One loop in eleven dimensions

M.B. Green; M. Gutperle; P. Vanhove

REVIEW

One loop in eleven dimensions

M.B. Green, M. Gutperle, P. Vanhove

High Energy Physics - Theory · Submitted 25 Jun 1997 · 19 references

Pith's verdict: The one-loop computation is solid and reproduces the string R^4 sector exactly; the soft spot is the unproved non-renormalization premise, which the authors flag themselves.

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arXiv IDhep-th/9706175 (version 2)

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CitationM.B. Green, M. Gutperle, P. Vanhove. "One loop in eleven dimensions." arXiv:hep-th/9706175 (1997). https://pith.science/paper/BZ2RW3E7

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Claim paper

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Editor's note

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This is the paper that put the eleven-dimensional origin of the type II R^4 terms on an explicit footing. The one-loop four-graviton amplitude in D=11 supergravity on T^n, Poisson-resummed into a winding sum, reproduces the IIA tree-level zeta(3)/R^3 term (eq. 11), the SL(2,Z)-invariant Maass waveform of eigenvalue 3/4 that encodes the IIB tree-level, one-loop and D-instanton contributions (eq. 12), and the Sl(3,Z) x Sl(2,Z) eight-dimensional assembly (eqs. 16-25). The chain from eq. (6) to eq. (10) is written out, and the coefficients land on the known string answers without fudging. That is the real content, and it holds up. The value C = pi/3 for the regularized zero-winding divergence is then fixed by T-duality in nine dimensions, and the paper is explicit that this is a matching argument, not a derivation: 'It is a challenge to Matrix theory to reproduce this number.' The wrapped-membrane term H5 is likewise fixed by consistency with T-duality plus one-loop string perturbation theory, and the text says so. So the two things a skeptic would flag are both acknowledged in the paper, not hidden. The stress-test note is right that the load-bearing assumption is the non-renormalization of the one-loop t8t8R4 coefficient, argued from the supersymmetry relation to C^(3) ^ X8 rather than proved. That is a genuine gap, and 'strong reason to believe' is doing real work. But it is the kind of gap that a 1997 paper in this area could not close, and the anomaly-normalization argument is at least a principled reason rather than hand-waving. Minor soft spots: the light-cone computation is done in a special frame with external momenta set to zero in the integrand, and the covariant extension is asserted to be 'easy to check' rather than checked; the step from eq. (8) to eq. (9) (the replacement of the tau-integral of F by tau^4) is stated rather than derived. Neither affects the leading t8t8R4 coefficient. Who should read it: anyone working on M-theory effective actions, D-instantons, or the R^4 sector of string duality. It is a standard reference and deserves to be. Recommend sending it out; the referees will want the non-renormalization premise spelled out, but the computation itself is not in doubt.

Referee report

Significance

If correct, this is an influential observation: it identifies the exact R^4 couplings of type II strings with a one-loop computation in eleven-dimensional supergravity and gives a sharp, falsifiable target for Matrix theory, namely the value C = π/3. The computation is explicit and the coefficients match known string theory results exactly, with no adjustable parameters beyond C. A particularly nice internal check, which the authors do not emphasize, is that H1 and H5 of Eqs. (16) and (25) collapse to -ln|η(T)|^4, so that the same C = π/3 is required for the T^3 result to reduce to the known one-loop string answer; this is an independent confirmation of the T-duality determination of Eq. (14). The paper is also candid about what it does not derive, explicitly leaving the microscopic origin of C to Matrix theory.

Major objections (3)

