Here $|\mathcal{G}|$ is the order of a discrete one-form symmetry group, if such a symmetry exists, which would lead to a multiplicity of the saddle.

${\alpha }^{\prime }$ corrections to the entropy for specific ${\mathrm{AdS}}_5\times S^5$ black holes were shown to vanish directly in $D=10$ [19].

We also assume that any such solution, which in general is complex, is a bona fide saddle point of the Euclidean gravitational path integral for computing the superconformal index; for some further discussion see Refs. [29, 30].

One can gauge fix in $D=5$ and then multiply by an overall factor of $\frac{1}{16\pi G_{(5)}}$.

For the higher derivative theories, the boundary terms are under less control.

Additional discussion of reality conditions in localization appear in [24, 26, 27].

The gluing construction can be generalized to $\partial M_5=S_{\tau }^1\times M_3$, where $M_3$ are squashed three-spheres/lens spaces [40].

The latter condition may be dropped, at the expense of introducing factors of $\gcd (n_1,n_2)$ below.

This is consistent with the explicit Killing spinor given in [9], which is charged under the Hopf (and uncharged under the anti-Hopf) direction in the $S^3$. We can also take ${\sigma }_0={\sigma }_2=-1$, ${\sigma }_1$ is the $\pm$ sign on the right hand side of (2) and with $r_I\leftrightarrow -\frac{1}{2}{\zeta }_I$, associated with the conjugate Killing spinor.

In the two-derivative computation, for the special choice of vector $\ell =(0,1,1)$, there is no contribution from the ${\mathrm{AdS}}_5$ factor, but in the four-derivative computation there still is. This possibility was not taken into account in [45].

Alternatively, [22, 23] compute the two-derivative $D=5$ on shell action using a “transverse” form of the Berline–Vergne–Atiyah–Bott formula [48]. We expect this approach to extend to higher derivatives, leading to equivalent formulas to those presented here.