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For an infinitesimal Weyl transformation $\delta h^{ab}=-2\omega h^{ab}$, the definition of the stress tensor givesThe insertions are Weyl invariant by assumption, so varying the normalized path integral givesConsequently the stated condition $⟨h^{ab}{T}_{ab}{\varphi }_1\cdots {\varphi }_n⟩=0$ makes the correlation function Weyl invariant. Quantum failure of this condition is the worldsheet Weyl anomaly.

The $D$ free embedding scalars contribute $c_X=D$. The ordinary diffeomorphism ghosts contribute $c_{bc}=-26$. The fermionic $(m,n)$ pair has $h_n=0$ and contributes $-2$. Each of the $D/2$ fermionic $({\chi }_a,{\psi }^a)$ pairs has $h_{\psi }=1/2$ and contributes $+1$. Each bosonic $({\beta }_r,{\gamma }^r)$ pair has $h_{\gamma }=-1/2$ and contributes $+11$, and there are two such pairs. The total holomorphic central charge is thereforeCancellation of the worldsheet Weyl anomaly requires $c_{\mathrm{tot}}=0$, soThe antiholomorphic sector gives the same condition.

Take $D$ embedding coordinates and give one coordinate the background charge $q$. The remaining $D-1$ free bosons contribute $D-1$, while the distinguished coordinate contributes $1+6{\alpha }^{\prime }{q}^2$, so the matter central charge isThe worldsheet ghosts contributes $-26$. Cancellation of the worldsheet Weyl anomaly therefore requireswhich is real for $D<26$ and produces a noncritical bosonic string.

In target-space language the dilaton is linear in this coordinate, so the local string coupling $g_s=e^{\Phi }$ changes exponentially. One end of the target direction is weakly coupled and the other is strongly coupled. Consequently string perturbation theory is trustworthy only in the weak-coupling region, rather than throughout the full background.

The target metric is a coupling of the string nonlinear sigma model. Quantum consistency requires the gauge-fixed worldsheet theory to preserve Weyl invariance, so its sigma-model beta functions must vanish. With no B-field or varying dilaton, the metric beta function begins asTherefore vanishing of the worldsheet Weyl anomaly requiresto leading order in ${\alpha }^{\prime }$: the target-space metric must be Ricci-flat at this order.