[Section 1, p. 2; final paragraph, p. 10] The identification of the one-loop amplitude with the full R^4 effective action rests on the claim that the one-loop t8t8R4 coefficient is protected from higher-loop corrections by an eleven-dimensional non-renormalization theorem following from the supersymmetry relation to C(3) ∧ X8. The authors describe this as a 'strong reason to believe', and the concluding paragraph repeats the argument via anomaly normalization, but no argument is given in this paper and the relation is cited to Ref. [4], which is listed as an unpublished preprint (hep-th/9704145). Because this premise is precisely what licenses comparing a one-loop result in a non-renormalizable theory to the exact string effective action, the paper should either sketch the normalization argument or point to a published derivation, and should state explicitly which part of the comparison would fail if the premise were wrong.
[Eq. (14) and the paragraph following Eq. (12)] The value C = π/3 is obtained by matching the coefficients of the V2 and Ω2^{1/2} V2^{-1/2} terms in Eq. (12) to the known IIA and IIB one-loop coefficients, that is, by a T-duality consistency condition rather than by a computation. The abstract's phrasing that the divergence 'must be regularized so that it has a precisely determined finite value' reads as a derivation and should be softened to say that the value is determined by matching to known string coefficients. Relatedly, Eq. (16) reuses the same C for the zero-winding term of the T^3 amplitude; since the zero-winding integral ∫dτ̂ τ̂^{1/2} is formally identical in all dimensions this is natural, but the paper should say so explicitly, because the assumption that a single regularization is consistent across n = 1, 2, 3 is part of what makes the eight-dimensional result work.
[p. 9, following Eq. (26); Eq. (23)] The statement that H = Σ H_i is invariant under Sl(3,Z) × Sl(2,Z) is asserted rather than shown. H3 in Eq. (23) contains a term -ln T2, which is not by itself invariant under the Sl(2,Z) acting on T, and H4 in Eq. (24) is written in IIA variables with an explicit K1 Bessel function; it is not manifest how the pieces combine into an invariant. A short demonstration, or a pointer to where it is given, would substantially strengthen the T^3 section, which is otherwise the least self-contained part of the paper. The same applies to the assertion in the paragraph after Eq. (23) that the logarithmic divergence 'may be regularized in a unique manner that is consistent with modular invariance'; the uniqueness should be justified or explicitly qualified as an assumption.

Minor objections (5)

[Eq. (13)] Ω2 is written as R10R11, but Ω2 is defined as the imaginary part of the complex structure of the torus, which should be R10/R11. Please check this together with the frame conventions for r_B: with Ω2 = R10/R11 and V2 = R10R11, the stated V2 = e^{φB/3} r_B^{-4/3} does not reproduce an R11 = e^{2φB/3} that is independent of r_B, so at least one of these relations appears to be mistyped.
[Eqs. (6) and (9)] Please show the intermediate step explicitly. As printed, the τ exponent n/2 - 13/2 in Eq. (6), combined with the momentum integration τ^{-(11-n)/2} and the four vertex insertions τ^4, does not obviously yield the n/2 - 5/2 of Eq. (9); the bookkeeping matters because the n = 0 case must reproduce the cubic divergence referred to in footnote 4.
[Eq. (10)] The change of variable τ̂ = 1/τ and the π factors absorbed into τ between Eqs. (6) and (9) should be stated explicitly, since the normalization of the constant C, and hence the value C = π/3, depends on this convention.
[Eq. (8)] The index structure of t8t8R4 as printed should be checked against the definition in Ref. [10]; since the notation t8t8R4 is used throughout as the name of the coupling, a precise definition at this point would help the reader.
[References and typography] There are several typographical slips: 'exactyl' in the first line of the paragraph following Eq. (3) on p. 2, 'ulta-violet' in the first paragraph of p. 10, and the spacing in 'M.T. Grisaru , A.E.M Van de Ven' in Ref. [15]. Ref. [4] is also given without journal status, which should be updated if it has since appeared.

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The paper

The paper claims

A one-loop calculation in eleven-dimensional supergravity, compactified on tori, reproduces the full torus dependence of the four-graviton $t_8t_8R^4$ terms in the ten-, nine- and eight-dimensional type II string effective actions.

The reading

The paper asks whether a one-loop computation in eleven-dimensional supergravity — a theory with no small coupling constant and a badly divergent ultraviolet — can still carry exact information about string theory. The authors scatter four gravitons at one loop, compactify the loop on one-, two- and three-dimensional tori, and rewrite the sum over loop momenta as a sum over how many times the circulating particle winds around the compact directions. The winding sum turns out to have exactly the right dependence on the torus geometry to reproduce the known tree-level, one-loop and D-instanton contributions to the $t_8t_8R^4$ terms of type IIA and type IIB strings in ten, nine and eight dimensions. The one free number left over, the value assigned to the divergent zero-winding term, is then fixed by demanding T-duality in nine dimensions, giving $C = 1/3$ times $3.14159...$; the same value is what eleven-dimensional supersymmetry requires through the relation to the eleven-form $C^{(3)}$ term. If this holds, a non-renormalizable theory's one-loop amplitude is not merely a formal artifact but a precise encoding of stringy quantum corrections.

