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We investigate topological properties and classification of mean-fieldtheories of stable bosonic systems. Of the three standard classifyingsymmetries, only time-reversal represents a real symmetry of the many-bosonsystem, while the other two, particle-hole and chiral, are simply constraintsthat manifest as symmetries of the effective single-particle problem. Forgapped systems in arbitrary space dimension we establish three fundamentalno-go theorems that prove the absence of: parity switches,symmetry-protected-topological quantum phases, and localized bosonic zero modesunder open boundary conditions. We then introduce a squaring, kernel-preservingmap connecting non-interacting Hermitian theories of fermions and stable bosonsystems, which serves as a playground to reveal the role of topology in bosonicphases and their localized midgap boundary modes. Finally, we determine thesymmetry classes inherited from the fermionic tenfold-way classification,unveiling an elegant threefold-way topological classification ofnon-interacting bosons. We illustrate our main findings in one- andtwo-dimensional bosonic lattice and field-theory models.

Squaring the fermion: The threefold way and the fate of zero modes