Longitudinal 5-branes as 4-spheres in Matrix theory

TL;DR

This work constructs longitudinal 5-branes with a spherical four-dimensional transverse geometry (a fuzzy 4-sphere) in Matrix theory using an explicit $N\times N$ matrix realization. Five matrices $X_i$ are built from the $n$-fold symmetric tensor product of 4×4 gamma matrices, yielding $N = (n+1)(n+2)(n+3)/6$ and $X_i = (r/n) G_i^{(n)}$, and the construction enforces geometric and charge properties that reproduce the L5-brane's expected behavior. The authors compute the energy, 4-brane charge, absence of 2-brane charge, and one-loop graviton interactions, finding agreement with 11D supergravity at leading order in $n$ and showing local flatness and $SO(5)$ invariance. They also discuss fluctuations and conclude that extending to general fluctuating L5-brane geometries is challenging within this framework, hinting at the need for a direct 5-brane action approach for arbitrary shapes.

Abstract

We present a construction in Matrix theory of longitudinal 5-branes whose geometry in transverse space corresponds to a 4-sphere. We describe these branes through an explicit construction in terms of N*N matrices for a particular infinite series of values of N. The matrices used in the construction have a number of properties which can be interpreted in terms of the 4-sphere geometry, in analogy with similar properties of the SU(2) generators used in the construction of a spherical membrane. The physical properties of these systems correspond with those expected from M-theory; in particular, these objects have an energy and a leading long-distance interaction with gravitons which agrees with 11D supergravity at leading order in N.