Simple-Sum Giant Graviton Expansions for Orbifolds and Orientifolds

TL;DR

This paper investigates simplified giant graviton expansions of the superconformal index in certain 4d orbifold and orientifold theories, revealing conditions under which these expansions reduce to a single sum, especially for orbifolds of bc^3.

Contribution

It demonstrates that simple-sum giant graviton expansions occur in specific orbifold and orientifold theories and identifies geometric conditions for their applicability.

Findings

Simple-sum expansion applies to bc^3 orbifolds and certain orientifolds.

Reduction to a simple sum depends on the toric diagram being a triangle.

The expansion simplification is linked to the geometry of the underlying Calabi-Yau.

Abstract

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 $\mathbb{C}^3$.