The giant graviton expansion in AdS$_5\times$SE$_5$

We study giant graviton-like D3-branes as probe configurations in type IIB supergravity backgrounds AdS$_5\times$SE$_5$, with emphasis on the structure of the five-dimensional Sasaki-Einstein manifolds SE$_5$. These configurations wrap supersymmetric three-cycles in SE$_5$ and rotate at the speed of light along the Reeb direction. We formulate the general problem in terms of the transverse K\"ahler potential and show that configurations carrying maximal angular momentum can be described by loci where the transverse K\"ahler potential diverges in suitably chosen coordinates. We quantize a particular set of excitations, similar to those considered in the AdS$_5\times S^5$ case, and show that they are governed by a Fock-Darwin problem with a conical deficit, which generalizes the Landau problem obtained for $S^5$. Succinctly, the information about the geometry of SE$_5$ is encoded in the form of the external potential and the conical deficit. We compute a protected contribution to the superconformal index arising from quantized fluctuations of supersymmetric giant graviton configurations. For the explicit case of AdS$_5\times T^{1,1}$, we recover the finite-N protected index of the dual quiver ${\cal N}=1$ superconformal field theory in the sector captured by a non-Abelian generalization of the quantized degrees of freedom.