In this thesis we study N = 6 superconformal field theories (SCFTs) in three dimensions. Such theories are highly constrained by supersymmetry, allowing many quantities to be computed exactly. Yet though constrained, N = 6 SCFTs still exhibit a rich array of behaviors, and in various regimes can be dual holographically to M-theory on AdS4 ×S7, IIA string theory on AdS4 ×CP3, and higher- spin gravity on AdS4. We will use tools from conformal bootstrap and supersymmetric localization to study N = 6 theories, both in general and in holographic regimes.
We begin in Chapter 2 by deriving the supersymmetric Ward identities and the superconformal block expansion for the four-point correlator ⟨SSSS⟩ of stress tensor multiplet scalars S. Chapter 3 then studies the mass-deformed sphere partition function, which can be computed exactly using supersymmetric localization, and relates derivatives of this quantity to specific integrals of ⟨SSSS⟩.
In Chapter 4 we study the IIA string and M-theory limits of the ABJ family of N = 6 SCFTs. Using the supersymmetric Ward identities and localization results, we are able to fully determine the R4 corrections to the ⟨SSSS⟩ correlator in both limits. By taking the flat space limit, we can compare to the known R4 contribution to the IIA and M-theory S-matrix, allowing us to perform a check of AdS/CFT at finite string coupling.
In Chapter 5 we study the higher-spin limit of N = 6 theories. Using the weakly broken higher- spin Ward identity, we completely determine the leading correction to ⟨SSSS⟩ in this limit up to two free parameters, which for ABJ theory we then fix using localization. Finally, in Chapter 6 we perform the first numerical bootstrap study of N = 6 superconformal field theories, allowing us to derive non-perturbative bounds on the CFT data contributing to ⟨SSSS⟩.