Commuting quantum matrix models

Abstract

We study a quantum system of p commuting matrices and find that such a quantum system requires an explicit curvature dependent potential in its Lagrangian for the system to have a finite energy ground state. In contrast it is possible to avoid such curvature dependence in the Hamiltonian. We study the eigenvalue distribution for such systems in the large matrix size limit. A critical rôle is played by p = 4. For p ≥ 4 the competition between eigenvalue repulsion and the attractive potential forces the eigenvalues to form a sharp spherical shell.

Multi-matrix correlators and localization

“…unlikely to provide a solution to our differential equation, because our effective potential appears to contain three-body interactions. Finding such a formula would amount to finding eigenfunctions of ∇ A,B along the lines of [16,29], but in our case we are only interested in the ground state wavefunction in the effective potential. While a complete analysis is beyond the scope of this paper, we may still lay out a prescription for finding an analytical solution to our modified integral.…”

“…If we want to replace the characters in (2.1) with restricted characters, we must either change the domain of integration or integrate against an appropriate measure factor that is sensitive to this information. This is equivalent to finding an analytic formula for restricted characters, which may be recast as a Schrödinger problem over the space of commuting matrices [18,29]. The point is that the norm of the quarter-BPS coherent state is related to the heat kernel over the space of commuting matrices, or equivalently to the Green's function of the Schrödinger equation.…”

Membrane Matrix models and non-perturbative checks of AdS/CFT

“…This was studied in [24] where it was found that for an ensemble of p commuting matrices, X a , in a quadratic potential, the X a are concentrated on a sphere for p ≥ 4.…”

Extremal chiral ring states in the AdS/CFT correspondence are described by free fermions for a generalized oscillator algebra

“…Indeed, the free fermion picture has become even more important in describing the correct physics for these D-branes as coherent states [15,16]. Another problem that has arisen recently is that the geometry arising from commuting matrix models of many matrices seems to be renormalized and essentially collapses when curvature corrections to the effective dynamics on moduli space are included [17]. This suggests that many of these ideas on the dual geometry for gauge theories being based on an approximately commuting matrix model should be reformulated.…”