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

TL;DR

This paper proves the Zhang-Kawazumi invariant is an eigenfunction of the Laplace-Beltrami operator, enabling explicit calculation of the two-loop $D^6 \mathcal{R}^4$ interaction coefficient in type II superstring theory, confirming duality and supersymmetry predictions.

Contribution

It demonstrates that the Zhang-Kawazumi invariant is an eigenfunction of the Laplace-Beltrami operator, allowing explicit evaluation of the two-loop $D^6 \mathcal{R}^4$ interaction coefficient.

Findings

The ZK invariant is an eigenfunction with eigenvalue 5 of the Laplace-Beltrami operator.

Explicit evaluation of the integral matches S-duality and supersymmetry predictions.

Provides a review of $D^{2p} \mathcal{R}^4$ interactions in compactified type II superstring theory.

Abstract

The coefficient of the $D^6 {\cal R}^4$ 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 $D^6 {\cal R}^4$ interaction predicted on the basis of S-duality and supersymmetry. A review of the current understanding of the $D^{2p} {\cal R}^4$ interactions in type II superstring theory compactified on a torus $T^d$ with $p \leq 3$ and $d \leq 4$ is included.