Heterotic tree-level four-dimensional gauge-coupling reduction data

Use natural units $\mathrm{\hslash }=c=1$. On an unwarped product $M_4\times X$ with ten-dimensional string-frame metric $G^{(s)}=g^{(4,s)}\oplus g^{(X,s)}$, finite internal string-frame volume ${\mathrm{Vol}}_s(X)$, and constant dimensionless heterotic dilaton ${\mathrm{\Phi }}_0$, choose one declared gauge trace and a ten-dimensional kinetic normalization $g_{10,0}$ by writing the selected gauge term as $S_{10,\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}=-\frac{1}{4g_{10,0}^2}{\int }_{M_4\times X}{d}^{10}x\sqrt{-G^{(s)}}{e}^{-2{\mathrm{\Phi }}_0}\mathrm{Tr}(F_{MN}{F}^{MN})\mathrm{.}$ For an external gauge field independent of $X$ and with no internal components, define the scalar four-dimensional Yang–Mills coupling $g_{\mathrm{Y}\mathrm{M},4}>0$ by the coefficient of $-\frac{1}{4}{\int }_{M_4}{d}^{4}x\sqrt{-g^{(4,s)}}\mathrm{Tr}(F_{\mu \nu }{F}^{\mu \nu })$, with external indices raised by $g^{(4,s)}$. Require $0<g_{10,0}^2<\mathrm{\infty }$. Then the tree-level reduction datum is $\frac{1}{g_{\mathrm{Y}\mathrm{M},4}^2}=e^{-2{\mathrm{\Phi }}_0}\frac{{\mathrm{Vol}}_s(X)}{g_{10,0}^2}\mathrm{.}$ Here $[g_{10,0}^2]={\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^6$, so the right side is dimensionless. The exact value of $g_{10,0}$, factors of $2\pi$ and ${\alpha }^{\mathrm{\prime }}$, and representation embedding index are fixed only after the action coefficient and trace are specified. The formula excludes threshold corrections, loops, flux corrections, warping, and localized sources.

Perturbative heterotic Calabi–Yau bundle compactification data

Supergravity Conventions And Normalizations

Heterotic Type I Ten Dimensional N1 Supergravity Action