Cancellation of divergences up to three loops in exceptional field theory

Understand

We consider the tetrahedral three-loop diagram in $E_d$ exceptional field theory evaluated as a scalar diagram for four external gravitons.

• At lowest order in momenta, this diagram contributes to the $\nabla^6 R^4$ term in the low-energy effective action for M-theory.
• We evaluate explicitly the sums over the discrete exceptional field theory loop momenta that become sums over 1/2-BPS states in the compact exceptional space.
• These sums can be rewritten as Eisenstein series that solve the homogeneous differential equations that supersymmetry implies for the $\nabla^6 R^4$ coupling.