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

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Unlike the \mathcal{R}^4ℛ4 and \nabla^4\mathcal{R}^4∇4ℛ4 couplings, whose coefficients are Langlands–Eisenstein series of the U-duality group, the coefficient \mathcal{E}^{(d)}_{(0,1)}ℰ(0,1)(d) of the \nabla^6\mathcal{R}^4∇6ℛ4 interaction in the low-energy effective action of type II strings compactified on a torus T^dTd belongs to a more general class of automorphic functions, which satisfy Poisson rather than Laplace-type equations. In earlier work [1], 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 dd. 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=4d=4, the result is shown to coincide with an alternative proposal [2] 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.