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Certain phase transitions between topological quantum field theories (TQFT)are driven by the condensation of bosonic anyons. However, as bosons in a TQFTare themselves nontrivial collective excitations, there can be topologicalobstructions that prevent them from condensing. Here we formulate such anobstruction in the form of a no-go theorem. We use it to show that nocondensation is possible in SO(3)$_k$ TQFTs with odd $k$. We further show thata layered theory obtained by tensoring SO(3)$_k$ TQFT with itself any integernumber of times does not admit condensation transitions either. This includes(as the case $k=3$) the noncondensability of any number of layers of theFibonacci TQFT.

No-Go Theorem for Boson Condensation in Topologically Ordered Quantum Liquids