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After a suitable gauge fixing, the local gravitational degrees of freedom ofthe Gowdy $S^1\times S^2$ and $S^3$ cosmologies are encoded in an axisymmetricfield on the sphere $S^2$. Recently, it has been shown that a standard fieldparametrization of these reduced models admits no Fock quantization with aunitary dynamics. This lack of unitarity is surpassed by a convenientredefinition of the field and the choice of an adequate complex structure. Theresult is a Fock quantization where both the dynamics and the SO(3)-symmetriesof the field equations are unitarily implemented. The present work proves thatthis Fock representation is in fact unique inasmuch as, up to equivalence,there exists no other possible choice of SO(3)-invariant complex structureleading to a unitary implementation of the time evolution.

Uniqueness of the Fock representation of the Gowdy $S^1\times S^2$ and $S^3$ models