  Core-excitations on different atoms are highly localized and therefore
decoupled. By placing molecules in an X-ray cavity the core-transitions become
coupled via the exchange of cavity photons and form delocalized hybrid
light-matter excitations known as core-polaritons. We demonstrate these effects
for the two inequivalent carbon atoms in 1,1-difluoroethylene. Polariton
signatures in the X-ray absorption, two-photon absorption, and multidimensional
four-wave mixing, signals are predicted.
