We emphasize that these corrections are distinct from the corrections obtained by considering higher derivative terms in the action [32, 33], which have recently been computed using localization [16].

Note that we have set the gauge-invariant world volume field strength on the brane to vanish since it plays no role in the following.

In general, the one-form $\eta$ needs only be well defined in a patch covering the leaf, and one must carefully track the $\mathrm{U}(1{)}^3$ transformations between patches, as in [13]. For our applications to the black hole, $\eta$ is globally well defined and so we can ignore this subtlety and work with a globally well-defined one-form.

This expression is considerably simplified by the assumption that $\eta$ is globally well defined. When it is necessary to introduce patches, one also receives contributions from the interfaces between the patches, which are not relevant for our applications.

For reference, the map between the parameters here and in [29, 30] is $({\omega }_1,{\omega }_2,\phi {)}^{\mathrm{here}}=2\pi \mathrm{i}(\sigma ,\tau ,\mathrm{\Delta }{)}^{\mathrm{there}}$. In the language of [29], we are on the “first branch” of solutions. There is also a “second branch” of solutions where the left-hand side of (29) is $-2\pi \mathrm{i}$.

As written, the result (35) is the action of a D3-brane, which is in fact only supersymmetric on the “second branch,” where the parameters satisfy ${\omega }_1+{\omega }_2-3\phi =-2\pi \mathrm{i}$ [in contrast to (29)] [29]. On the first branch, only anti-D3-branes are supersymmetric. By construction, this changes the sign of the WZ term in (8), but also the sign in the $\kappa$-symmetry condition, and hence the sign in the calibration condition (9) (see [38]) and that of the DBI term in (8). The result is that the relative sign between the two terms does not change, preserving the form of (14), but the overall sign does, matching the result of [29] on the first branch.

We emphasize that the auxiliary vector field $Y$ is not a choice of regularization, rather it is a requirement for odd-dimensional localization, see [26], and should drop out after including all possible fixed point and boundary terms.