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It is well-known that one signature of the three-dimensional electrontopological insulator is the Witten effect: if the system is coupled to acompact electromagnetic gauge field, a monopole in the bulk acquires ahalf-odd-integer polarization charge. In the present work, we propose acorresponding phenomenon for the topological insulator of bosons in 3dprotected by particle number conservation and time-reversal symmetry. We claimthat although a monopole inside a topological insulator of bosons can remainelectrically neutral, its statistics are transmuted from bosonic to fermionic.We demonstrate that this ``statistical Witten effect" directly implies that ifthe surface of the topological insulator is neither gapless, nor spontaneouslybreaks the symmetry, it necessarily supports an intrinsic two-dimensionaltopological order. Moreover, the surface properties cannot be fully realized ina purely 2d system. We also confirm that the surface phases of the bosonictopological insulator proposed by Vishwanath and Senthil (arXiv:1209.3058)provide a consistent termination of a bulk exhibiting the statistical Witteneffect. In a companion paper, we will provide an explicit field-theoretic,lattice-regularized, construction of the 3d topological insulator of bosons,employing a parton decomposition and subsequent condensation of parton-monopolecomposites.

Bosonic topological insulator in three dimensions and the statistical Witten effect