_{k+n}$. Their Poisson brackets close without a central term, so they are first-class constraints. Quantization requires normal ordering and can produce the Virasoro central extension.