Supergiant Fast X-ray Transients (SFXTs) are a new class of HMXBs discovered thanks to the monitoring of the Galactic plane performed with the INTEGRAL satellite in the last 5 years. These sources display short outbursts (significantly shorter than typical Be/X-ray binaries) with a peak luminosity of a few 1E36 erg/s. The quiescent level, measured only in a few sources, is around 1E32 erg/s. We are performing a monitoring campaign with Swift of four SFXTs (IGRJ16479-4514, XTEJ1739-302, IGRJ17544-2619 and AXJ1841.0-0536/IGRJ18410-0535). We report on the first four months of Swift observations, started on 2007 October 26. We detect a low level X-ray activity in all four SFXTs which demonstrates that these transient sources accrete matter even outside their outbursts. This fainter X-ray activity is composed of many flares with a large flux variability, on timescales of thousands of seconds. The lightcurve variability is also evident on larger timescales of days, weeks and months, with a dynamic range of more than one order of magnitude in all four SFXTs. The X-ray spectra are typically hard, with an average 2-10 keV luminosity during this monitoring of about 1E33-1E34 erg/s. We detected pulsations from the pulsar AXJ1841.0-0536, with a period of 4.7008+/-0.0004 s. This monitoring demonstrates that these transients spend most of the time accreting matter, although at a much lower level (~100-1000 times lower than during the bright outbusts), and that the true quiescence, characterized by a soft spectrum and a luminosity of a few 1E32 erg/s, observed in the past only in a couple of members of this class, is probably a very rare state.
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