Two loops in eleven dimensions

Hwang-h. Kwon; Michael B. Green; Pierre Vanhove (DAMTP; UK)

REVIEW

Two loops in eleven dimensions

Michael B. Green, Hwang-h. Kwon, Pierre Vanhove (DAMTP, UK)

High Energy Physics - Theory · Submitted 6 Oct 1999 · 23 references

Pith's verdict: A real two-loop computation with a genuinely new moduli-space technique; the headline coefficient is solid, but two of its constants are fixed by consistency rather than computed.

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arXiv IDhep-th/9910055 (version 3)

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CitationHwang-h. Kwon, Michael B. Green, Pierre Vanhove (DAMTP, UK). "Two loops in eleven dimensions." arXiv:hep-th/9910055 (1999). https://pith.science/paper/T2J7TZQA

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Editor's note

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The thing worth knowing: this paper actually does the two-loop four-graviton diagram of eleven-dimensional supergravity on T^2, and it does it with a trick that turned out to matter. The three Schwinger parameters get repackaged as the volume and complex structure of an auxiliary torus, the integration domain comes out as the Γ0(2) fundamental domain, and the divergences sort themselves into SL(2,Z) orbits — singular, degenerate, non-degenerate. That classification is the paper's real contribution, and it is done cleanly. The finite piece from the non-degenerate orbits, Eq. (4.37), is computed directly by unfolding and comes out as 2π^4 ζ(3)ζ(4) l11^{-8} V^{-4}. That number is not put in by hand; it falls out of the integral. The result is Eq. (5.1), with the ζ(5) V^{-5/2} E_{5/2}(Ω) structure that everyone now quotes for D^4R^4, plus a constant term and the D-instanton series. That is the paper's durable content and it holds up on reading. The stress-test note is right that §4.4's cross-check routes through the IIA/IIB equality of §2.3, but it overstates the damage: the unfolding computation of I_fin never uses that equality, so the coefficient is derived, not fitted. What the equality buys is the string-theory interpretation of the number, not the number. Where I would push back harder than the reader does: §2.3 itself is an argument from supermoduli counting credited to conversations with Berkovits, with no published derivation behind it, and it is load-bearing twice over — it fixes c1 in (3.14) and, via the decompactification requirement, forces c2 + a(Λl11)^8 = 0 in (4.48). The coefficient a of the primitive divergence is never computed. So 'exact expression' carries two constants fixed by consistency and one left undetermined. The authors are upfront about D^6R^4 being a mess; they are less upfront that the no-higher-loop claim for D^4R^4 is asserted on the same supersymmetry-counting grounds they decline to trust one order up. That asymmetry deserves a referee's attention. Who should read it: anyone working on M-theory derivative corrections, modularity in string effective actions, or the D^4R^4 / E_{5/2} story. The technique generalizes and is worth knowing independently of the coefficient. Send it out. The computation is sound, the method is new, and the open questions are the kind a referee can actually sharpen.

Referee report

Significance

If the result holds, this is a substantial and technically impressive piece of work. The auxiliary-torus construction is a genuine structural insight: it converts an apparently intractable two-loop eleven-dimensional Feynman integral into a one-loop-like modular integral whose divergences are organized by the boundaries of a fundamental domain, and it makes the SL(2,Z) symmetry of the integrand manifest rather than accidental. The paper also ships several things a referee should credit explicitly. First, the finite term of Eq. (4.37) is obtained by an unfolding computation that does not use the string-theoretic input, and it independently reproduces the r_A^2 term that Eq. (3.32) derives from the one-loop amplitude; that is a non-trivial cross-check between two different Feynman diagrams. Second, the Lambda^3 sub-divergence of Eq. (4.34) cancels exactly against the one-loop counterterm diagram of Eq. (4.40) once c_1 is taken from Eq. (3.14); I checked this cancellation and it works. Third, the zeta(3) zeta(4) structure of Eq. (4.37) follows correctly from the unfolding integral, with the m-sum giving zeta(3) and the n-sum giving zeta(4). Fourth, the result is falsifiable: the zeta(

Major objections (3)

