  For bosons with flat energy dispersion, condensation can occur in different
symmetry sectors. Here, we consider bosons in a Kagome lattice with $\pi$-flux
hopping, which in the presence of mean-field interactions exhibit degenerate
condensates in the $\Gamma$- and the $K$-point. We analyze the excitation above
both condensates and find strikingly different properties: For the $K$-point
condensate, the Bogoliubov-de Gennes (BdG) Hamiltonian has broken particle-hole
symmetry (PHS) and exhibits a topologically trivial quasiparticle band
structure. However, band flatness plays a key role in breaking the time
reversal symmetry (TRS) of the BdG Hamiltonian for a $\Gamma$-point condensate.
Consequently, its quasiparticle band structure exhibits non-trivial topology,
characterized by non-zero Chern numbers and the presence of edge states.
Although quantum fluctuations energetically favor the $K$-point condensate, the
interesting properties of the $\Gamma$-point condensate become relevant for
anisotropic hopping. The topological properties of the $\Gamma$-point
condensate get even richer in the presence of extended Bose-Hubbard
interactions. We find a topological phase transition into a topological
condensate characterized by high Chern number and also comment on the
realization and detection of such excitations.
