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.
Highlights
Exceptional field theory [1,2,3,4,5,6,7,8,9,10,11,12] provides in principle a framework to compute manifestlyU-duality invariant amplitudes in perturbation theory
We study certain four-graviton amplitudes in exceptional field theory in dimensions D ≥ 4 up to two loops
In the low energy expansion, we find terms in the M-theory effective action of type R4, ∇4R4 and ∇6R4 with automorphic coefficient functions in agreement with independent derivations from string theory
Summary
Exceptional field theory [1,2,3,4,5,6,7,8,9,10,11,12] provides in principle a framework to compute manifestly. We will perform the one- and two-loop calculation of the four-graviton amplitude in exceptional field theory This will automatically produce U-duality invariant expressions and include full multiplets of. There is an additional divergence that occurs at 2-loops in six dimensions in exceptional field theory, which is associated to the 1-loop R4 type form factor divergence in supergravity (as advocated in [32]) We analyse this amplitude in detail and show that the divergences cancel out in this case as well in the complete amplitude. Appendix B we study certain properties of SO(n, n) Eisenstein series that are used in analysing the ∇6R4 correction and appendix C discusses the possibility of defining automorphic functions from a sum over charges
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