We present the first set of cosmological baryonic zoom-in simulations of galaxies including dissipative self-interacting dark matter (dSIDM). These simulations utilize the Feedback In Realistic Environments (FIRE-2) galaxy formation physics, but allow the dark matter to have dissipative self-interactions analogous to Standard Model forces, parameterized by the self-interaction cross-section per unit mass, $(sigma/m)$, and the dimensionless degree of dissipation, $0<f_{rm diss}<1$. We survey this parameter space, including constant and velocity-dependent cross-sections, and focus on structural and kinematic properties of dwarf galaxies with $M_{rm halo} simeq 10^{10-11} {rm M}_{odot}$. Central density profiles of simulated dwarfs become cuspy when $(sigma/m)_{rm eff} gtrsim 0.1,{rm cm^{2},g^{-1}}$ (and $f_{rm diss}=0.5$ as fiducial). The power-law slopes asymptote to $alpha approx -1.5$ in low-mass dwarfs independent of cross-section, which arises from a dark matter cooling flow. Through comparisons with dark matter only simulations, we find the profile in this regime is insensitive to the inclusion of baryons. However, when $(sigma/m)_{rm eff} ll 0.1,{rm cm^{2},g^{-1}}$, baryonic effects can produce cored density profiles comparable to non-dissipative cold dark matter (CDM) runs but at smaller radii. Simulated galaxies with $(sigma/m) gtrsim 10,{rm cm^{2},g^{-1}}$ develop significant coherent rotation of dark matter, accompanied by halo deformation, but this is unlike the well-defined thin dark disks often attributed to baryon-like dSIDM. The density profiles in this high cross-section model exhibit lower normalizations given the onset of halo deformation. For our surveyed dSIDM parameters, halo masses and galaxy stellar masses do not show appreciable difference from CDM, but dark matter kinematics and halo concentrations/shapes can differ.
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