Baryons, monopoles and dualities in Chern-Simons-matter theories

There is significant evidence for a duality between (non-supersymmetric) U(N) Chern-Simons theories at level k coupled to fermions, and U(k) Chern-Simons theories at level N coupled to scalars. Most of the evidence comes from the large N 't Hooft limit, where many details of the duality (such as whether the gauge group is U(N) or SU(N), the precise level of the U(1) factor, and order one shifts in the level) are not important. The main evidence for the validity of the duality at finite N comes from adding masses and flowing to pure Chern-Simons theories related by level-rank duality, and from flowing to the non-supersymmetric duality from supersymmetric dualities, whose finite N validity is well-established. In this note we clarify the implications of these flows for the precise form of the duality; in particular we argue that in its simplest form the duality maps SU(N) theories to U(k) theories, though there is also another version relating U(N) to U(k). This precise form strongly affects the mapping under the duality of baryon and monopole operators, and we show, following arguments by Radicevic, that their mapping is consistent with our claims. We also discuss the implications of our results for the additional duality between these Chern-Simons matter theories and (the UV completion of) high-spin gravity theories on AdS_4. The latter theories should contain heavy particles carrying electric and/or magnetic charges under their U(1) gauge symmetry.