Energy of toroidal M2 brane in flat 11d background

Abstract: The supermembrane action is non-linear and a priori non-renormalizable. Still, some quantities in this theory may be free of log UV divergences and thus defined unambiguously. To explore this possibility we compute the energy of an M2 brane wrapped on a 2-torus in flat 11d space to two loops in the inverse-tension expansion. The result is free of logarithmic divergences and the same should hold also at higher loop orders. For fermions periodic around both circles the quantum corrections cancel, consistent with the BPS nature of the wrapped M2 brane state. For fermions antiperiodic around one or both circles the energy is a non-trivial function of the radii given by infinite sums of modified Bessel functions. We also compute 3-loop correction to the energy of the bosonic membrane on $\mathbb R^2\times S<sup>1$. In contrast to the string case, it contains, besides powers of the 1-loop $ζ(3)$ coefficient, a new term proportional to $ζ(9)$. Summing contributions of all-loop bubble graphs leads to a simple cubic equation for the membrane energy. The analogous resummation in the Nambu or GS string t case reproduces the familiar exact square-root expression ${\mathcal E}=\sqrt{(2πR\,T_1)<sup>2+m_0<sup>2}$. We find that the bubble-graph part of the membrane energy does not vanish for any value of the radius $R_{11}$. If contributions of other non-trivial diagrams in the supermembrane case do not qualitatively change this conclusion, this would disfavour the conjectured relation between 11d theory on an antiperiodic circle and the strong-coupling limit of type 0A string theory.