  We study the multiplicity of host stars to known transiting extra-solar
planets to test competing theories on the formation mechanisms of hot Jupiters.
We observed 45 exoplanet host stars using VLT/SPHERE/IRDIS to search for
potential companions. For each identified candidate companion we determined the
probability that it is gravitationally bound to its host by performing common
proper motion checks and modelling of synthetic stellar populations around the
host. We detected new candidate companions around K2-38, WASP-72, WASP-80,
WASP-87, WASP-88, WASP-108, WASP-118, WASP-120, WASP-122, WASP123, WASP-130,
WASP-131 and WASP-137. The closest candidates were detected at separations of
$0.124''\pm0.007''$ and $0.189''\pm0.003''$ around WASP-108 and WASP-131; the
measured $K$ band contrasts indicate that these are stellar companions of
$0.35\pm0.02\,M_{\odot}$ and $0.62^{+0.05}_{-0.04}\,M_{\odot}$, respectively.
Including the re-detection and confirmation of previously known companions in
13 other systems we derived a multiplicity fraction of
$55.4^{+5.9}_{-9.4}\,\%$. For the representative sub-sample of 40 hot Jupiter
host stars among our targets, the derived multiplicity rate is
$54.8^{+6.3}_{-9.9}\,\%$. Our data do not confirm any trend that systems with
eccentric planetary companions are preferably part of multiple systems. On
average, we reached a magnitude contrast of $8.5\pm0.9$ mag at an angular
separation of 0.5''. This allows to exclude additional stellar companions with
masses larger than $0.08$ M$_\odot$ for almost all observed systems; around the
closest and youngest systems this sensitivity is achieved at physical
separations as small as 10 au. The presented study shows that SPHERE is an
ideal instrument to detect and characterize close companions to exoplanetary
host stars.
