  The bulk electronic structure of $T_d$-MoTe$_2$ features large hole Fermi
pockets at the Brillouin zone center ($\Gamma$) and two electron Fermi surfaces
along the $\Gamma-X$ direction. However, the large hole pockets, whose
existence has important implications for the Weyl physics of $T_d$-MoTe$_2$,
has never been conclusively detected in quantum oscillations. This raises doubt
about the realizability of Majorana states in $T_d$-MoTe$_2$, because these
exotic states rely on the existence of Weyl points, which originated from the
same band structure predicted by density functional theory (DFT). Here, we
report an unambiguous detection of these elusive hole pockets via Shubnikov-de
Haas (SdH) quantum oscillations. At ambient pressure, the quantum oscillation
frequencies for these pockets are 988 T and 1513 T, when the magnetic field is
applied along the $c$-axis. The quasiparticle effective masses $m^*$ associated
with these frequencies are 1.50 $m_e$ and 2.77 $m_e$, respectively, indicating
the importance of Coulomb interactions in this system. We further measure the
SdH oscillations under pressure. At 13 kbar, we detected a peak at 1798 T with
$m^*$ = 2.86 $m_e$. Relative to the oscillation data at a lower pressure, the
amplitude of this peak experienced an enhancement, which can be attributed to
the reduced curvature of the hole pockets under pressure. Combining our
experimental data with DFT + $U$ calculations, where $U$ is the Hubbard
parameter, our results shed light on why these important hole pockets have not
been detected until now.
