Local punctured-moduli perturbative amplitude form

Let $m,n,g\in {\mathbb{Z}}_{\ge 0}$, let $U\subset {\mathbb{C}}^m$ be a connected open chart, and let $\pi :\mathcal{C}\to U$ be a specified analytic family of pointed compact worldsheets in the sense of Pointed analytic worldsheet and family, with connected fibers of common genus $g$ and exactly $n$ marked sections. In this item, a fiber is called stable when $2g-2+n>0$; only smooth fibers are considered, not nodal curves. Assume the family has local coordinates and tangent representatives ${\mu }_a,{\bar{\mu }}_a$. Assume an instance supplies vertex representatives, the bosonic $bc$ system, and a multilinear local CFT correlator functional on their ordered products. Require its evaluated coefficients to be $C^{\mathrm{\infty }}$ functions on $U$ and require worldsheet-coordinate equivariance: simultaneous passive coordinate transport of all field and density inputs, including the fixed ghost zero-mode normalization, leaves the evaluation on the same tangent vectors unchanged. Thus any anomaly multiplier on these overlaps is assumed trivial in the chosen local trivialization. This is additional local model data, not implied by the individual primary-field laws. Fix a partition of the external insertions into unintegrated representatives used to fix residual conformal-Killing zero modes and integrated representatives carrying their stated insertion forms. Define the chartwise amplitude form by its evaluation on the tangent basis as the declared correlator functional with the ordered insertions $⟨{\prod }_{i\in I_0}c\bar{c}{V}_i{\prod }_{j\mathrm{\notin }{I}_0}{V}_j^{\mathrm{i}\mathrm{n}\mathrm{t}}{\prod }_{a=1}^m(b,{\mu }_a)(\bar{b},{\bar{\mu }}_a){⟩}_U,$ where $I_0$, the ghost/picture conventions, the correlator functional, and any puncture-coordinate integrations are fixed as part of the instance. If the model uses a different but equivalent gauge-fixed representative, its conversion is additional input. This defines only the local form on $U$; it does not construct a global determinant-line section, compactified moduli space, regulator, or integral.

Pointed analytic worldsheet and family

The pointed-worldsheet supplier's current readiness hash is stale, and [A1] is an explicit per-instance input rather than a proved global construction. The definition is intentionally weaker than a global string measure.