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$1/8$BPS Couplings and Exceptional Automorphic Functions
by Guillaume Bossard, Axel Kleinschmidt, Boris Pioline
This Submission thread is now published as
Submission summary
Submission information 
Preprint Link: 
scipost_202003_00049v1
(pdf)

Date accepted: 
20200326 
Date submitted: 
20200320 01:00 
Submitted by: 
Pioline, Boris 
Submitted to: 
SciPost Physics 
Ontological classification 
Academic field: 
Physics 
Specialties: 
 HighEnergy Physics  Theory

Approach: 
Theoretical 
Abstract
Unlike the $R^4$ and $\nabla^4 R^4$ couplings, whose coefficients are LanglandsEisenstein series of the Uduality group, the coefficient $\mathcal{E}^{(d)}_{(0,1)}$ of the $\nabla^6 R^4$ interaction in the lowenergy 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 Laplacetype equations. In earlier work, it was proposed that the exact coefficient is given by a twoloop integral in exceptional field theory, with the full spectrum of mutually 1/2BPS states running in the loops, up to the addition of a particular LanglandsEisenstein 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 nonperturbative corrections
which have the expected form for 1/8BPS instantons and bound states of 1/2BPS instantons and antiinstantons. The additional LanglandsEisenstein series arises from a subtle cancellation between the twoloop amplitude with 1/4BPS states running in the loops, and the threeloop amplitude with mutually 1/2BPS states in the loops. For $d=4$, the result is shown to coincide with an alternative proposal in terms of a covariantised genustwo string amplitude, due to interesting identities between the KawazumiZhang invariant of genustwo curves and its tropical limit, and between double lattice sums for the particle and string multiplets, which may be of independent mathematical interest.
Published as
SciPost Phys. 8, 054 (2020)