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#### 1/8-BPS Couplings and Exceptional Automorphic Functions

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##### Citation

Bossard, G., Kleinschmidt, A., & Pioline, B. (2020). 1/8-BPS Couplings and Exceptional
Automorphic Functions.* SciPost Physics,* *8*(4): 054. doi:10.21468/SciPostPhys.8.4.054.

Cite as: https://hdl.handle.net/21.11116/0000-0005-A589-6

##### 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.

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.