A toy model for the AdS/CFT correspondence

We study the large-N gauged quantum mecchanics for a single hermitean 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. This is interpreted as an exact open-closed duality. We connect this model with a decoupling limit of = 4 SYM and the study of giant gravitons in AdS₅ × S⁵. We show that the two giant gravitons that expand along AdS₅ and S⁵ 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.