Abstract
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.
Highlights
Much of our current understanding of superstring theory is based on two different expansion schemes: the genus expansion, and the low-energy expansion
The coefficient of the D6R4 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 genustwo 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
Summary
Much of our current understanding of superstring theory is based on two different expansion schemes: the genus expansion, and the low-energy expansion. The situation for the coefficient E(0,1) of the D6R4 interaction is more challenging, as it is clear from the analysis of 2-loop supergravity amplitudes that E(0,1) cannot be an eigenmode of the Laplacian, rather it must satisfy an inhomogeneous Laplace equation, eq (2.31) below, with a source proportional to the square of E(0,0), see [12, 13, 17] The solution of this equation contains perturbative terms that correspond to zero-, one-, two-, and three-loop contributions in superstring theory. We shall evaluate the integral in (1.2) and thereby find agreement between the value predicted by S-duality, (1.1), and the result of two-loop superstring perturbation theory This matching is highly non-trivial, and involves some novel mathematics. It would be interesting to understand its connection with (1.4) and (1.5), if any exists, especially since the ZK invariant for genus h ≥ 3 does not satisfy a simple equation of the type (1.4), as will be shown in appendix C
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