  We investigate the ground state of the $d^1$ spin-orbital model for triply
degenerate $t_{2g}$ orbitals on a triangular lattice which unifies intrinsic
frustration of spin and orbital interactions with geometrical frustration.
Using full or Lanczos exact diagonalization of finite clusters we establish
that the ground state of the spin-orbital model which interpolates between the
superexchange and direct exchange interactions on the bonds is characterized by
valence-bond correlations. In the absence of Hund's exchange the model
describes a competition between various possible valence-bond states. By
considering the clusters with open boundary conditions we demonstrate that
orbital interactions are always frustrated, but this frustration is removed by
pronounced spin singlet correlations which coexist with supporting them dimer
orbital correlations. Such local configurations contribute to the disordered
ground states found for the clusters with periodic boundary conditions which
interpolate between a highly resonating, dimer-based, entangled spin-orbital
liquid phase, and a valence-bond state with completely static spin-singlet
states. We argue that these states are also realized for the infinite lattice
and anticipate that pronounced transitions between different regimes found for
particular geometries will turn out to smooth crossovers in the properties of
the spin-orbital liquid in the thermodynamic limit. Finally, we provide
evidence that the resonating spin-orbital liquid phase involves entangled
states on the bonds. In such a phase classical considerations based on the
mean-field theory cannot be used, spin exchange interactions do not determine
spin bond correlations, and quantum fluctuations play a crucial role in the
ground states and magnetic transitions.