[Section 2.2 and Section 5] There is a tension between the protection criterion stated in Section 2.2 and the outcome reported in Section 5, and it bears directly on the word 'exact' in the abstract. Section 2.2 states that 'terms with less than eight powers of momentum acting on R^4 should be protected and may be determined in this manner', and then explicitly says 'This includes the S^3R^4 ~ D^6R^4 term which should therefore also be determined by similar considerations.' But Section 5 reports that the D^6R^4 expressions 'do not possess modular invariant integrands when expressed in terms of the integration variables V and tau and we have not made sense of the integrals', and concludes that 'extra contributions from higher-loop supergravity amplitudes are needed'. If the stated criterion is satisfied by D^6R^4 and yet the two-loop computation fails to produce a sensible answer for it, the criterion is not by itself a reliable indicator of protection. Since D^4R^4 has four derivatives and D^6R^4 has six, both lie below the stated threshold of eight, and the criterion as written does not distinguish them. The paper needs either a sharper argument for why D^4R^4 is protected while D^6R^4 is not, or an explicit statement in the abstract and in Section 5 that the D^4R^4 coefficient of Eq. (5.1) is exact only under the assumption that no higher-loop eleven-dimensional diagrams contribute. The introduction already concedes that 'it is not known if the D^4R^4 and D^6R^4 terms ... are protected from corrections arising from higher-loop diagrams', so the abstract's 'exact expression' (abstract sentence s3) is stronger than what the body establishes.
[Section 2.3] The statement that the type IIA and type IIB four-graviton amplitudes are equal up to and including two loops is load-bearing: it fixes the one-loop counterterm c_1 in Eq. (3.14), and through Eq. (4.48) it controls the renormalized coefficient quoted in Eq. (5.1). Section 2.3 argues this by counting contractions between the two epsilon tensors and the supermoduli insertions, and the subsection is explicitly attributed to 'conversations with Nathan Berkovits'. The argument as written is a heuristic paragraph, not a derivation: the reader is told that at one loop 'there are no contractions between the two epsilon tensors' and that at two loops 'this is still not enough to allow the sixteen remaining indices of these tensors to be saturated', but the counting is not exhibited. I note in the authors' favour that c_1 is also fixed independently by the supersymmetry argument of reference [10], as the text says, so the result does not rest on this input alone; that mitigates the concern considerably. Still, the paper should either supply the supermoduli counting in an appendix or state plainly in Section 2.3 and in the conclusion that the IIA/IIB equality at two loops is being used as an assumption whose detailed justification is deferred.
[Section 4.2 and Section 4.3, Eqs. (4.25)-(4.34)] The symbol V is used for two different quantities in this passage: the volume of the target torus T^2 and the volume of the auxiliary torus T-hat^2 introduced in Eq. (4.17). The collision is not cosmetic, because the power of Lambda in Eq. (4.32) is determined by how the integrand of Eq. (4.30) scales in the integration variable, and with a single symbol V the reader cannot tell which volume appears in the exponent of Eq. (4.25), in the measure dV/V^3 of Eq. (4.20), or in the factor l_11^2 V/Omega_2 of Eq. (4.37). Relatedly, the exponent of Eq. (4.25) contains a term -2V det A that has no counterpart in Eq. (4.20), where the corresponding expression is manifestly positive definite; its origin should be explained. Since Eq. (4.32) is the equation that fixes the coefficient of E_{5/2} in the final result, and since the step from Eq. (4.30) to Eq. (4.32) cannot be checked as written, the authors should rewrite this passage with distinct symbols for the two volumes and display the intermediate integration explicitly.

Minor objections (7)

[Eq. (4.47)] The left-hand side reads 'A^{(2)}_4 + delta A^{(1)}_4', but the surrounding text and Eq. (4.46) indicate that the two-loop counterterm contribution delta A^{(2)}_4 is intended. Please correct.
[Eq. (3.21)] The left-hand side is written as I(S,T) + I(T,U) + I(U,S), whereas Eq. (3.1) uses I(S,T) + I(S,U) + I(U,T). The two are of course the same set, but the inconsistent ordering is distracting in an equation that is used repeatedly later.
[Eqs. (3.29) and (3.30)] The overall factor (4 pi^8 l_11^15 r_A^{-1}) is immediately multiplied by a further r_A, so the net power of r_A in front of the bracket should be stated once and checked against the corresponding IIB expression.
[Eq. (4.12)] The passage from the planar and non-planar expressions of Eqs. (4.10) and (4.11) to the symmetric form of Eq. (4.12) is asserted rather than shown, and the two starting expressions carry different powers of Delta (Delta^{(11-n)/2} with sigma^2 lambda^2, versus Delta^{(11-n)/2} with 2 sigma lambda^2 rho). Since Eq. (4.12) is the foundation of the entire modular analysis, the explicit combination of diagrams that produces it should be displayed.
[Appendix A, Eq. (A.6)] The vanishing one-loop coefficient of l_s^4(s^2+t^2+u^2) is written as a bare '0 times'. Section 5 uses this vanishing as a check on the framework ('the one string loop contribution to the s^2R^4 interaction in ten dimensions is absent as it should be according to [20]'), so it deserves a sentence of explanation rather than a placeholder. In the same equation, the factor 2 pi kappa^2_{(10)} appears in the first line but is dropped in the second; please check the normalization.
[Fig. 4] The fundamental domain of Gamma_0(2) is described as covering 'precisely three copies of F', with regions labelled F, F', F'', f, f', g, g'. The figure would be much easier to use if these labels were tied explicitly to the generators named in the text (S, T, ST^{-1}, TS, T^{-1}ST^{-1}, T^{-1}).
[Section 4.4 and Appendix A] Several coefficients of the low-energy expansion of the one-loop string amplitude are quoted from the companion paper [20] (hep-th/9910056). Since these coefficients are used as the benchmark against which the eleven-dimensional results are checked, the paper would be more self-contained if the relevant terms were reproduced here rather than left to the reader to look up.

