Heterotic Type I Ten Dimensional N1 Supergravity Action

Adopt two separate perturbative ten-dimensional $N=1$ supergravity-plus-Yang–Mills low-energy branches on a smooth oriented Lorentzian ten-manifold $(M,g)$ with signature $(-,+,\dots ,+)$. The metric $g$ is smooth, and in each branch the dilaton is a smooth real dimensionless scalar field $\mathrm{\Phi }:M\to \mathbb{R}$; denote the separate heterotic and Type-I dilatons by ${\mathrm{\Phi }}_H$ and ${\mathrm{\Phi }}_I$. The heterotic branch has gauge group $G_H=E_8\times E_8$ or $\mathrm{Spin}(32)\mathrm{/}{\mathbb{Z}}_2$; the $SO(32)$ Type-I branch has gauge group $G_I=SO(32)$. For the group $G$ of either branch, let $P_G\to M$ be a supplied smooth principal $G$-bundle with a smooth connection $A$. On a local trivialization $U\subseteq M$, $A$ is a $\mathfrak{g}$-valued one-form and its curvature is the $\mathfrak{g}$-valued two-form $F_A=dA+A\wedge A$; globally $F_A\in {\mathrm{\Omega }}^2(M,\mathrm{ad}{P}_G)$. We use this geometric connection normalization. In natural units with length-valued spacetime coordinates, the connection one-form is dimensionless, so its components have dimension length${}^{-1}$ and the curvature components have dimension length${}^{-2}$. A source-specific rescaling of component gauge fields, and all trace and kinetic-term normalizations, are left unspecified unless separately supplied. A spin structure is supplied in each branch; no existence of such a structure is asserted. If $S_{+}$ and $S_{-}$ are its real Majorana–Weyl spin bundles in the convention of Supergravity Conventions And Normalizations, the smooth gravitino, dilatino, and adjoint gaugino have types $\psi \in \mathrm{\Gamma }(S_{+}\otimes T^{\ast }M)$, $\lambda \in \mathrm{\Gamma }(S_{-})$, and $\chi \in \mathrm{\Gamma }(S_{+}\otimes \mathrm{ad}{P}_G)$, respectively. The two-form sector has local smooth potentials: in the heterotic branch the NS–NS potential $B_2\in {\mathrm{\Omega }}^2(U)$, and in the Type-I branch the Ramond–Ramond potential $C_2\in {\mathrm{\Omega }}^2(U)$. Both heterotic closed gravity/dilaton terms and the gauge kinetic term have tree-level multiplier $e^{-2{\mathrm{\Phi }}_H}$. In the Type-I branch the closed gravity/dilaton terms have tree-level multiplier $e^{-2{\mathrm{\Phi }}_I}$, and the open-string disk gauge kinetic term has multiplier $e^{-{\mathrm{\Phi }}_I}$. Here tree-level means the leading perturbative string-coupling contribution: the sphere for closed-string terms and the disk for the Type-I open-string gauge term. This postulate fixes the stated fields and dilaton weights, not all component coefficients or trace normalizations; it does not assert an S-duality map or equality of the two branch actions.

Washington Taylor, §§2.2, 2.4, and 3.3.4

Mohammad R. Garousi, §1

The two actions are adopted as source-reported low-energy frameworks. Garousi's separate field dictionary is context only; no duality map is assumed in this item. Higher-derivative gauge and curvature terms, their trace conversions, and a complete fermionic component action require their own exact conventions before use.