2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/52207A certain sector of the matrix model for M-theory on a plane wave background has recently been shown to produce the transverse five-brane. We consider this theory at finite temperature. We find that, at a critical temperature it has a Gross-Witten phase transition which corresponds to deconfinement of the matrix model gauge theory. We interpret the phase transition as the Hagedorn transition of M-theory and of type II string theory in the five-brane background. We also show that there is no Hagedorn behaviour in the transverse membrane background case.1+34 pages, JHEP style. v2: typos, refs correctedHigh Energy Physics - Theorytext