On Stable Sector in Supermembrane Matrix Model

TL;DR

This paper analyzes the spectrum of SU(2) x SO(2) matrix supersymmetric quantum mechanics, deriving asymptotic formulas and exploring the spectrum's structure based on quantum numbers, with implications for model stability.

Contribution

It provides an explicit solution to the Gauss law constraints and characterizes the spectrum's structure for different quantum numbers in the supermembrane matrix model.

Findings

Energy levels are four-fold degenerate with respect to n.

For odd n_q, the spectrum is forbidden; for even n_q, it has both continuous and discrete parts.

Half-integer n_q yields a purely discrete spectrum, stabilizing the model.

Abstract

We study the spectrum of SU(2) x SO(2) matrix supersymmetric quantum mechanics. We use angular coordinates that allow us to find an explicit solution of the Gauss law constraints and single out the quantum number n (the Lorentz angular momentum). Energy levels are four-fold degenerate with respect to n and are labeled by n_q, the largest n in a quartet. The Schr\"odinger equation is reduced to two different systems of two-dimensional partial differential equations. The choice of a system is governed by n_q. We present the asymptotic solutions for the systems deriving thereby the asymptotic formula for the spectrum. Odd n_q are forbidden, for even n_q the spectrum has a continuous part as well as a discrete one, meanwhile for half-integer n_q the spectrum is purely discrete. Taking half-integer n_q one can cure the model from instability caused by the presence of continuous spectrum.