  BiFeO$_3$ is a multiferroic material featuring ferroelectricity and
noncollinear antiferromagnetism, the latter manifested as a cycloid of spin
density. Definitive and efficient control of the characteristic spin texture of
BiFeO$_3$ is attractive for emerging quantum devices. In this regard,
crystal-field $d\rightarrow d$ excitations localized on Fe atomic sites in
BiFeO$_3$ provide an avenue for manipulation of the spin texture as they induce
a complex interplay among the spin, charge, and lattice degrees of freedom. In
this work, the ab initio GW-BSE method is used to characterize these
excitations within an excitonic picture. We find that the $d-d$ transitions
appear as strongly bound, chiral, spin-flip excitons deep within the electronic
band gap as a result of the intricate competition between the lattice
potential, the antiferromagnetic ordering, the spin-orbit coupling, and the
electron-hole interaction. Most crucially, these excitons are composed of
electron-hole pairs with opposite spins that constitute almost all of their
$\pm \hbar$ total angular momentum. These excitons of specific angular momentum
can be selectively excited using circularly polarized light, consequently
modulating the local magnetic moment giving rise to transient ferrimagnetism.
