LLL vs. LLM: Half BPS sector of SYM ≡ quantum Hall system

Abstract

In this paper we elaborate on the correspondence between the quantum Hall system with filling factor equal to one and the N = 4 SYM theory in the 1/2 BPS sector, previously mentioned in the [hep-th/0409174, 0409115]. We show the equivalence of the two in various formulations of the quantum Hall physics. We present an extension of the noncommutative Chern-Simons Matrix theory which contains independent degrees of freedom (fields) for particles and quasiholes. The BPS configurations of our model, which is a model with explicit particle-quasihole symmetry, are in one-to-one correspondence with the 1/2 BPS states in the N = 4 SYM. Within our model we shed light on some less clear aspects of the physics of the N = 4 theory in the 1/2 BPS sector, like the giant dual-giant symmetry, stability of the giant gravitons, and stringy exclusion principle and possible implications of the (fractional) quantum Hall effect for the AdS/CFT correspondence.

Dynamics of giant-gravitons in the LLM geometry and the fractional quantum Hall effect

“…Ref. [9] made a remarkable inspiring attempt in this direction. The 1/2 BPS sector for a single chiral scalar in N = 4 SYM compactified on S 3 is a matrix model.…”

Fractional quantum Hall effect on the two-sphere: A matrix model proposal

“…Note also that an overall phase in this matrix is irrelevant in the transformations (15) . Then we can show that the above lagrangian changes by a total derivative,…”

“…Alternative approaches have been suggested in [13] where two complex matrices were used to parametrize the coordinates of the electrons and in [14] where a fermionic matrix model was introduced to describe a Hall fluid at filling fraction one. See also [15] for further discussions. The lack of a Chern-Simons type action is surprising, since Laughlin's treatment [16] of the fractional quantum Hall effect was easily extended by Haldane [17] to the two-sphere.…”

BPSR-balls in Script N = 4 SYM onR×S3, quantum Hall analogy and AdS/CFT holography

“…In this case, the wavefunction is nothing but the Laughlin wavefunction for a quantum Hall droplet with filling factor ν = 1, an incompressible quantum fluid forming a circular disk. (For an introduction of the QHE for particle physicists see, e.g., [28]; for that for string theorists see, e.g., [13,10].) The general states described by a wavefunction (5.23) represent planar fluid composed of discontinuous components of the form of concentric rings.…”