What carries the argument

The load-bearing mechanism is Poisson resummation of the loop momentum sum into a winding sum. In equation (9) the amplitude is a sum over Kaluza-Klein integers $l_I$ weighted by $e^{-tau G^{(n)}_{IJ} l^I l^J}$; inverting the torus metric turns it into equation (10), a sum over winding integers $hat l_I$ weighted by $e^{-tau G^{(n)}_{IJ} hat l^I hat l^J}$, where the circulating particle's world-line wraps the compact directions. That rewriting is what converts a supergravity loop into a sum over Euclidean D-particle world-lines: the massless winding reproduces perturbative string one-loop effects, the massive windings reproduce D-instantons, and the single-winding term on a circle reproduces the tree-level string term. The second piece of machinery is the modular function $f$, the Maass waveform of eigenvalue $3/4$ satisfying $Omega_2^2 partial partial-bar f = (3/4) f$, which packages the tree-level, one-loop and instanton series into one object and is reproduced term by term by the winding sum.

Extended reading notes

Core claim

The central claim is that the one-loop four-graviton amplitude of eleven-dimensional supergravity, reduced on $T^n$ and Poisson-resummed into a sum over windings $G^{(n)}_{IJ}$-weighted integers, reproduces the complete dependence of the $t_8t_8R^4$ term on the toroidal moduli. On a single circle, the nonzero-winding sum gives the tree-level type IIA contribution $1/3$ times $3.14159...$ times $1/R_{11}^3$; on a two-torus, the nonzero-winding sum gives exactly the modular-invariant function $f(Omega,bar Omega)$ that is an eigenfunction of the Laplacian with eigenvalue $3/4$, whose large-$Omega_2$ expansion contains the type IIB tree-level term, the one-loop term and the whole D-instanton series. On a three-torus, the one-loop amplitude supplies the $Sl(3,Z)$-symmetric Kaluza-Klein piece, while the wrapped-membrane piece is fixed separately by U-duality and one-loop string theory. The ultraviolet divergence sits entirely in the zero-winding term, a constant $C$ times the volume, and the authors determine $C = 1/3$ times $3.14159...$ by matching the coefficients of the two power-behaved terms under T-duality in nine dimensions.

Load-bearing premise

The argument requires that the one-loop coefficient of the $t_8t_8R^4$ term receives no corrections from higher loops, so that a computation in a non-renormalizable theory can be compared directly to the exact string effective action; the paper asserts this on the grounds of an eleven-dimensional non-renormalization theorem tied to the anomaly-fixed eleven-form, but does not prove the theorem here.

Editorial extensions

If this is right

The complete $t_8t_8R^4$ term of ten-dimensional type IIB string theory follows from the one-loop eleven-dimensional amplitude in the zero-volume limit $V_2 to 0$, including its infinite series of D-instanton contributions.
The tree-level $1/3$ times $3.14159...$-weighted $1/R_{11}^3$ term of type IIA string theory is recovered from a single winding of the supergraviton world-line around the eleventh dimension, not from a string-tree computation.
The logarithmic divergence of the eight-dimensional one-loop amplitude is regularized in a way fixed by modular invariance, producing the combination $-ln(U_2 |eta(U)|^4) - ln(T_2)$ in the moduli-dependent piece.
The value of the divergent constant is fixed to $C = 1/3$ times $3.14159...$, which the authors note is also the value required by eleven-dimensional supersymmetry through the $C^{(3)}$ term, and which they pose as a target for Matrix theory to reproduce.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the one-loop coefficient really is protected, then the eleven-dimensional amplitude is a rare example of a divergent quantity in a non-renormalizable theory whose finite part is physically meaningful; that would make similar protected couplings in lower-dimensional compactifications a practical testing ground for any candidate microscopic theory.
The fact that the same constant is fixed independently by T-duality and by the supersymmetry relation to the eleven-form suggests a broader pattern: anomaly-fixed couplings may act as normalizations for whole towers of higher-derivative terms, which would let one read off the coefficient of a term without computing the loop at all.
The paper's split on $T^3$ — Kaluza-Klein effects from the one-loop amplitude, wrapped-membrane effects fixed by duality — suggests that on $T^4$ and beyond, where the U-duality group no longer factors, the one-loop result alone will underdetermine the answer and genuinely new input about wrapped branes will be needed.
A concrete check that would sharpen the claim: compute the zero-winding constant directly in a microscopic formulation of eleven-dimensional dynamics and see whether it lands on $1/3$ times $3.14159...$ rather than on some other finite value.

