  We study the superconducting pairing instabilities and gap functions for
prototypical two-dimensional (2D) transition-metal dichalcogenides (TMDCs)
WS$_2$, MoTe$_2$, and MoS$_2$ in the 2H phase under both hole and electron
doping at 10 K. Our first-principles quantum many-body Green's function
approach allows us to treat the full $d$ and $p$ manifold of orbitals with
strong spin-orbit coupling, yielding pairing predictions with material specific
detail. The resulting gap functions exhibit a variety of mixed-parity
superconducting states, including $s$, $p$, $d$, $f$, $d\pm id$, and $p\pm ip$
pairing modes. In particular, we predict 3% and 4% hole-doped WS$_2$ to be a
chiral $p\pm ip$ topological superconductor. For 1% hole-doped MoS$_2$, we find
a competition between three doubly degenerate chiral and non-chiral
instabilities. Overall, the relative pairing strengths are found to follow the
Fermi surface topology, due to nesting between the Fermi surface sheets.
Finally, we discuss our predictions in relation to available experimental data
and classify the topology of the predicted superconducting pairing symmetries.

Superconductivity;Materials Science;Strongly Correlated Electrons