String vertex datum and covariant insertion

Fix a pointed analytic worldsheet with its local coordinates. The local BRST model is input data: complex vector spaces $C_{\sigma }^{r,p}(k)$ of local field representatives, indexed by ghost number $r\in \mathbb{Z}$, picture $p$ in a specified subset of $\frac{1}{2}\mathbb{Z}$ (omit $p$ in a model without pictures), sector $\sigma$ in a specified finite set $S$, and momentum $k$. Supply linear maps $Q:C_{\sigma }^{r,p}(k)\to C_{\sigma }^{r+1,p}(k)$ satisfying $Q^2=0$. Define the BRST cohomology by $H_{\sigma }^{r,p}(k):=\ker (Q{\mathrm{\mid }}_{C_{\sigma }^{r,p}(k)})\mathrm{/}\mathrm{im}(Q{\mathrm{\mid }}_{C_{\sigma }^{r-1,p}(k)})$; square zero makes the denominator a subspace of the numerator. The field realization, its coordinate laws, and any products used below are additional supplied model data, not consequences of this quotient.

For target labels supply a finite-dimensional real vector space $E$ with a nondegenerate Lorentzian bilinear form $\eta$ of signature $(-,+,\dots ,+)$, its complexification when complex momenta are used, and $k\in E^{\ast }$ or $E^{\ast }\otimes \mathbb{C}$, respectively. Use $\mathrm{\hslash }=c=1$, so momentum has units of inverse length and $\eta$ is dimensionless. For each sector supply a polarization space $P_{\sigma }$, specified as a subspace of a finite tensor product of copies of $E\otimes \mathbb{C}$ and its dual (the scalar case is $\mathbb{C}$), and an exact admissibility subset $M_{\sigma }\subseteq (E^{\ast }\otimes \mathbb{C})\times P_{\sigma }$ encoding its sector constraints. Supply a real squared-mass label ${\mu }_{\sigma }$ of units length${}^{-2}$; negative values are permitted. Here the mass-shell predicate is ${\eta }^{-1}(k,k)+{\mu }_{\sigma }=0$. A model with additional level labels supplies these data separately for each label. The chosen physical classes form a specified subset $H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s},\sigma }^{r,p}(k)\subseteq H_{\sigma }^{r,p}(k)$.

A vertex datum is a field $V_i\in C_{{\sigma }_i}^{r_i,p_i}(k_i)$ with polarization ${ϵ}_i\in P_{{\sigma }_i}$, conformal weights $(h_i,{\bar{h}}_i)\in {\mathbb{C}}^2$, its listed indices, and a declared local-coordinate transformation law. For a primary field, its insertion density is $V_i(z,\bar{z})(dz{)}^{h_i}(d\bar{z}{)}^{{\bar{h}}_i}$, interpreted weakly under smearing for distributional fields, on the chosen local chart or specified branch data. For an integrated $(1,1)$ closed vertex use $V_i^{(1,1)}{d}^{2}z$, with $d^{2}z=dxdy$ for $z=x+iy$; for an unintegrated insertion supply a separately typed local operator $U_i\in C_{{\sigma }_i}^{r_i^U,p_i^U}(k_i)$ with its own weights $(h_i^U,{\bar{h}}_i^U)$ and coordinate law, with residual conformal-Killing zero modes fixed separately. In the elementary closed bosonic dressing convention, the supplied odd ghosts $c,\bar{c}$ have ghost number $1$, picture $0$, and weights $(-1,0),(0,-1)$; their product with $V_i$ is defined as model data and $U_i=c\bar{c}{V}_i$ has ghost number $r_i+2$, picture $p_i$, and weights $(h_i-1,{\bar{h}}_i-1)$ when that product is primary. In particular a ghost-number-zero $(1,1)$ coefficient gives a ghost-number-two $(0,0)$ unintegrated operator. Other sectors supply their own dressing or descent relation. On a boundary the instance supplies the one-dimensional measure and the primary coefficient law $V_y=(dy\mathrm{/}dx{)}^{-h}{V}_x$ on orientation-preserving boundary coordinate overlaps, with branch data if needed. A datum already presented as a BRST representative $V_i$ is physical when $Q{V}_i=0$, $[V_i]\in H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s},{\sigma }_i}^{r_i,p_i}(k_i)$, $(k_i,{ϵ}_i)\in M_{{\sigma }_i}$, and its mass-shell predicate holds. If $V_i$ instead denotes an integrated coefficient, its physical interpretation requires an associated $U_i$ satisfying $Q{U}_i=0$ and $[U_i]\in H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s},{\sigma }_i}^{r_i^U,p_i^U}(k_i)$, the same sector and mass-shell conditions, and the model-supplied dressing/descent relation between the two insertions; it does not require the bare coefficient to obey $Q{V}_i=0$. This definition does not make every primary physical and does not construct the input BRST model, normalization, or path-integral measure.

The current primary-field supplier record is owner-escalated because its pointed-worldsheet prerequisite was not closed; the pointed-worldsheet supplier has a ready label but a hash-stale current record. Their exact existing statements are used conditionally, and this vertex carrier remains escalated until the actual inputs are current.