Original abstract

Four-graviton scattering in eleven-dimensional supergravity is considered at one loop compactified on one, two and three-dimensional tori. The dependence on the toroidal geometry determines the known perturbative and non-perturbative terms in the corresponding processes in type II superstring theories in ten, nine and eight dimensions. The ultra-violet divergence must be regularized so that it has a precisely determined finite value that is consistent both with T-duality in nine dimensions and with eleven-dimensional supersymmetry.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.37 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.45 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Four-graviton scattering in eleven-dimensional supergravity is considered at one loop compactified on one, two and three-dimensional tori. The dependence on the toroidal geometry determines the known perturbative and non-perturbative terms in the corresponding processes in type II superstring theories in ten, nine and eight dimensions. The ultra-violet divergence must be regularized so that it has a precisely determined finite value that is consistent both with T-duality in nine dimensions and with eleven-dimensional supersymmetry.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Four-graviton scattering in eleven-dimensional supergravity is considered at one loop compactified on one, two and three-dimensional tori.	contribution 0.94	0.77	0.06	asserted 0.88
s1	The dependence on the toroidal geometry determines the known perturbative and non-perturbative terms in the corresponding processes in type II superstring theories in ten, nine and eight dimensions.	interpretation 0.55	0.47	0.57	asserted 1.00
s2	The ultra-violet divergence must be regularized so that it has a precisely determined finite value that is consistent both with T-duality in nine dimensions and with eleven-dimensional supersymmetry.	method 0.68	0.51	0.21	asserted 0.97

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.98.

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References

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E.S. Fradkin and A.A. Tseytlin, Quantum properties of higher dimensional and di- mensionally reduced supersymmetric theories , Nucl. Phys. B227 (1983) 252

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T. Banks, W. Fischler, S.H. Shenker and L. Susskind, M-Theory as a matrix model: a conjecture, hep-th/9610043, Phys. Rev. D55 (1997) 5112

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C. Bachas and E. Kiritsis, F 4 Terms in N=4 String Vacua , Proceedings of Trieste Spring School and Workshop, April 1996 hep-th/9611205

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C. Bachas, C. Fabre, E. Kiritsis, N.A. Obers and P. Vanho ve Heterotic/ type-I duality, D-brane instantons and generalized prepotentials (to appear)

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M.T. Grisaru , A.E.M Van de Ven and D. Zanon, Two-dimensional supersymmetric sigma models on Ricci ﬂat K¨ ahler manifolds are not ﬁnite , Nucl. Phys. B277 (1986) 388 ; Four loop divergences for the N=1 supersymmetric nonlinear sigma model in two-dimensions, Nucl. Phys. B277 (1986) 409

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D.J. Gross and E. Witten, Superstring modiﬁcations of Einstein ’s equations , Nucl. Phys. B277 (1986) 1

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Automorphic properties of low energy string amplitudes in various dimensions
hep-th 2010-01 conditional novelty 6.0 of 10

The coefficients of R^4, d^4R^4, and d^6R^4 interactions in toroidally compactified maximal supergravity are determined as automorphic functions satisfying Laplace eigenvalue equations, with supergravity ultraviolet d...
D-instantons and twistors
hep-th 2008-12 conditional novelty 6.0 of 10

