Constant H field, cosmology and faster than light solitons

We analyze the possibility of having a constant spatial NS-NS field, $H_{123}$. Cosmologically, it will act as stiff matter, and there will be very tight constraints on the possible value of $H_{123}$ today. However, it will give a noncommutative structure with an {\em associative} star product of the type $\theta^{ij}=\alpha \epsilon^{ijk} x^k$. This will be a fuzzy space with constant radius slices being fuzzy spheres. We find that gauge theory on such a space admits a noncommutative soliton with galilean dispersion relation, thus having speeds arbitrarily higher than c. This is the analogue of the Hashimoto-Itzhaki construction at constant $\theta$, except that one has fluxless solutions of arbitrary mass. A holographic description supports this finding. We speculate thus that the presence of constant (yet very small) $H_{123}$, even though otherwise virtually undetectable could still imply the existence of faster than light solitons of arbitrary mass (although possibly quantum-mechanically unstable). The spontaneous Lorentz violation given by $H_{123}$ is exactly the same one already implied by the FRW metric ansatz.