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A bstract We study the mass-deformed sphere free energy of three-dimensional $$ \mathcal{N} $$ N = 2 superconformal field theories with holographic duals. Building on previous observations, we conjecture a proportionality relation between the sphere free energy on the boundary and the prepotential of the four-dimensional $$ \mathcal{N} $$ N = 2 supergravity theory in the bulk. We verify this formula by explicit computation in several examples of supergravity theories with vector multiplets and hypermultiplets. 

A bstract We study which bulk couplings contribute to the S 3 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. 

Abstract The UV finiteness found in calculations of the 4‐point amplitude insupergravity at loop orderhas not been explained, which motivates our study of the relevant superspace invariants and on‐shell superamplitudes for bothand. The local 4‐point superinvariants forare expected to have nonlinear completions whose 6‐point amplitudes have non‐vanishing SSL's (soft scalar limits), violating the behavior required of Goldstone bosons. For, we find atthat local 6‐point superinvariant and superamplitudes, which might cancel these SSL's, do not exist. This rules out the candidate 4‐point counterterm and thus gives a plausible explanation of the observedfiniteness. However, atwe construct a local 6‐point superinvariant with non‐vanishing SSL's, so the SSL argument does not explain the observedUV finiteness. Forsupergravity there are no 6‐point invariants at eitheror 4, so the SSL argument predicts UV finiteness.