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Choose a reference metric in each conformal class. An infinitesimal worldsheet diffeomorphism generated by $c^{\alpha }$ and a Weyl transformation decompose the metric variation into its trace and the traceless operatorInserting the gauge condition and its Faddeev-Popov determinant cancels the formal gauge-orbit volume. Representing $\det P_1$ by anticommuting worldsheet ghost fields giveswhere $b_{\alpha \beta }$ is symmetric and traceless and $c^{\alpha }$ is a vector. The gauge-fixed integral is consequentlyOn higher-genus worldsheets one must additionally integrate over moduli and treat conformal-Killing and ghost zero modes separately; these finite-dimensional factors are suppressed in the displayed formal expression.

For positive lapse, eliminating momentum gives $(\dot{X}-u{X}^{\prime }{)}^2/(2e)-e{T}^2{X}^{\prime 2}/2$. This is the Polyakov action for ${\gamma }_{ab}=\left(\begin{array}{cc}{u}^2-e^2{T}^2& u\\ u& 1\end{array}\right)$, up to a Weyl transformation. Thus $e=1/T,u=0$ is conformal gauge.

Replacing the flat target metric in the Polyakov action by a spacetime metric $g_{\mu \nu }(X)$ gives the displayed string sigma model. The embedding fields $X^{\mu }$ map the worldsheet into target spacetime, while $h_{ab}$ is an independent worldsheet metric. Classical worldsheet diffeomorphisms and Weyl transformations permit conformal gauge; quantum consistency additionally requires vanishing sigma-model beta functions.

The Polyakov path integral integrates over string worldsheet metrics as well as embedding fields, modulo worldsheet diffeomorphisms and Weyl transformations. Summing surface topologies gives perturbative two-dimensional quantum gravity coupled to the coordinate matter fields. Gauge fixing retains worldsheet moduli and introduces ghosts; a conformal gauge is locally flat, not a global flat metric on every genus. Cancellation of the worldsheet Weyl anomaly makes the critical bosonic version consistent.

Under ${\gamma }_{\mu \nu }\mapsto e^{2\omega }{\gamma }_{\mu \nu }$, the quadratic kinetic density of a $p$-brane scales as $e^{(p-1)\omega }$ and its constant density scales as $e^{(p+1)\omega }$. At $p=1$ the first factor is one and the constant term in the auxiliary-metric brane action vanishes. The traced metric equation is then an identity, leaving a Weyl transformation freedom. For $p\ne 1$, the metric equation instead identifies the auxiliary and induced worldvolume metrics.

A Weyl transformation locally rescales a metric tensor, $g\mapsto e^{2\omega }g$, while retaining the same points and coordinates. Applied to a worldsheet metric, it leaves the embedding $X^{\mu }$ unchanged, and the classical Polyakov action is invariant under these Weyl transformations. Applied to a spacetime metric, a Weyl rescaling can relate different frame conventions, such as a string metric and an Einstein metric; the accompanying field normalizations must be transformed as well. It is not itself a coordinate transformation.

The intrinsic Lorentzian metric in the Polyakov action is independent of the string embedding map. Its equation relates it conformally to the induced worldsheet metric. A Weyl transformation changes the intrinsic metric without changing the classical action.

A worldsheet Weyl anomaly is the quantum failure of the gauge-fixed string path integral to remain invariant under a Weyl transformation. Its local coefficients are the sigma-model beta functions, while its constant part is controlled by the total central charge.