Matching the D 6ℛ4 interaction at two-loops

The coefficient of the D 6ℛ4 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 6ℛ4 interaction predicted on the basis of S-duality and super-symmetry. A review of the current understanding of the D 2p ℛ4 interactions in type II superstring theory compactified on a torus T d with p ≤ 3 and d ≤ 4 is included.