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

Abstract

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 $D=6$. This proposal hinges on a systematic re-analysis of the weak coupling and large radius of the $D^6{R}^4$ in all dimensions $D\ge 3$, which fills in some gaps and resolves some inconsistencies in earlier studies.

18 pages (+ 26 page appendix), three Mathematica files included with submission; v2: minor corrections, version to appear in JHEP

References in corpus (9)

• Manifest Ultraviolet Behavior for the Three-Loop Four-Point Amplitude of N=8 Supergravity
• Matching the $D^6{\mathcal{R}}^4$ interaction at two-loops
• The D^4 R^4 term in type IIB string theory on T^2 and U-duality
• Modular properties of two-loop maximal supergravity and connections with string theory
• Johnson's homomorphisms and the Arakelov-Green function
• Minimal unitary representations from supersymmetry
• The D^6 R^4 term from three loop maximal supergravity
• SL(2,Z)-invariance and D-instanton contributions to the $D^6{R}^4$ interaction
• $\mathcal{E}{\nabla }^4{R}^4$ type invariants and their gradient expansion