= Solution
The <Polyakov action> introduces an independent <worldsheet metric> $h_{ab}$:
$$
S_P=-\frac1{4\pi\alpha'}\int d^2\sigma\,\sqrt{-h}\,h^{ab}\partial_aX\cdot\partial_bX.
$$
Its redundancies are <worldsheet diffeomorphisms> and <Weyl transformations>. A <Polyakov path integral> over all metrics without dividing by this gauge volume would count physically equivalent configurations repeatedly. Fixing <conformal gauge> therefore needs the <Faddeev-Popov determinant>, not merely the substitution of a flat metric into the action.
After using the <Weyl transformation> to remove the trace of a metric variation, the infinitesimal <worldsheet diffeomorphism ghost operator> is
$$
(P_1c)_{ab}=\nabla_ac_b+\nabla_bc_a-h_{ab}\nabla_dc^d.
$$
The <Faddeev-Popov determinant> of this operator is represented by anticommuting fields: a vector <Faddeev-Popov ghost> $c^a$ and a symmetric traceless <Faddeev-Popov antighost field> $b^{ab}$. In Euclidean coordinates their <worldsheet ghost action> is, up to a consistent overall normalization,
$$
S_{\mathrm{gh}}=\frac1{2\pi}\int\sqrt h\,b^{ab}(P_1c)_{ab}\,d^2\sigma
=\frac1{2\pi}\int d^2z\,(b\bar\partial c+\bar b\partial\bar c).
$$
The two chiral copies of this <bc system> have <conformal weights> $(2,-1)$ and <anticommutators> $\{b_m,c_n\}=\delta_{m+n,0}$. They are auxiliary fields for the gauge determinant, not additional spacetime particles. Their use is different from the unwanted timelike <negative-norm string states> of covariant matter quantization.
The chiral <holomorphic stress-energy tensor> of the <bc system> is
$$
T_{\mathrm{gh}}=-2:b\partial c:-:(\partial b)c:.
$$
Double contractions in its <operator product expansion> give the <central charge of reparameterization ghosts> $c_{\mathrm{gh}}=-26$. Equivalently, a fermionic <bc system> with antighost weight $\lambda$ has $c=1-3(2\lambda-1)^2$, and $\lambda=2$ gives $-26$. The free coordinate matter contributes $c_{\mathrm m}=D$. Thus the total <worldsheet Weyl anomaly> vanishes for the flat critical <bosonic string theory> precisely when
$$
\boxed{c_{\mathrm{tot}}=D-26=0}.
$$
This statement assumes no additional matter or compensating <Liouville field theory>.
<BRST symmetry> expresses the gauge symmetry after gauge fixing by replacing its infinitesimal parameter with the <Grassmann variable> $c$. In one chiral conformal patch, its graded action has the form
$$
sX^\mu=c\partial X^\mu,\qquad sc=c\partial c,\qquad sb=T_{\mathrm{tot}};
$$
there is an analogous barred copy for a <closed string>. The <Grassmann parity> makes $s^2X=0$: the term from $sc$ cancels the term from applying $s$ to $\partial X$, while the term with $c^2$ vanishes. The gauge-fixed chiral transformations are on shell when their conservation equations are used. At the quantum level, <BRST nilpotence> also tests the anomaly.
For example, an <open string> <BRST charge> can be written using the matter <Virasoro generators> as
$$
Q_B=\sum_n c_{-n}L_n^{\mathrm m}
-\frac12\sum_{m,n}(m-n):c_{-m}c_{-n}b_{m+n}:
-a c_0.
$$
Use the <ghost oscillator vacuum> $b_{n\geq0}|0\rangle_{mathrm{gh}}=c_{n>0}|0\rangle_{mathrm{gh}}=0$ and the corresponding oscillator <normal ordering>. In this convention the last term is $-c_0$ in the critical theory. With a differently shifted ghost $L_0$, the displayed intercept must be shifted consistently; one must not count the same shift twice. Squaring $Q_B$ shows why both critical conditions are needed. Its anomalous terms, in this convention, have coefficients
$$
Q_B^2=\frac12\sum_{n\in\mathbb Z}\left[\frac{D-26}{12}(n^3-n)+2(a-1)n\right]c_{-n}c_n.
$$
The independent cubic and linear terms vanish at $D=26$, $a=1$. The equivalent chiral <BRST current> is $j_B=c(T_{mathrm m}+T_{mathrm{gh}}/2)+(3/2)\partial^2c$, with its contour integral defining the charge.
The graded identity $\{Q_B,b_n\}=L_n^{\mathrm{tot}}$ recovers the gauge constraints. In particular, on $|\psi\rangle\otimes|0\rangle_{mathrm{gh}}$, <BRST-closed> imposes $L_{n>0}^{\mathrm m}|\psi\rangle=0$ and $(L_0^{\mathrm m}-1)|\psi\rangle=0$. Physical states are classes in <BRST cohomology>,
$$
\boxed{\mathcal H_{\mathrm{phys}}=\ker Q_B/\operatorname{im}Q_B},
\qquad |\psi\rangle\sim|\psi\rangle+Q_B|\chi\rangle,
$$
at the appropriate ghost number: one for the usual open-string vertex and two for the closed-string vertex, with the closed-string zero-mode and <closed-string level matching> conditions also imposed. Since $Q_B^2=0$, an exact shift preserves closure. <BRST Ward identities> make exact insertions decouple from physical amplitudes and establish gauge independence. Together with the <no-ghost theorem for the critical bosonic string>, this explains how the auxiliary ghosts enforce gauge invariance while the physical spectrum retains positive norm.
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