Can Neutrinos Probe Extra Dimensions?

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