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We use the superconformal bootstrap to derive exact relations between OPEcoefficients in three-dimensional superconformal field theories with ${\cal N}\geq 4$ supersymmetry. These relations follow from a consistent truncation ofthe crossing symmetry equations that is associated with the cohomology of acertain supercharge. In ${\cal N} = 4$ SCFTs, the non-trivial cohomologyclasses are in one-to-one correspondence with certain half-BPS operators,provided that these operators are restricted to lie on a line. The relations wefind are powerful enough to allow us to determine an infinite number of OPEcoefficients in the interacting SCFT ($U(2)_2 \times U(1)_{-2}$ ABJ theory)that constitutes the IR limit of $O(3)$ ${\cal N} = 8$ super-Yang-Mills theory.More generally, in ${\cal N} = 8$ SCFTs with a unique stress tensor, we are ledto conjecture that many superconformal multiplets allowed by group theory mustactually be absent from the spectrum, and we test this conjecture in known${\cal N} = 8$ SCFTs using the superconformal index. For generic ${\cal N} = 8$SCFTs, we also improve on numerical bootstrap bounds on OPE coefficients ofshort and semi-short multiplets and discuss their relation to the exactrelations between OPE coefficients we derived. In particular, we show that thekink previously observed in these bounds arises from the disappearance of acertain quarter-BPS multiplet, and that the location of the kink is likely tiedto the existence of the $U(2)_2 \times U(1)_{-2}$ ABJ theory, which can beargued to not possess this multiplet.

Exact Correlators of BPS Operators from the 3d Superconformal Bootstrap