Observation of the Semileptonic Decay ${D}^{0}\to {a}_{0}\left(980{\right)}^{-}{e}^{+}{\nu }_{e}$ and Evidence for ${D}^{+}\to {a}_{0}\left(980{\right)}^{0}{e}^{+}{\nu }_{e}$

Using an ${e}^{+}{e}^{-}$ collision data sample of $2.93\text{}\text{}{\mathrm{fb}}^{-1}$ collected at a center-of-mass energy of 3.773 GeV by the BESIII detector at BEPCII, we report the observation of ${D}^{0}\to {a}_{0}\left(980{\right)}^{-}{e}^{+}{\nu }_{e}$ and evidence for ${D}^{+}\to {a}_{0}\left(980{\right)}^{0}{e}^{+}{\nu }_{e}$ with significances of $6.4\sigma $ and $2.9\sigma $, respectively. The absolute branching fractions are determined to be $B\left({D}^{0}\to {a}_{0}\left(980{\right)}^{-}{e}^{+}{\nu }_{e}\right)×B\left({a}_{0}\left(980{\right)}^{-}\to \eta {\pi }^{-}\right)=\left[1.3{3}_{-0.29}^{+0.33}\left(\mathrm{stat}\right)±0.09\left(\mathrm{syst}\right)\right]×{10}^{-4}$ and $B\left({D}^{+}\to {a}_{0}\left(980{\right)}^{0}{e}^{+}{\nu }_{e}\right)×B\left({a}_{0}\left(980{\right)}^{0}\to \eta {\pi }^{0}\right)=\left[1.6{6}_{-0.66}^{+0.81}\left(\mathrm{stat}\right)±0.11\left(\mathrm{syst}\right)\right]×{10}^{-4}$. This is the first time the ${a}_{0}\left(980\right)$ meson has been measured in a ${D}^{0}$ semileptonic decay, which would open one more interesting page in the investigation of the nature of the puzzling ${a}_{0}\left(980\right)$ states.