Giant gravitons and the emergence of geometric limits in $\beta$-deformations of ${\cal N}=4$ SYM

TL;DR

This paper investigates how the geometry of the dual space in $eta$-deformed ${ m SYM}$ theories can be inferred from the spectrum of open strings between giant gravitons, revealing conditions for geometric limits based on the deformation parameter.

Contribution

It introduces a method to determine the geometric nature of the dual space by analyzing open string spectra in $eta$-deformed ${ m SYM}$, highlighting the role of number-theoretic properties of $eta$.

Findings

When $eta$ is a root of unity, the dual space is an orbifold.

Near roots of unity, the space is a finite deformation of the orbifold.

Irrational $eta$ can lead to sporadic light states.

Abstract

We study a one parameter family of supersymmetric marginal deformations of ${\cal N}=4$ SYM with $U(1)^3$ symmetry, known as $\beta$-deformations, to understand their dual $AdS\times X$ geometry, where $X$ is a large classical geometry in the $g_{YM}^2N\to \infty$ limit. We argue that we can determine whether or not $X$ is geometric by studying the spectrum of open strings between giant gravitons states, as represented by operators in the field theory, as we take $N\to\infty$ in certain double scaling limits. We study the conditions under which these open strings can give rise to a large number of states with energy far below the string scale. The number-theoretic properties of $\beta$ are very important. When $\exp(i\beta)$ is a root of unity, the space $X$ is an orbifold. When $\exp(i\beta)$ close to a root of unity in a double scaling limit sense, $X$ corresponds to a finite deformation of the orbifold. Finally, if $\beta$ is irrational, sporadic light states can be present.