The Holographic Contributions to the Sphere Free Energy

TL;DR

This paper investigates which bulk couplings in holographic duals affect the sphere free energy of 3D ${ m N}=2$ SCFTs, revealing that many couplings do not influence $F(rak{m})$, while others do, based on super-algebra properties.

Contribution

It identifies the specific bulk couplings that influence the sphere free energy in holographic duals of 3D ${ m N}=2$ SCFTs, highlighting the insensitivity to many couplings.

Findings

$F(rak{m})$ is independent of non-chiral F-terms and D-terms.

$F(rak{m})$ depends on chiral F-terms and real-mass terms.

Massive vector multiplet couplings do not affect $F(rak{m})$.

Abstract

We study which bulk couplings contribute to the $S^3$ free energy $F(\mathfrak{m})$ of three-dimensional ${\cal N} = 2$ superconformal field theories with holographic duals, potentially deformed by boundary real-mass parameters $\mathfrak{m}$. In particular, we show that $F(\mathfrak{m})$ 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, $F(\mathfrak{m})$ 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, $\mathfrak{osp}(2|4)$. 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 $F(\mathfrak{m})$ 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 $F(\mathfrak{m})$ does not contain information about scaling dimensions or OPE coefficients of single-trace long scalar ${\cal N} = 2$ superconformal multiplets.