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 is known that the expansion can be reduced to a simple sum for the ${\cal N}=4$U(N) supersymmetric Yang-Mills (SYM) by choosing appropriate expansion variables. We find such a reduction occurs for a few examples of orbifold and orientifold theories: the $\mathbb {Z}_k$ 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, i.e. the Calabi-Yau is an orbifold of $\mathbb {C}<^>3$.