We emphasize that in this background field approach the target space is taken to be off-shell. This approach is not to be confused with the off-shell methods of string field theory.

In the case of giant graviton expansions one can understand the brane expansion to arise from trace relations in the dual quantum field theory. For generic boundary conditions there is, however, no radial quantization and as such one would have to understand the field theory origin of such an expansion from a Lagrangian point of view, similar to what was suggested in [15].

Throughout we will use the four-dimensional indices $\mu ,\nu =1,\dots ,4$, the seven-dimensional indices $\alpha ,\beta =5,\dots ,11$, the eleven-dimensional indices $M,N=1,\dots ,11$, and the M2-brane world volume indices $a,b=1,\dots ,3$. Explicit numbered indices (appearing on gamma matrices and frames) are flat but when there is an ambiguity flat indices are hatted.

The eleven-dimensional $\mathrm{\Gamma }$ matrices are given as a direct product: ${\mathrm{\Gamma }}_{\widehat{\mu }}=(i{\gamma }_{(4)}{\gamma }_{\widehat{\mu }})\otimes 1$, and ${\mathrm{\Gamma }}_{\widehat{\alpha }}={\gamma }_{(4)}\otimes {\gamma }_{\widehat{\alpha }}$.

We are not being careful with boundary terms in this discussion and refer to [16, 17] for a more complete discussion.

Note that $s(z)$ is nothing more than the standard 3D $\mathcal{N}=2$ chiral multiplet contribution to supersymmetric $S^3$ partition function.

This is in line with the localization in the moduli space of the pointlike brane, i.e., supergravity [17], and with the absence of additional zero modes, implying that no factors of the target space length scale should be introduced when evaluating the integral.