What controls star formation in the central 500 pc of the Galaxy?

The star formation rate (SFR) in the Central Molecular Zone (CMZ, i.e. the central 500 pc) of the Milky Way is lower by a factor of >= 10 than expected for the substantial amount of dense gas it contains, which challenges current star formation theories. In this paper, we quantify which physical mechanisms could be responsible. On scales larger than the disc scaleheight, the low SFR is found to be consistent with episodic star formation due to secular instabilities or possibly variations of the gas inflow along the Galactic bar. The CMZ is marginally Toomre-stable when including gas and stars, but highly Toomre-stable when only accounting for the gas, indicating a low condensation rate of self-gravitating clouds. On small scales, we find that the SFR in the CMZ may be caused by an elevated critical density for star formation due to the high turbulent pressure. The existence of a universal density threshold for star formation is ruled out. The HI-H-2 phase transition of hydrogen, the tidal field, a possible underproduction of massive stars due to a bottom-heavy initial mass function, magnetic fields, and cosmic ray or radiation pressure feedback also cannot individually explain the low SFR. We propose a self-consistent cycle of star formation in the CMZ, in which the effects of several different processes combine to inhibit star formation. The rate-limiting factor is the slow evolution of the gas towards collapse - once star formation is initiated it proceeds at a normal rate. The ubiquity of star formation inhibitors suggests that a lowered central SFR should be a common phenomenon in other galaxies. We discuss the implications for galactic-scale star formation and supermassive black hole growth, and relate our results to the star formation conditions in other extreme environments.