  We conjecture that the topological structure of the gauge symmetries required
by the Calabi-Yau vacuum and the dualities in string/D-branes considered in the
world with some additional dimensions can lead to an extension of the main
principles of the Special Theory of Relativity. The link between the
topological structure of the vacuum and the, hierarchy of the gauge symmetries
could be checked by the existence of the ``almost massless'', ''sterile''
particles. These particles could have flying properties different from the
standard predictions of the STR. It is natural to consider this property for
neutrinos, known up to now ``$U(1)_{em}$ sterile'' particles. Here we discuss
two such effects depending on the possible existence of large and/or small
extra dimensions and what is the maximal speed for both cases.
  The effect of large extra dimensions can be directly connected with the
existence of a hidden boost, $c_{hid} >> c_{em}$, excited by new global
dimensions and can lead to the monotonous rise of the neutrino velocity at high
energies, $v_{\nu} > c$.
  The effect of small extra dimensions, universal with respect to all
particles, can be connected with the gravitation recoils of the propagation of
neutrino in the space-time vacuum-foam and it leads to the effect of
diminishing the neutrino velocity at high energies.
  We propose to check these conjectures for neutrinos of different energies and
species. Limits for existing neutrino data and expected sensitivities of
possible experiments for accelerator - produced neutrinos are considered. It is
pointed out that CERN has the unique opportunity of measuring the indicated
effects.
