2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/183010The partition function of the membrane is investigated. In particular, the case relevant to perturbative string theory of a membrane with topology $S^1 \times Σ$ is examined. The coupling between the string world sheet Euler character and the dilaton is shown to arise from a careful treatment of the membrane partition function measure. This demonstrates that the M-theory origin of the dilaton coupling to the string world sheet is quantum in nature.12 pages, latexHigh Energy Physics - Theorytext