Unique structural and optical properties of atomically thin transition metal
dichalcogenides (TMDs) enable in principle their efficient coupling to photonic
cavities having the optical mode volume below the diffraction limit. So far,
this has only been demonstrated by coupling TMDs with plasmonic modes in
metallic nano-structures, which exhibit strong energy dissipation limiting
their potential applications in devices. Here, we present an alternative
approach for realisation of ultra-compact cavities interacting with
two-dimensional semiconductors: we use mono- and bilayer TMD WSe$_2$ coupled to
low-loss high-refractive-index gallium phosphide (GaP) nano-antennas. We
observe a photoluminescence (PL) enhancement exceeding 10$^4$ compared with
WSe$_2$ placed on the planar GaP, and trace its origin to a combination of
enhancement of the spontaneous light emission rate, favourable modification of
the PL directionality and enhanced optical excitation efficiency, all occurring
as a result of WSe$_2$ coupling with strongly confined photonic modes of the
nano-antennas. Further effect of the coupling is observed in the polarisation
dependence of WSe$_2$ PL, and in the Raman scattering signal enhancement
exceeding 10$^3$. Our findings reveal high-index dielectric nano-structures as
a promising platform for engineering light-matter coupling in two-dimensional
semiconductors.