Abstract
We analyze the constraints imposed by unitarity and crossing symmetry on the four-point function of the stress-tensor multiplet of $$ \mathcal{N}=8 $$ superconformal field theories in three dimensions. We first derive the superconformal blocks by analyzing the superconformal Ward identity. Our results imply that the OPE of the primary operator of the stress-tensor multiplet with itself must have parity symmetry. We then analyze the relations between the crossing equations, and we find that these equations are mostly redundant. We implement the independent crossing constraints numerically and find bounds on OPE coefficients and operator dimensions as a function of the stress-tensor central charge. To make contact with known $$ \mathcal{N}=8 $$ superconformal field theories, we compute this central charge in a few particular cases using supersymmetric localization. For limiting values of the central charge, our numerical bounds are nearly saturated by the large N limit of ABJM theory and also by the free U(1) Ă U(1) ABJM theory.
Highlights
The conformal bootstrap [2,3,4] is an old idea that uses the associativity of the operator algebra to provide an infinite set of constraints on the operator dimensions and the operator product expansion (OPE) coefficients of abstract conformal field theories (CFTs)
In such cases the theory has more than one stress tensor, and our localization computations are only sensitive to the sum of the central charges corresponding to the different decoupled CFTs
We find that the central charges computed below for these product CFTs are given by the appropriate sum of central charges corresponding to the irreducible CFTs
Summary
The conformal bootstrap [2,3,4] is an old idea that uses the associativity of the operator algebra to provide an infinite set of constraints on the operator dimensions and the operator product expansion (OPE) coefficients of abstract conformal field theories (CFTs). The authors of [24] studied the implications of unitarity and crossing symmetry on the four-point function of the superconformal primary operator O20 of the N = 4 stress-tensor multiplet.. The authors of [24] studied the implications of unitarity and crossing symmetry on the four-point function of the superconformal primary operator O20 of the N = 4 stress-tensor multiplet.5 This superconformal primary is a Lorentz scalar that transforms in the 20 irrep under the so(6) R-symmetry. We study the analogous question in three-dimensional N = 8 SCFTs. In particular, we analyze the four-point function of the superconformal primary O35c of the N = 8 stress-tensor multiplet.
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