Matrix theory, AdS/CFT and Higgs-Coulomb equivalence

We discuss the relation between the Matrix theory definitions of a class of decoupled theories and their AdS/CFT description in terms of the corresponding near-horizon geometry. The near horizon geometry, naively part of the Coulomb branch, is embedded in the Higgs branch via a natural change of variables. The principles of the map apply to all DLCQ descriptions in terms of hyper-Kähler quotients, such as the ADHM quantum mechanics for the D1-D5 system. We then focus the (2, 0) field theory, and obtain an explicit mapping from all states in the N₀ = 1 momentum sector of N₄ M5-branes to states in (a DLCQ version of) AdS₇ × S⁴. We show that, even for a single DO-brane, the space-time coordinates become non-commuting variables, suggesting an inherent non-commutativity of space-time in the presence of field strengths even for theories with gravity.