  Understanding the nature of the excitation spectrum in quantum spin liquids
is of fundamental importance, in particular for the experimental detection of
candidate materials. However, current theoretical and numerical techniques have
limited capabilities, especially in obtaining the dynamical structure factor,
which gives a crucial characterization of the ultimate nature of the quantum
state and may be directly assessed by inelastic neutron scattering. In this
work, we investigate the low-energy properties of the $S=1/2$ Heisenberg model
on the triangular lattice, including both nearest-neighbor $J_1$ and
next-nearest-neighbor $J_2$ super-exchanges, by a dynamical variational Monte
Carlo approach that allows accurate results on spin models. For $J_2=0$, our
calculations are compatible with the existence of a well-defined magnon in the
whole Brillouin zone, with gapless excitations at $K$ points (i.e., at the
corners of the Brillouin zone). The strong renormalization of the magnon branch
(also including roton-like minima around the $M$ points, i.e., midpoints of the
border zone) is described by our Gutzwiller-projected state, where Abrikosov
fermions are subject to a non-trivial magnetic $\pi$-flux threading half of the
triangular plaquettes. When increasing the frustrating ratio $J_2/J_1$, we
detect a progessive softening of the magnon branch at $M$, which eventually
becomes gapless within the spin-liquid phase. This feature is captured by the
band structure of the unprojected wave function (with $2$ Dirac points for each
spin component). In addition, we observe an intense signal at low energies
around the $K$ points, which cannot be understood within the unprojected
picture and emerges only when the Gutzwiller projection is considered,
suggesting the relevance of gauge fields for the low-energy physics of spin
liquids.
