String channel propagator on a plumbing collar

Use the CFT plumbing relation $zw=q$ and a specified nondegenerate state pairing. In a closed bosonic channel, let $L_0^m,{\bar{L}}_0^m$ denote matter conformal weights and fix intercepts $a,\bar{a}$ (equal to $1$ in the critical bosonic convention). Define the local collar contribution for a state pair by $\frac{d^{2}q}{\mathrm{\mid }q{\mathrm{\mid }}^2}{q}^{L_0^m-a}{\bar{q}}^{{\bar{L}}_0^m-\bar{a}},d^{2}q=d(\mathrm{Re}q)d(\mathrm{Im}q),$ with a branch of $\log q$ fixed on the collar chart. For an open channel use a real plumbing parameter $0<q<ϵ$ and $dq\mathrm{/}q{q}^{L_0^m-a}$. The full local channel is the state-pairing sum of these terms only when that sum is declared complete and converges as required by the chosen sewing input. Set $t=-\log \mathrm{\mid }q\mathrm{\mid }$; this is the logarithmic long-tube coordinate. For an RNS sector, the chosen state/picture convention must supply its own $L_0$ and intercept; the bosonic value is not silently reused.

The following schematic records only the two local coordinate disks and their collar identification; it does not depict or assert a global compactified moduli space.

The circles indicate punctured components with small coordinate disks removed; the double arrow labels the annular collar identification from Plumbing family for punctured Riemann surfaces. No amplitude proof below relies on this schematic.

Plumbing family for punctured Riemann surfaces

The CFT plumbing carrier is draft, and its global analytic sewing consequences are not used as established facts here. In the bosonic massless channel $L_0^m={\bar{L}}_0^m=1$ with $a=\bar{a}=1$, so the radial collar measure is $d^{2}q\mathrm{/}\mathrm{\mid }q{\mathrm{\mid }}^2$; convergence then depends on the state coefficient and regulator.