All D-instanton corrections to the hypermultiplet moduli space metric in Type II Calabi-Yau compactifications can be packaged as a sum of dilogarithms controlled by a single holomorphic function, in the leading instan...
Manifest Ultraviolet Behavior for the Three-Loop Four-Point Amplitude of N=8 Supergravity
hep-th 2008-08 accept novelty 6.0 of 10

A manifestly ultraviolet-finite-looking reorganization of the three-loop N=8 supergravity amplitude shows a nonzero divergence in D=6, fixing the critical dimension at D_c=6.
The low energy expansion of the one-loop type II superstring amplitude
hep-th 1999-10 conditional novelty 6.0 of 10

The one-loop four-graviton amplitude in type II string theory is expanded to fourth order in momentum, yielding the coefficient of the s^3 R^4 term and showing the s^2 term vanishes.
On the Species Scale, Modular Invariance and the Gravitational EFT expansion
hep-th 2023-10 accept novelty 5.0 of 10

In maximally supersymmetric string setups the leading R4 (or R2) curvature correction is suppressed by the species scale, while higher-derivative operators are suppressed by the lightest tower mass.
The Emergence Proposal in Quantum Gravity and the Species Scale
hep-th 2022-12 conditional novelty 5.0 of 10

One-loop sums over towers of light states, cut off at the species scale, reproduce the field dependence of moduli metrics, gauge kinetic functions, and flux potentials in several string vacua.
U-duality and M-Theory
hep-th 1998-09 accept novelty 5.0 of 10

Taking U-duality as an exact symmetry of toroidally compactified M-theory fixes the BPS multiplet structure and yields mass formulae for 1/2- and 1/4-BPS states, at the cost of predicting states in D<=3 that have no i...
Lorentzian Regularization of the Type IIB Superstring Torus Vacuum
hep-th 2026-06 unverdicted novelty 4.0 of 10

A first direct regularized construction of the unprojected spin sectors of the Type IIB superstring torus vacuum is given via sector-resolved modular integrals.
Primordial black holes from a type-IIB/F theory inflaton potential
gr-qc 2026-10 reject novelty 3.0 of 10

Adding a Gaussian bump tuned by three free parameters to a string-derived inflaton potential can produce a primordial black hole population near 10^{-6} solar masses, but the bump's shape and amplitude are not derived...
Review of the No-Boundary Wave Function
hep-th 2023-03 accept novelty 3.0 of 10

The no-boundary wave function, implemented via a regular momentum condition at the South Pole rather than a literal sum over compact metrics, yields stable semiclassical saddle points that prefer low inflationary ener...
On Legacy of Starobinsky Inflation
gr-qc 2025-01 unverdicted novelty 2.0 of 10

A memorial review of the Starobinsky inflation model that proposes a deformation for primordial black hole production and discusses superstring quantum corrections plus universal reheating.
The Geometry, Branes and Applications of Exceptional Field Theory
hep-th 2020-06 accept novelty 2.0 of 10

Exceptional field theory recasts the maximal supergravities as a single extended geometry whose local symmetry is the exceptional U-duality group Ed(d), with extra coordinates conjugate to brane winding modes.
Duality Symmetric String and M-Theory
hep-th 2013-06 accept

The paper is a review that connects the doubled worldsheet, Double Field Theory, and exceptional generalised geometry as one program for making T- and U-duality manifest.

How the literature uses it

99 arXiv papers cite this in 236 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 164 sentences without a confident label

Builds on it 2 papers

The explicit $^(D)_(0,0)$ coefficients in dimensions $D≥6$ will be obtained in terms of Eisenstein series that satisfy Laplace eigenvalue equations on moduli space space, building on the work of [ref] .

Automorphic properties of low energy string amplitudes in various dimensions · Introduction · 0.99
This was achieved by extending the original computation of Green-Gutperle-Vanhove [ref] for toroidal compactifications of M-theory to include also light particle-like towers of transverse $M2$- and $M5$-branes wrapped on appropriate cycles.

Emergence of $R^4$-terms in M-theory · Conclusions · 1.00

Uses its method or tool 9 papers

We compute $11$D amplitudes in the superparticle formalism compactified on a $2$-torus [ref] .

