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1/8-BPS Couplings and Exceptional Automorphic Functions
High Energy Physics - Theory · Submitted 15 Jan 2020
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 0.94 | 0.18 | 0.18 | asserted 1.00 |
| s1 | 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. | background 1.00 | 0.06 | 0.07 | hedged possibility 0.61 |
| s2 | Here we compute the weak coupling and large radius expansions of these automorphic functions for any $d$. | contribution 0.96 | 0.96 | 0.11 | asserted 0.98 |
| s3 | 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. | result 1.00 | 0.95 | 0.40 | asserted 1.00 |
| s4 | 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. | interpretation 0.52 | 0.67 | 0.13 | asserted 1.00 |
| s5 | 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. | result 0.95 | 0.74 | 0.07 | asserted 0.96 |
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The paper
Original 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.
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References
How the literature uses it
10 arXiv papers cite this in 11 sentences
Papers citing it · 9 sentences without a confident label
Cites it as background 2 papers
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The 1/8 BPS couplings have also been analyzed from the worldsheet perspective in [ref] , and from the spacetime point of view in [ref] .
Worldsheet (anti)instanton bound states in type II on $T^2$ · Introduction · 0.97
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However, the half-integer cases have also shown up in many contexts and we refer to [ref] for related work.
To the cusp and back: Resurgent analysis for modular graph functions · Conclusions · 0.96
What it cites · 137 sentences without a confident label
Builds on 2 papers
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Using this observation, we can obtain Siegel modular functions of $Sp(4,)$ from theta liftings of vector-valued modular forms under $SL(2,)$, generalising earlier constructions of the log-norm of the Igusa cusp form $Ψ_10$ [ref] and of the genus-two Kawazumi–Zhang invariant [ref] .
A Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces · Theta series representation for real-analytic Siegel modular forms · 0.97
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Using this observation, we can obtain Siegel modular functions of $Sp(4,)$ from theta liftings of vector-valued modular forms under $SL(2,)$, generalising earlier constructions of the log-norm of the Igusa cusp form $Ψ_10$ [ref] and of the genus-two Kawazumi–Zhang invariant [ref] .
$N=2$ heterotic string threshold correction, $K3$ surface and generalized Kac-Moody superalgebra · Theta series representation for real-analytic Siegel modular forms · 0.97
Uses the method or tool of 14 papers
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To prove [ref] for the particle multiplet sum, one uses the same Poincaré sum representation [ref], and the restriction of the tensorial equation [ref] on functions of $GL(2)$, which appeared in Eqs. (4.94) and (4.96) of [ref] .
Loops in exceptional field theory · Tensorial differential equations · 0.95
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Implementing an infrared cut-off $μ$ as in [ref] , one obtains [The term $16 π^-ϵ Γ(ϵ) L^2ϵ$ has not been derived but must be there for cancelling the first order pole.] ℐ_2,ϵ + π/3 π^-ϵ Γ(ϵ) μ^-2ϵ ^2_00, ϵ + π^2/3 ( π^-2ϵ Γ(ϵ)^2 + 16 π^-ϵ Γ(ϵ) + 𝒪(ϵ^0) ) μ^-4ϵ ϵ→0∼ 4 π^2/27 log(g_8)^2 + 2π/9 log(g_8) ( 2ζ(2)/g_8^2 + 4 ζ(4) ( E_ Λ_1^0.7$SL(2)$(U)+E_ Λ_1^0.7$SL(2)$(T) ) + π/3 )+ … + 4π^2/3 log( 2πμ)^2 - 2π/3 log(2πμ) ( 2ζ(2)/g_8^2 + 4 ζ(4) ( E_ Λ_1^0.7$SL(2)$(U)+E_ Λ_1^0.7$SL(2)$(T) ) + 4π/3 log(g_8) + π/3 )
Cancellation of divergences up to three loops in exceptional field theory · Divergences and threshold terms in the weak-coupling limit · 0.86
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The renormalised value will be defined as the value at $ϵ=0$, after subtracting a specific Eisenstein series canceling the poles that occur when $d=4,5,6$ [ref] .
Supergravity divergences, supersymmetry and automorphic forms · Double lattice sums and regularised integrals · 0.83
Compares with 11 papers
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Taking into account the divergence coming from the supergravity amplitude and the $ℛ^4$ form-factor as in [ref], one obtains instead ℐ_2,ϵ + π/3 π^-ϵ Γ(ϵ) μ^-2ϵ ^2_00, ϵ + π^2/3 ( π^-2ϵ Γ(ϵ)^2 + 16 π^-ϵ Γ(ϵ) + 𝒪(ϵ^0) ) μ^-4ϵ ∼ 49 π^2/27 log(R)^2 - 7π/9 log(R) ( R ^1_00 + π3 ) + 4π^2/3 log(2πμ)^2 - 2π/3 log(2πμ) ( R ^1_00 - 14 π3 log(R) + π3 ) which is finite as $ϵ→0$ and reproduces the expected logarithmic terms from [ref] .
$D^6 R^4$ amplitudes in various dimensions · Divergences and threshold terms in the large radius limit · 0.97
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The function [ref] is divergent at the given value of the parameter [ref] and we consider instead its regularised version defined in [ref].]
Loops in exceptional field theory · Introduction and summary · 0.85
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Before evaluating [ref] further, we note the consistency of the known expression with the genus-two contribution to the $∇^6^4$ coupling, given by [ref]
Zhang-Kawazumi Invariants and Superstring Amplitudes · Contributions from 1/4-BPS states · 0.92
Cites as support 1 paper
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However, the supersymmetric Ward identity for the renormalised coupling requires that such Fourier coefficients must vanish [ref] .
The two D6R4 type invariants and their higher order generalisation · 5) The fifth layer · 0.98
Cites as background 9 papers
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In physics terminology, Fourier coefficients describe effects from non-perturbative states coupling to axions, corresponding to coordinates along nilpotent generators in the symmetric moduli space $E_d+1/K(E_d+1)$ in a given parabolic decomposition [ref] .
$R^4$ couplings and automorphic unipotent representations · Nilpotent orbits and BPS states · 0.86
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In physics terminology, Fourier coefficients describe effects from non-perturbative states coupling to axions, corresponding to coordinates along nilpotent generators in the symmetric moduli space $E_d+1/K(E_d+1)$ in a given parabolic decomposition [ref] .
Small representations, string instantons, and Fourier modes of Eisenstein series (with an appendix by D. Ciubotaru and P. Trapa) · Nilpotent orbits and BPS states · 0.86
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In physics terminology, Fourier coefficients describe effects from non-perturbative states coupling to axions, corresponding to coordinates along nilpotent generators in the symmetric moduli space $E_d+1/K(E_d+1)$ in a given parabolic decomposition [ref] .
Effects of D-instantons · Nilpotent orbits and BPS states · 0.86
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