Near-IR period-luminosity relations for pulsating stars in $\omega$
  Centauri (NGC 5139)
  $\omega$ Centauri (NGC 5139) hosts hundreds of pulsating variable stars of
different types, thus representing a treasure trove for studies of their
corresponding period-luminosity (PL) relations. Our goal in this study is to
obtain the PL relations for RR Lyrae, and SX Phoenicis stars in the field of
the cluster, based on high-quality, well-sampled light curves in the
near-infrared (IR). $\omega$ Centauri was observed using VIRCAM mounted on
VISTA. A total of 42 epochs in $J$ and 100 epochs in $K_{\rm S}$ were obtained,
spanning 352 days. Point-spread function photometry was performed using DoPhot
and DAOPHOT in the outer and inner regions of the cluster, respectively. Based
on the comprehensive catalogue of near-IR light curves thus secured, PL
relations were obtained for the different types of pulsators in the cluster,
both in the $J$ and $K_{\rm S}$ bands. This includes the first PL relations in
the near-IR for fundamental-mode SX Phoenicis stars. The near-IR magnitudes and
periods of Type II Cepheids and RR Lyrae stars were used to derive an updated
true distance modulus to the cluster, with a resulting value of $(m-M)_0 =
13.708 \pm 0.035 \pm 0.10$ mag, where the error bars correspond to the adopted
statistical and systematic errors, respectively. Adding the errors in
quadrature, this is equivalent to a heliocentric distance of $5.52\pm 0.27$
kpc.
