The non-perturbative phase diagram of the BMN matrix model

Yuhma Asano,Samuel Kováčik,Denjoe O’Connor,Veselin G Filev

doi:10.1007/jhep07(2018)152

Abstract

We study the maximally supersymmetric plane wave matrix model (the BMN model) at finite temperature, T, and locate the high temperature phase boundary in the (μ, T) plane, where μ is the mass parameter. We find the first transition, as the system is cooled from high temperatures, is from an approximately SO(9) symmetric phase to one where three matrices expand to form fuzzy spheres. For μ > 3.0 there is a second distinct transition at a lower temperature. The two transitions approach one another at smaller μ and merge in the vicinity of μ = 3.0. The resulting single transition curve then approaches the gauge/gravity prediction as μ is further decreased. We find a rough estimate of the transition, for all μ, is given by a Padé resummation of the large-μ, three-loop perturbative predictions. We find evidence that the transition at small μ is to an M5-brane phase of the theory.

Highlights

We study the maximally supersymmetric plane wave matrix model at finite temperature, T, and locate the high temperature phase boundary in the (μ, T ) plane, where μ is the mass parameter
Since the corresponding supergravity solutions are supposed to be obtained at low energy, the geometry should be constructed by a low-energy moduli operator in the gauge theory
We map the phase diagram in the (μ, T ) plane and determine the phase boundary as the system cools from the high temperature phase

Summary

The BMN model

The covariant derivative is defined by Dτ · = ∂τ · −i[A, ·] and C is the charge conjugation matrix of Spin(9). In (2.1) we have rescaled Xi, ψ, τ and A to absorb the dependence on the ’t Hooft coupling defined as λ = N g2 and we use β = λ1/3/T and μ = μ0/λ1/3 with μ0 the mass parameter of the plane wave geometry. This result, while reliable for large μ becomes untrustworthy as μ decreases and passes through zero for μ 13.4. We use the Pade resummed result (2.3) as a guide to where one might expect the transition in the full model as μ is decreased and study the system using the rational hybrid Monte Carlo algorithm and a novel lattice discretisation described in [25]. These are BPS states and are protected ground states of the quantum Hamiltonian

Lattice formulation

Observables

Nβ β dτ Tr

Results and phase diagram

Conclusions