Warped $AdS_{2}$ and $SU(1,1|4)$ symmetry in Type IIB

We investigate the existence of solutions with 16 supersymmetries to Type IIB supergravity on a spacetime of the form $AdS_{2}\times S^{5}\times S^{1}$ warped over a two-dimensional Riemann surface $\Sigma$. The existence of the Lie superalgebra $SU(1,1|4) \subset PSU(2,2|4)$, whose maximal bosonic subalgebra is $SO(2,1)\oplus SO(6)\oplus SO(2)$, motivates the search for half-BPS solutions with this isometry that are asymptotic to $AdS_{5} \times S^{5}$. We reduce the BPS equations to the Ansatz for the bosonic fields and supersymmetry generators compatible with these symmetries, then show that the only non-trivial solution is the maximally supersymmetric solution $AdS_{5}\times S^{5}$. We argue that this implies that no solutions exist for fully back-reacted D7 probe or D7/D3 intersecting branes whose near-horizon limit is of the form $AdS_{2}\times S^{5}\times S^{1}\times \Sigma$.