Abstract
We perform a systematic study of commutative SO(p) invariant matrix models with quadratic and quartic potentials in the large N limit. We find that the physics of these systems depends crucially on the number of matrices with a critical rôle played by p = 4. For p ≤ 4 the system undergoes a phase transition accompanied by a topology change transition. For p > 4 the system is always in the topologically trivial phase and the eigenvalue distribution is a Dirac delta function spherical shell. We verify our analytic work with Monte Carlo simulations.
Highlights
All of these conjectured formulations of M-theory are regularised versions of the supermembrane
We perform a systematic study of commutative SO(p) invariant matrix models with quadratic and quartic potentials in the large N limit
We find that the physics of these systems depends crucially on the number of matrices with a critical role played by p = 4
Summary
2.1 The model We consider a commuting SO(p) invariant p-matrix model with partition function:. Equation (2.6) determines the eigenvalue distribution in the large N limit and admits rotationally invariant shell solutions. The only shell solution consistent with SO(p) invariance is a p − 1 dimensional spherical shell These solutions have been considered in refs. Equation (2.6) admits p-dimensional (“fat”) rotationally invariant solutions, which may or may not be energetically favoured relative to the shell solution. To explore these solutions we consider a course grained approximation: Λi → x ,. Upon variation with respect to ρ we obtain the integral equation: μp + Vp(|x|) = dpx ρp(x ) log(x − x )2 ,.
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