  We report ALMA observations of CO(3-2) emission in the Seyfert galaxy NGC613,
at a spatial resolution of 17pc, as part of our NUclei of GAlaxies sample. Our
aim is to investigate the morphology and dynamics of the gas inside the central
kpc, and to probe nuclear fueling and feedback phenomena. The morphology of
CO(3-2) line emission reveals a 2-arm trailing nuclear spiral at
$r\lesssim$100pc and a circumnuclear ring at ~350pc radius, that is coincident
with the star-forming ring seen in the optical images. The molecular gas in the
galaxy disk is in a remarkably regular rotation, however, the kinematics in the
nuclear region is very skewed. The nuclear spectrum of CO and dense gas tracers
HCN(4-3), HCO+(4-3), and CS(7-6) show broad wings up to $\pm$300km/s,
associated with a molecular outflow emanating from the nucleus (r~25pc). We
derive a molecular outflow mass $M_{out}$=2x10$^6$M$_\odot$ and a mass outflow
rate of $\dot{M}_{out}=$27$\rm M_\odot yr^{-1}$. The molecular outflow
energetics exceed the values predicted by AGN feedback models: the kinetic
power of the outflow corresponds to 20%L_AGN and the momentum rate is
$\dot{M}_{out}v\sim400L_{AGN}/c$. The outflow is mainly boosted by the AGN
through entrainment by the radio jet, but given the weak nuclear activity of
NGC613, we might be witnessing a fossil outflow, resulted from a strong past
AGN that now has already faded. The nuclear trailing spiral observed in CO
emission is inside the ILR ring of the bar. We compute the gravitational
torques exerted in the gas to estimate the efficiency of the angular momentum
exchange. The gravity torques are negative from 25 to 100pc and the gas loses
its angular momentum in a rotation period, providing evidence of a highly
efficient inflow towards the center. This phenomenon shows that the massive
central black hole has a significant dynamical influence on the gas, triggering
its fueling.
