Realizing the Quantum Hall System in String Theory

In a recent paper Bernevig, Brodie, Susskind and Toumbas constructed a brane realization of the Quantum Hall fluid. Since then it has been realized that the Quantum Hall system is very closely related to non–commutative Chern Simons theory and this suggests alternative brane constructions which we believe are more reliable and clear. In this paper a brane construction is given for the non–commutative Chern Simons Matrix formulation of the Quantum Hall system as described by in recent papers by Susskind, Polychronakos and by Hellerman and Van Raamsdonk. The system is a generalized version of Berkooz’s “Rigid Light Cone Membrane which occurs as an excition of the DLCQ description of the M5–brane in a background 3– form field. The original construction of Berkooz corresponds to the fully filled ν = 1 state of the QH system. To change the filling fraction to ν = 1/(k + 1) a system of k background D8-branes is required. Quasi–hole excitations can be generated by passing a D6-brane though the Rigid Membrane. 1 Chern Simons Matrix Theory and the Quantum Hall System According to [1, 2, 3], the quantum hall system at filling fraction ν = 1/(k + 1) can be described by Abelian non–commutative Chern Simons Theory at level k.