  In conventional and high transition temperature copper oxide and iron
pnictide superconductors, the Cooper pairs all have even parity. As a rare
exception, Sr$_2$RuO$_4$ is the first prime candidate for topological chiral
p-wave superconductivity, which has time-reversal breaking odd-parity Cooper
pairs known to exist before only in the neutral superfluid $^3$He. However,
there are several key unresolved issues hampering the microscopic description
of the unconventional superconductivity. Spin fluctuations at both large and
small wavevectors are present in experiments, but how they arise and drive
superconductivity is not yet clear. Spontaneous edge current is expected but
not observed conclusively. Specific experiments point to highly band- and/or
momentum-dependent energy gaps for quasiparticle excitations in the
superconducting state. Here, by comprehensive functional renormalization group
calculations with all relevant bands, we disentangle the various competing
possibilities. In particular we show the small wavevector spin fluctuations,
driven by a single two-dimensional band, trigger p-wave superconductivity with
quasi-nodal energy gaps.