Type IIB at eight derivatives: insights from Superstrings, Superfields and Superparticles · 11D superparticle amplitudes · 0.89
As explained in [ref] , the invariance under the discrete subgroup $SL(2,)$ can be restored by summing over images, using similar techniques as the ones used for $R^4$ couplings in toroidal compactifications [ref] .

D-instantons and twistors · S-duality and A-type D-instanton corrections · 0.86
Thus, using the worldline formalism [ref] , the amplitude reads as

The Double EFT Expansion in Quantum Gravity · Decompactification limits and the EFT breaking scale · 1.00

Uses its data or values 6 papers

First recall the analytic terms in the derivative expansion of one-loop supergravity ($L=1$) compactified on $^1$ up to the order of interest in this paper are [ref] A_L=1^an = iκ_11^4(2π)^11 l_11^3 4π^4 ^4 [ 2ζ(3)R_11^3+4ζ(2) +4ζ(4)3 R_11σ_24^2 + 4ζ(6)27 R_11^3 σ_34^3 +64ζ(8)2835 R_11^5 σ_2^24^4 +16ζ(10)1125 R_11^7 σ_2σ_34^5 +64ζ(12)31185·691 R_11^9 (675σ_2^3+872σ_3^2)4^6 ⋯ ] .

Modular properties of two-loop maximal supergravity and connections with string theory · Analytic terms · 0.85
At low orders, however, one can distinguish between tree and loop terms by the different power of $ℓ_11$ that multiply them.] terms $_R$, $_R^4$, and $_D^6R^4$ are protected, and so can be computed from Type IIA string theory by compactifying on a circle [ref] [$_D^4R^4$ can also be computed in this way, but it vanishes and so we did not write it.] to get SGtoR4 𝒜_R^4/𝒜_R=stu/3·2^7 , 𝒜_D^6R^4/𝒜_R=(stu)^2/15·2^15 .

6d (2,0) and M-theory at 1-loop · Introduction · 0.82
Its dependence with respect to the moduli space parametrised by ${𝒱_2, τ}$ has been already computed in [ref] , and it is captured by the following non-trivial function (see also [ref] )

On the Species Scale, Modular Invariance and the Gravitational EFT expansion · M-theory on $T^2$ · 0.84

Compares with it 7 papers

We compare the results of conventional ten-dimensional ($10$D) superstring amplitudes with eleven-dimensional ($11$D) superparticle amplitudes compactified on a two-torus [ref] , and superspace approaches to Type IIB $=2$ supergravity [ref] .

Type IIB at eight derivatives: insights from Superstrings, Superfields and Superparticles · Introduction and summary · 0.99
However, let us remind that in GGV [ref] such a term had somehow to be added by hand, while here we can reproduce it by integrating out a physical quantity and imposing the desired modular behavior.

Emergence of $R^4$-terms in M-theory · TEXT-term in eight dimensions · 0.93
For $R|R$, we precisely match the 1-loop correction to 11d supergravity [ref] , as was done for $𝒩=4$ SYM and 10d supergravity in [ref] .

6d (2,0) and M-theory at 1-loop · Introduction · 0.97

Cites it as support 1 paper

Evidence for this comes from the protected $R^4$ corrections to the supergravity action [ref] .

D-branes · SL(2, $ℤ$) duality of type IIB · 1.00

Cites it as background 34 papers

Many D-instanton induced terms were predicted using S-duality invariances of various compactified string theories [ref] , but except for the early attempts [ref] , the direct systematic computation of these amplitudes from first principles has not been carried out.

Normalization of Type IIB D-instanton Amplitudes · Introduction · 0.97
In this section we will provide some background for the calculation of the $R^4$-term à la GGV [ref] .

Emergence of $R^4$-terms in M-theory · One-loop TEXT-terms in perturbative string theory · 0.93
Although the coefficients, $^(D)_(p,q)$, have not been determined in generality, a significant amount of information has accumulated for the first three terms for the cases with $D≥6$ (duality groups $E_d+1$ with $d ≤4$) [ref] , and there are various conjectures concerning the coefficient of the $^4$ interaction, $^(D)_(0,0)$, for higher-rank groups [ref] .

Eisenstein series for higher-rank groups and string theory amplitudes · Introduction · 0.80

Discussion

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