Plumbing family for punctured Riemann surfaces

Let $(C;p_1,\dots ,p_N;z_1,\dots ,z_N)$ be a pointed compact analytic worldsheet with the pointed-worldsheet conventions. Choose two distinct marked points with disjoint coordinate disks $D_r$ and $D_s$, using coordinates $z$ and $w$ that vanish at their respective points. For a complex parameter $q$ with $\mathrm{\mid }q\mathrm{\mid }<rs$, the local plumbing family is the analytic hypersurface

For $q\mathrm{\ne }0$, the overlap between the two punctured coordinate disks is identified by $w=q\mathrm{/}z$; its annular chart domains are $\mathrm{\mid }q\mathrm{\mid }\mathrm{/}s<\mathrm{\mid }z\mathrm{\mid }<r$ and $\mathrm{\mid }q\mathrm{\mid }\mathrm{/}r<\mathrm{\mid }w\mathrm{\mid }<s$. Define the sewn fiber precisely as follows. Let $K_z(q)$ be the closed coordinate subdisk $\mathrm{\mid }z\mathrm{\mid }\le \mathrm{\mid }q\mathrm{\mid }\mathrm{/}s$ and let $K_w(q)$ be the closed coordinate subdisk $\mathrm{\mid }w\mathrm{\mid }\le \mathrm{\mid }q\mathrm{\mid }\mathrm{/}r$. Start with the open surface $C_q^{\circ }=C\setminus (K_z(q)\cup K_w(q))$, and take its quotient by $z\sim w=q\mathrm{/}z$ on the two open annuli just specified. The equivalence relation is generated by these identifications and is the identity elsewhere. Give this quotient the topology and holomorphic charts induced from $C_q^{\circ }$; thus the small closed disks are excised before the annuli are glued. Equivalently, for any $0<a<r$, $0<b<s$ with $\mathrm{\mid }q\mathrm{\mid }<ab$, attach the closed fixture $zw=q$, $\mathrm{\mid }z\mathrm{\mid }\le a$, $\mathrm{\mid }w\mathrm{\mid }\le b$ to the complement of the open coordinate disks of radii $a,b$ along its two boundary circles. Collar coordinates extend across each seam, and the maps $w=q\mathrm{/}z$ identify this fixture description with the open-annulus quotient. This equivalent description specifies the compact Hausdorff surface without retaining duplicate annular boundary limits. For $q=0$, the local fiber is the nodal curve with equation $zw=0$; this statement is local and makes no assertion of a global stable-curve compactification or proper compactified moduli space.

A multi-plumbing consists of a finite ordered list of pairwise disjoint pairs of marked points, coordinate disks for each pair, and independent parameters $q_1,\dots ,q_R$ satisfying $\mathrm{\mid }{q}_j\mathrm{\mid }<r_j{s}_j$. The smooth sewn family is restricted to the punctured polydisk where each $q_j\mathrm{\ne }0$. The geometric identifications use the complex numbers $q_j$ themselves and require no branch of a logarithm. Any analytic expression containing powers $q_j^{L(0)}$ or $\log q_j$ additionally specifies a branch on its parameter domain.

If the coordinates are linearly rescaled to $z^{\mathrm{\prime }}=az$ and $w^{\mathrm{\prime }}=bw$ with $a,b\in {\mathbb{C}}^{\times }$, the same local relation is written $z^{\mathrm{\prime }}{w}^{\mathrm{\prime }}=q^{\mathrm{\prime }}$ with $q^{\mathrm{\prime }}=abq$. No exact finite rescaling formula for nonlinear coordinate changes is asserted by this definition.

Pointed analytic worldsheet and family

Complex manifolds, holomorphic vector bundles and Hermitian metrics

The quotient charts and the reduced nodal fiber are checked in Local smoothing and nodal fiber of plumbing. For pairwise disjoint plumbing neighborhoods, the simultaneous family is independent, up to the canonical biholomorphism fixing the complement and the labeled local data, of the order in which its pairs are sewn; this is verified in Order independence of disjoint analytic plumbing. The order-independence statement is local to the declared disjoint pairs and adds no compactified-moduli or analytic CFT claim.