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 ${\cal N}=4$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 $\mathbb {C}^3$.