$R^4$ couplings, the fundamental membrane and exceptional theta correspondences

TL;DR

This work seeks a first-principles derivation of non-perturbative $R^4$ couplings in M-theory by treating the eleven-dimensional membrane as the fundamental degrees of freedom. It constructs a covariant membrane amplitude with $Sl(3,\mathbb{Z})$ modular invariance and analyzes its relation to the known one-loop string results, revealing that while the BPS spectrum is correctly captured, the instanton summation measure is not, hinting at the need for an exceptional theta correspondence to lift $Sl(3,\mathbb{Z})$ data to $E_{d+1(d+1)}(\mathbb{Z})$ automorphic forms. The authors propose an explicit theta-correspondence program involving $\Xi_{d+1}$ invariant under $Sl(3,\mathbb{Z})\times E_{d+1(d+1)}(\mathbb{Z})$ that would reproduce the full $R^4$ couplings and potentially unify membranes with five-branes, offering a path toward a finite, consistent membrane quantization. This exceptional framework, if realized, could reveal the fundamental degrees of freedom of M-theory and deepen connections between non-perturbative string/M-theory and automorphic forms.

Abstract

This letter is an attempt to carry out a first-principle computation in M-theory using the point of view that the eleven-dimensional membrane gives the fundamental degrees of freedom of M-theory. Our aim is to derive the exact BPS $R^4$ couplings in M-theory compactified on a torus $T^{d+1}$ from the toroidal BPS membrane, by pursuing the analogy with the one-loop string theory computation. We exhibit an $Sl(3,\Zint)$ modular invariance hidden in the light-cone gauge (but obvious in the Polyakov approach), and recover the correct classical spectrum and membrane instantons; the summation measure however is incorrect. It is argued that the correct membrane amplitude should be given by an exceptional theta correspondence lifting $Sl(3,\Zint)$ modular forms to $\exc(\Zint)$ automorphic forms, generalizing the usual theta lift between $Sl(2,\Zint)$ and $SO(d,d,\Zint)$ in string theory. The exceptional correspondence $Sl(3)\times E_{6(6)}\subset E_{8(8)}$ offers the interesting prospect of solving the membrane small volume divergence and unifying membranes with five-branes.