The holographic contributions to the sphere free energy

Abstract

Abstract We study which bulk couplings contribute to the S3 free energy F($$ \mathfrak{m} $$ m ) of three-dimensional $$ \mathcal{N} $$ N = 2 superconformal field theories with holographic duals, potentially deformed by boundary real-mass parameters m. In particular, we show that F($$ \mathfrak{m} $$ m ) is independent of a large class of bulk couplings that include non-chiral F-terms and all D-terms. On the other hand, in general, F($$ \mathfrak{m} $$ m ) does depend non-trivially on bulk chiral F-terms, such as prepotential interactions, and on bulk real-mass terms. These conclusions can be reached solely from properties of the AdS super-algebra, $$ \mathfrak{osp} $$ osp (2|4). We also consider massive vector multiplets in AdS, which in the dual field theory correspond to long single-trace superconformal multiplets of spin zero. We provide evidence that F($$ \mathfrak{m} $$ m ) is insensitive to the vector multiplet mass and to the interaction couplings between the massive vector multiplet and massless ones. In particular, this implies that F($$ \mathfrak{m} $$ m ) does not contain information about scaling dimensions or OPE coefficients of single-trace long scalar $$ \mathcal{N} $$ N = 2 superconformal multiplets.