Matrix Theory on ALE Spaces and Wrapped Membranes

TL;DR

This paper investigates wrapped membranes in matrix theory on ALE spaces, revealing that only roots of the A-D-E groups form BPS states, and provides explicit solutions and spectra consistent with supergravity.

Contribution

It identifies the BPS bound states as roots of A-D-E groups and constructs explicit solutions for the A_{n-1} series, linking noncommutative geometry with wrapped membranes.

Findings

Bound states correspond to roots of A-D-E groups.

Explicit classical solutions for A_{n-1} series.

Spectra around solutions match supergravity predictions.

Abstract

We study the properties of wrapped membranes in matrix theory on ALE spaces. We show that the only BPS bound states of wrapped membranes that can form are roots of the $A$-$D$-$E$ group. We determine a bound on the energy of a bound state and find the correct dependence on the blow-up parameters and longitudinal momentum expected from M-Theory. For the $A_{n-1}$ series, we construct explicit classical solutions for the wrapped membrane bound states. These states have a very rich structure and have a natural interpretation in terms of noncommutative geometry. In the $A_1$ case, we examine the spectrum of excitations around the wrapped membrane solution and provide an explicit calculation of their energies. The results agree exactly with supergravity calculations.