A toy model for the AdS/CFT correspondence
David Berenstein
TL;DR
This work analyzes a large-N gauged one-matrix quantum mechanics in a harmonic oscillator potential as a toy model for AdS/CFT, proposing a two-dimensional string-like dual with string-scale curvature. It develops three equivalent descriptions of the spectrum—closed-string traces, an open-string eigenvalue basis, and Schur-polynomial representations—demonstrating an exact open-closed duality and mapping to half-BPS sectors of N=4 SYM. The authors connect giant gravitons to eigenvalue excitations and holes in the Fermi sea, offering semiclassical perspectives and insights into non-planar corrections, while acknowledging the absence of a fully geometric dual due to strong curvature. The results provide a tractable framework to study AdS/CFT-like dualities, non-perturbative effects, and the interplay between D-branes and gauge-invariant operators, with connections to the c=1 string model.
Abstract
We study the large N gauged quantum mechanics for a single Hermitian matrix in the Harmonic oscillator potential well as a toy model for the AdS/CFT correspondence. We argue that the dual geometry should be a string in two dimensions with a curvature of stringy size. Even though the dual geometry is not weakly curved, one can still gain knowledge of the system from a detailed study of the open-closed string duality. We give a mapping between the basis of states made of traces (closed strings) and the eigenvalues of the matrix (D-brane picture) in terms of Schur polynomials. We connect this model with the study of giant gravitons in AdS_5 x S^5. We show that the two giant gravitons that expand along AdS_5 and S^5 can be interpreted in the matrix model as taking an eigenvalue from the Fermi sea and exciting it very much, or as making a hole in the Fermi sea respectively. This is similar to recent studies of the c=1 string. This connection gives new insight on how to perform calculations for giant gravitons.