Loops in exceptional field theory

Abstract

We study certain four-graviton amplitudes in exceptional field theory in dimensions $D\geq 4$ up to two loops. As the formulation is manifestly invariant under the U-duality group $E_{11-D}(\mathbb{Z})$, our resulting expressions can be expressed in terms of automorphic forms. In the low energy expansion, we find terms in the M-theory effective action of type $R^4$, $\nabla^4R^4$ and $\nabla^6 R^4$ with automorphic coefficient functions in agreement with independent derivations from string theory. This provides in particular an explicit integral formula for the exact string theory $\nabla^6 R^4$ threshold function. We exhibit moreover that the usual supergravity logarithmic divergences cancel out in the full exceptional field theory amplitude, within an appropriately defined dimensional regularisation scheme. We also comment on terms of higher derivative order and the role of the section constraint for possible counterterms.