Measurements of Absolute Branching Fractions of Fourteen Exclusive Hadronic $D$ Decays to $\eta $

Using $2.93\text{}\text{}{\mathrm{fb}}^{-1}$ of ${e}^{+}{e}^{-}$ collision data taken at a center-of-mass energy of 3.773 GeV with the BESIII detector, we report the first measurements of the absolute branching fractions of 14 hadronic ${D}^{0\left(+\right)}$ decays to exclusive final states with an $\eta $, e.g., ${D}^{0}\to {K}^{-}{\pi }^{+}\eta $, ${K}_{S}^{0}{\pi }^{0}\eta $, ${K}^{+}{K}^{-}\eta $, ${K}_{S}^{0}{K}_{S}^{0}\eta $, ${K}^{-}{\pi }^{+}{\pi }^{0}\eta $, ${K}_{S}^{0}{\pi }^{+}{\pi }^{-}\eta $, ${K}_{S}^{0}{\pi }^{0}{\pi }^{0}\eta $, and ${\pi }^{+}{\pi }^{-}{\pi }^{0}\eta $; ${D}^{+}\to {K}_{S}^{0}{\pi }^{+}\eta $, ${K}_{S}^{0}{K}^{+}\eta $, ${K}^{-}{\pi }^{+}{\pi }^{+}\eta $, ${K}_{S}^{0}{\pi }^{+}{\pi }^{0}\eta $, ${\pi }^{+}{\pi }^{+}{\pi }^{-}\eta $, and ${\pi }^{+}{\pi }^{0}{\pi }^{0}\eta $. Among these decays, the ${D}^{0}\to {K}^{-}{\pi }^{+}\eta $ and ${D}^{+}\to {K}_{S}^{0}{\pi }^{+}\eta $ decays have the largest branching fractions, which are $B\left({D}^{0}\to {K}^{-}{\pi }^{+}\eta \right)=\left(1.853±0.02{5}_{\text{stat}}±0.03{1}_{\text{syst}}\right)%$ and $B\left({D}^{+}\to {K}_{S}^{0}{\pi }^{+}\eta \right)=\left(1.309±0.03{7}_{\text{stat}}±\phantom{\rule{0ex}{0ex}}0.03{1}_{\text{syst}}\right)%$, respectively. The charge-parity asymmetries for the six decays with highest event yields are determined, and no statistically significant charge-parity violation is found.