Simple-Sum Giant Graviton Expansions for Orbifolds and Orientifolds

We study giant graviton expansions of the superconformal index of 4d orbifold/orientifold theories. In general, a giant graviton expansion is given as a multiple sum over wrapping numbers. It has been known that the expansion can be reduced to a simple sum for the $\mathcal{N}=4$ $U(N)$ SYM by choosing appropriate expansion variables. We find such a reduction occurs for a few examples of orbifold and orientifold theories: ${\mathbb{Z}}_k$ orbifold and orientifolds with $O3$ and $O7$. We also argue that for a quiver gauge theory associated with a toric Calabi-Yau $3$-fold the simple-sum expansion works only if the toric diagram is a triangle, that is, the Calabi-Yau is an orbifold of ${\mathbb{C}}^3$.