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

Abstract

We study a one parameter family of supersymmetric marginal deformations of $\mathcal{N}=4$ 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 $g_{YM}^{2}N\to \infty$ 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.

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