Matching the $D^6{\mathcal{R}}^4$ interaction at two-loops

Abstract

The coefficient of the $D^6{\mathcal{R}}^4$ interaction in the low energy expansion of the two-loop four-graviton amplitude in type II superstring theory is known to be proportional to the integral of the Zhang-Kawazumi (ZK) invariant over the moduli space of genus-two Riemann surfaces. We demonstrate that the ZK invariant is an eigenfunction with eigenvalue 5 of the Laplace-Beltrami operator in the interior of moduli space. Exploiting this result, we evaluate the integral of the ZK invariant explicitly, finding agreement with the value of the two-loop $D^6{\mathcal{R}}^4$ interaction predicted on the basis of S-duality and supersymmetry. A review of the current understanding of the $D^{2p}{\mathcal{R}}^4$ interactions in type II superstring theory compactified on a torus $T^d$ with $p\le 3$ and $d\le 4$ is included.

40 pages, various typos and coefficients corrected in version 2

References in corpus (7)

• Manifest Ultraviolet Behavior for the Three-Loop Four-Point Amplitude of N=8 Supergravity
• 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
• SL(2,Z)-invariance and D-instanton contributions to the $D^6{R}^4$ interaction
• Topics in Two-Loop Superstring Perturbation Theory
• Gross--Schoen Cycles and Dualising Sheaves

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• Generating series of all modular graph forms from iterated Eisenstein integrals
• Minimal unitary representations from supersymmetry
• Tetrahedral modular graph functions
• Non-planar corrections in orbifold/orientifold $\mathcal{N}=2$ superconformal theories from localization
• On Genus-one String Amplitudes on $Ad{S}_5\times S^5$
• Low momentum expansion of one loop amplitudes in heterotic string theory
• Two-loop superstring five-point amplitudes I: Construction via chiral splitting and pure spinors
• Poisson equation for genus two string invariants: a conjecture
• Integrating simple genus two string invariants over moduli space
• Two-loop Yang-Mills diagrams from superstring amplitudes