Quantum Mechanical Sectors in Thermal N = 4 Super Yang - Mills on R × S

We study the thermodynamics of U(N) N = 4 Super Yang-Mills (SYM) on R × S3 with non-zero chemical potentials for the SU(4) R-symmetry. We find that when we are near a point with zero temperature and critical chemical potential, N = 4 SYM on R × S3 reduces to a quantum mechanical theory. We identify three such critical regions giving rise to three different quantum mechanical theories. Two of them have a Hilbert space given by the SU(2) and SU(2|3) sectors of N = 4 SYM of recent interest in the study of integrability, while the third one is the half-BPS sector dual to bubbling AdS geometries. In the planar limit the three quantum mechanical theories can be seen as spin chains. In particular, we identify a near-critical region in which N = 4 SYM on R × S3 essentially reduces to the ferromagnetic XXX1/2 Heisenberg spin chain. We find furthermore a limit in which this relation becomes exact.