  Using cosmological hydrodynamic simulations with physically motivated models
of super-massive black hole (SMBH) formation and growth, we compare the
assembly of Milky Way-mass ( $M_{\mathrm{vir}} \approx 7 \times 10^{11}$
$M_{\odot}$ at $z = 0$) galaxies in cold dark matter (CDM) and self-interacting
dark matter (SIDM) models. Our SIDM model adopts a constant cross-section of 1
cm$^2$/g. We find that SMBH formation is suppressed in the early universe due
to SIDM coring. SMBH-SMBH mergers are also suppressed in SIDM as a consequence
of the lower number of SMBHs formed. Lack of initial merger-driven SMBH growth
in turn delays SMBH growth by billions of years in SIDM compared to CDM.
Further, we find that this delayed growth suppresses SMBH accretion in the
largest progenitors of the main SIDM galaxies during the first 5 Gyrs of their
evolution. Nonetheless, by $z = 0.8$ the CDM and SIDM SMBH masses differ only
by around 0.2 dex, so that both remain compatible with the $M_{BH}-M_{*}$
relation. We show that the reduced accretion causes the SIDM SMBHs to less
aggressively regulate star formation in their host galaxies than their CDM
counterparts, resulting in a factor of 3 or more stars being produced over the
lifetime of the SIDM galaxies compared to the CDM galaxies. Our results
highlight a new way in which SIDM can affect the growth and merger history of
SMBHs and ultimately give rise to very different galaxy evolution compared to
the classic CDM model.
