$D^6 R^4$ amplitudes in various dimensions

Understand

Four-graviton couplings in the low energy effective action of type II string vacua compactified on tori are strongly constrained by supersymmetry and U-duality.

• While the $R^4$ and $D^4 R^4$ couplings are known exactly in terms of Langlands-Eisenstein series of the U-duality group, the $D^6 R^4$ couplings are not nearly as well understood.
• Exploiting the coincidence of the U-duality group in $D=6$ with the T-duality group in $D=5$, we propose an exact formula for the $D^6 R^4$ couplings in type II string theory compactified on $T^4$, in terms of a genus-two modular integral plus a suitable Eisenstein series.
• The same modular integral computes the two-loop correction to $D^6 R^4$ in 5 dimensions, but here provides the non-perturbative completion of the known perturbative terms in $D=6$.