Zhang–Kawazumi invariants and superstring amplitudes

The four-graviton amplitude in Type IIB superstring theory is invariant under the S-duality group SL(2,Z)SL(2,Z) acting on the complex coupling T  . This amplitude contains a sum of effective interactions D2pR4D2pR4 with coefficients that are modular functions of T   whose form has been conjectured when p≤3p≤3. The weak coupling expansion of these coefficients can be calculated in superstring perturbation theory. We here show that the two-loop D6R4D6R4 term is proportional to the integral of an invariant   introduced by Zhang and Kawazumi that is related to the Faltings invariant. The conjectured value of the two-loop superstring contribution to D6R4D6R4 leads us to a prediction for the integral of the Zhang–Kawazumi invariant over the moduli space of genus-two surfaces. We propose invariants corresponding to p>3p>3, which generalize Zhang–Kawazumi invariants.