1/8-BPS Couplings and Exceptional Automorphic Functions

Abstract

Unlike the ${\mathcal{R}}^4$ and ${\nabla }^4{\mathcal{R}}^4$ couplings, whose coefficients are Langlands-Eisenstein series of the U-duality group, the coefficient ${\mathcal{E}}_{(0,1)}^{(d)}$ of the ${\nabla }^6{\mathcal{R}}^4$ 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.

Minor corrections

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Cited by in corpus (7)

• D-instantons in Type IIA string theory on Calabi-Yau threefolds
• Two-loop superstring five-point amplitudes II: Low energy expansion and S-duality
• Higher Derivative Supergravities in Diverse Dimensions
• Poincaré series for modular graph forms at depth two. I. Seeds and Laplace systems
• To the cusp and back: Resurgent analysis for modular graph functions
• Perturbative higher-derivative terms in N=6 asymmetric orbifolds
• Worldsheet (anti)instanton bound states in type II on $T^2$