By a mild abuse of terminology, we use the terms superconformal and off shell interchangeably when referring to the 5D supergravity formalism.

Here, “auxiliary” refers to fields without kinetic terms in the two-derivative theory.

Our perspective is off shell and therefore intrinsically different from the approach of [42, 49], which are based on on-shell models where the gauge coupling to the hypermultiplet effectively plays the role of a Lagrange multiplier. The on-shell nature of the hypermultiplets in those constructions makes it difficult to extend the analysis beyond the two-derivative level. One can rather view our results as complementary to the geometric perspective of $a$ maximization in terms of the internal Sasakian volume, developed in [50, 51, 52].

We leave the details of the spinorial representation of $\mathcal{Q}$ unspecified, except for its $R$ charge, as they play no role here. See Ref. [9] and references therein for details.

By a standard abuse of notation, we denote the complete multiplet in the same way as its lowest component. The distinction should be evident from the context in a given discussion or a formula.

Note that classifying superconformal invariants by their derivative order is not unique. Because gauged supergravity contains a natural length scale $l$, one may instead classify invariants according to their scaling in $l$, which can be read off from Table 1 and Eq. (2). From this perspective, the $\mathcal{R}F$ invariant exhibits a scaling behavior distinct from that of the standard two-derivative action.

In the present convention for curvature, ${\mathrm{AdS}}_5$ has a constant positive curvature.