Bosonic bc ghost and Beltrami moduli pairing

Let $\mathrm{\Sigma }$ be a fixed Riemann surface chart with a chosen Beltrami representative $\mu ={\mu }_{\bar{z}}^{z}d\bar{z}\otimes {\mathrm{\partial }}_z$ for a tangent direction to complex structure. Define its local antighost pairing by $(b,\mu )={\int }_{\mathrm{\Sigma }}{b}_{zz}{\mu }_{\bar{z}}^{z}dxdy,$ using a finite chart cover and compactly supported smooth representatives on each patch (and a partition of unity when more than one patch represents the same global tensor). Use dimensionless worldsheet and moduli coordinates, dimensionless Beltrami components, and the dimensionless $b,c$ normalization of the supplied ghost mode algebra; consequently $(b,\mu )$ is dimensionless. The numerical normalization of this pairing is exactly the displayed coefficient $1$ with area $dxdy$: in the ghost supplier's convention $d^{2}z=2dxdy$ this is $\frac{1}{2}\int b_{zz}{\mu }_{\bar{z}}^z{d}^{2}z$. No additional $2\pi$ or string-tension factor is included. For a tangent basis ${\mu }_1,\dots ,{\mu }_m$, the local moduli insertion is the ordered product ${\prod }_{a=1}^m(b,{\mu }_a)$ and its antiholomorphic counterpart; because $b$ is odd, the order is fixed and a change of tangent basis contributes its determinant. At fixed punctures, define the conformal-Killing space as $H^0(\mathrm{\Sigma },T_{\mathrm{\Sigma }}(-D))$, where $D$ is the marked divisor. Its zero modes are fixed by a stated gauge slice, represented by the corresponding $c$-ghost evaluations, or divided out by the stated automorphism-group volume. The convention and zero-mode saturation are data of each amplitude. No global determinant line, global gauge slice, or integration measure is included.

For a sphere with three fixed punctures the residual Möbius transformations are fixed by the three marked points; for a torus without punctures, translation remains a one-complex-dimensional conformal-Killing group and the quotient/zero-mode convention must be stated. No zero-mode count for other strata is asserted here without its actual surface and puncture data.