Flat Bosonic String Model

Fix an integer $D\ge 3$, target space $M={\mathbb{R}}^D$, and Minkowski metric $\eta =\mathrm{diag}(-1,+1,\dots ,+1)$. A worldsheet is an oriented smooth two-dimensional manifold $\mathrm{\Sigma }$, possibly with smooth boundary, equipped with an auxiliary Lorentzian metric $h$ of signature $(-,+)$. A classical Polyakov field configuration consists of this auxiliary metric and a $C^2$ map $X:\mathrm{\Sigma }\to M$, smooth to that order up to any boundary. Its induced symmetric bilinear form is $\gamma =X^{\ast }\eta$, with coordinate components ${\gamma }_{ab}={\eta }_{\mu \nu }{\mathrm{\partial }}_a{X}^{\mu }{\mathrm{\partial }}_b{X}^{\nu }$. No rank or nondegeneracy condition on $dX$ or $\gamma$ is imposed on the general Polyakov field domain; $h$ remains nondegenerate independently. The nondegenerate Lorentzian immersion branch on a specified region is the additional requirement that $dX$ have rank two and $\gamma$ be nondegenerate Lorentzian there. Global injectivity is not assumed; an embedding, when required, is an additional hypothesis. On a closed-string cylinder use dimensionless coordinates $\tau \in \mathbb{R}$ and $\sigma \in \mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$; on an open-string strip use $\tau \in \mathbb{R}$ and $\sigma \in [0,\pi ]$. The target coordinates $X^{\mu }$ have length dimension. The flat model excludes target curvature, a $B$-field, a dilaton, endpoint gauge fields, sources, and string interactions; these are separate model data. The nondegenerate induced-metric branch excludes null or collapsed regions where that extra condition is imposed, and is the branch used for differentiable Nambu–Goto action equivalence. Open Polyakov fields may have degenerate induced metric at their endpoints. For this endpoint illustration, assume boundary-adapted conformal coordinates: $h_{ab}=e^{2\varphi }\mathrm{diag}(-1,+1)$ for a smooth real function $\varphi$, with the boundary at $\sigma =0,\pi$. All-Neumann data in these coordinates mean ${\mathrm{\partial }}_{\sigma }{X}^{\mu }=0$ for every target component, so $dX$ has rank at most one there. If the metric constraints $\eta ({\mathrm{\partial }}_{\tau }X,{\mathrm{\partial }}_{\sigma }X)=0$ and $\eta ({\mathrm{\partial }}_{\tau }X,{\mathrm{\partial }}_{\tau }X)+\eta ({\mathrm{\partial }}_{\sigma }X,{\mathrm{\partial }}_{\sigma }X)=0$ are also imposed, then $\eta ({\mathrm{\partial }}_{\tau }X,{\mathrm{\partial }}_{\tau }X)=0$ and $\gamma$ vanishes at that endpoint. These endpoint degeneracies do not degenerate the independent auxiliary metric or exclude the field from the Polyakov domain. Neither dynamics nor boundary conditions are imposed merely by this kinematic definition.