$1/8$-BPS couplings and exceptional automorphic functions

Abstract

Unlike the $R^4$ and $\nabla^4 R^4$ couplings, whose coefficients are Langlands--Eisenstein series of the U-duality group, the coefficient $\mathcal{E}^{(d)}_{(0,1)}$ of the $\nabla^6 R^4$ interaction in the low-energy effective action of type II strings compactified on a torus $T^d$ belongs to a more general class of automorphic functions, which satisfy Poisson rather than Laplace-type equations. In earlier work, it was proposed that the exact coefficient is given by a two-loop integral in exceptional field theory, with the full spectrum of mutually 1/2-BPS states running in the loops, up to the addition of a particular Langlands--Eisenstein series. Here we compute the weak coupling and large radius expansions of these automorphic functions for any $d$. We find perfect agreement with perturbative string theory up to genus three, along with non-perturbative corrections which have the expected form for 1/8-BPS instantons and bound states of 1/2-BPS instantons and anti-instantons. The additional Langlands--Eisenstein series arises from a subtle cancellation between the two-loop amplitude with 1/4-BPS states running in the loops, and the three-loop amplitude with mutually 1/2-BPS states in the loops. For $d=4$, the result is shown to coincide with an alternative proposal in terms of a covariantised genus-two string amplitude, due to interesting identities between the Kawazumi--Zhang invariant of genus-two curves and its tropical limit, and between double lattice sums for the particle and string multiplets, which may be of independent mathematical interest.