Exact Results on Maximally Supersymmetric Conformal Field Theories in Three Dimensions

Abstract

In this thesis, we present exact results on \({\cal N} = 8\) superconformal field theories (SCFTs) in three dimensions. In the first part, we extend previous work by beginning the setup for the \({\cal N} = 8\) superconformal bootstrap in three dimensions. In particular, we express the four-point functions of superconformal primaries in the \((8,B,+)_{[0020]}\) and \((8,B,+)_{[0040]}\) multiplets as a sum of \(SO(8)\) harmonics multiplied by conformal blocks. Using the associativity of the operator algebra, as well as the Ward identities, we determine the maximal set of independent crossing equations relating the conformal blocks. In the second part of this work, we use the results of recent work on supersymmetric localization to obtain exact analytic expressions for the OPE coefficients of certain low-lying \(\frac12\)-BPS and \(\frac14\)-BPS operators in the \(U(N)_1 \times U(N)_{-1}\) ABJM theories \((N=2,3)\) and for the \(SU(2)_3 \times SU(2)_{-3}\) BLG theory. We also find that the partition function values, the normalized four-point functions, and the OPE coefficients all match for the interacting sector of the \(U(3)_1 \times U(3)_{-1}\) ABJM theory and for the \(SU(2)_3 \times SU(2)_{-3}\) BLG theory. We therefore conjecture a previously unknown duality between the two theories.