The D^6 R^4 term from three loop maximal supergravity

Abstract

We consider the D^6 R^4 term which is the leading contribution in the low momentum expansion of the three loop, four graviton amplitude in maximal supergravity. We calculate the moduli dependent coefficient of this term in 11 dimensional supergravity compactified on T^2. Only diagrams that involve the Mercedes skeleton contribute resulting in a compact expression involving an integral over 6 Schwinger parameters. We express this integral as an integral over the moduli of an auxiliary T^3. This includes an SL(3,Z) invariant integral over the shape moduli of the T^3. We discuss the renormalization of the ultraviolet divergences of this amplitude that arise from the boundaries of moduli space. The renormalized amplitude is simple which is a consequence of the fact that the D^6 R^4 term is BPS.

62 pages, 23 figures, LaTeX

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• Eisenstein series and automorphic representations
• Loops in exceptional field theory
• $D^6{R}^4$ amplitudes in various dimensions
• Poisson equation for the Mercedes diagram in string theory at genus one
• Constraining Higher Derivative Supergravity with Scattering Amplitudes
• Tetrahedral modular graph functions
• The two D6R4 type invariants and their higher order generalisation
• Higher Derivative Supergravities in Diverse Dimensions
• Poisson equation for the three loop ladder diagram in string theory at genus one
• Eigenvalue equation for the modular graph $C_{a,b,c,d}$
• Infrared divergences and harmonic anomalies in the two-loop superstring effective action
• Cancellation of divergences up to three loops in exceptional field theory
• Low momentum expansion of one loop amplitudes in heterotic string theory
• $D^6{R}^4$ curvature corrections, modular graph functions and Poincaré series
• Worldsheet (anti)instanton bound states in type II on $T^2$
• Some finite terms from ladder diagrams in three and four loop maximal supergravity