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We complete the classification of symmetry constraints on gapped quadraticfermion hamiltonians proposed by Kitaev. The symmetry group is supposed compactand can include arbitrary unitary or antiunitary operators in the Fock spacethat conserve the algebra of quadratic observables. We analyze the multiplicityspaces of {\em real} irreducible representations of unitary symmetries in theNambu space. The joint action of intertwining operators and antiunitarysymmetries provides these spaces with the structure of Clifford module: weprove a one-to-one correspondence between the ten Altland-Zirnbauer symmetryclasses of fermion systems and the ten Morita equivalence classes of real andcomplex Clifford algebras. The antiunitary operators, which occur in sevenclasses, are projectively represented in the Nambu space by unitary "chiralsymmetries". The space of gapped symmetric hamiltonians is homotopicallyequivalent to the product of classifying spaces indexed by the dual object ofthe group of unitary symmetries.

Clifford modules and symmetries of topological insulators