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The paper

The paper claims

The two-loop four-graviton amplitude of eleven-dimensional supergravity on a two-torus fixes the exact coefficient of the $D^4R^4$ term in the effective action, matching known string terms and predicting two-loop and D-instanton corrections

The reading

The paper asks what the two-loop Feynman diagrams of eleven-dimensional supergravity contribute to the low-energy effective action when two dimensions are curled into a torus. The authors reduce the leading momentum-dependent part of the four-graviton amplitude to scalar $��^3$ loop integrals and then reorganize the Schwinger parameters as the volume and complex structure of a second torus. That step exposes a hidden $SL(2,��)$ symmetry and turns ultraviolet divergences into boundary effects that can be regularized in a way compatible with string dualities. The payoff is an exact coefficient for the $D^4R^4$ interaction, Eq. (5.1), whose expansion reproduces known tree-level and one-loop string terms, predicts a two-loop string coefficient and an infinite family of D-instanton terms, and forces the leading two-loop divergence in $D^4R^4$ to cancel.

What carries the argument

The load-bearing device is a change of variables from the three Schwinger parameters of the two-loop integral to the volume $V$ and complex structure $��$ of an auxiliary torus $��T^2$, with $��_1=��/(��+��)$, $��_2=��{��}/(��+��)$ and $V=l_{11}^3��{��}$. This turns the integrand into a sum over maps of $��T^2$ into the compactification torus $T^2$ and exposes an $SL(2,��)$ action. The integration runs over the fundamental domain of $��_0(2)$, which is three copies of the standard fundamental domain. Boundary regions classify the divergences: $V��$ gives the primitive $��^8$ divergence, $��_2��$ and its images give the one-loop subdivergences, and non-degenerate orbits give the finite modular-invariant piece. Poisson resummation from Kaluza-Klein modes to winding numbers is what makes the orbit classification possible.

Extended reading notes

Core claim

The central result is Eq. (5.1), an exact expression for the renormalized two-loop contribution to the $D^4R^4$ term of the effective action. The $��$-dependent part is a non-holomorphic Eisenstein series $E_{5/2}(��,��)$; its large-$��_2$ expansion contains a tree-level term, a two-loop term and an infinite series of exponentially suppressed D-instanton terms. The $��$-independent piece matches the one-loop supergravity calculation, and the whole structure is consistent with type IIA/IIB equality through two loops. The paper also argues that the leading two-loop divergence cannot contribute to $D^4R^4$, so this interaction vanishes in the decompactification limit; the next term, $D^6R^4$, does not organize modularly and is left unresolved.

Load-bearing premise

The load-bearing premise is that the type IIA and type IIB four-graviton amplitudes are equal through two loops; the paper argues this by counting supermoduli contractions rather than by a completed string calculation, and that equality fixes the one-loop counterterm and forces the two-loop constant to cancel the leading divergence.

Editorial extensions

If this is right

Eq. (5.1) gives a definite coefficient for the $s^2R^4$ term at two loops in type IIA and type IIB string theory, which a direct two-loop string calculation can check.
The Eisenstein series $E_{5/2}$ predicts an infinite set of D-instanton corrections to $D^4R^4$ with fixed coefficients.
The $D^4R^4$ interaction is absent in the decompactified eleven-dimensional limit because the leading two-loop divergence is cancelled by the two-loop counterterm.
The $D^6R^4$ term is not captured by the same modular organization, which suggests that higher-loop supergravity effects contribute there.

Reading between the lines

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

If the predicted two-loop string coefficient is confirmed, it would show that a calculation in a non-renormalizable eleven-dimensional theory can capture a genuine two-loop string effect once the ultraviolet sensitivity is handled by duality.
The same auxiliary-torus method could be applied to other protected higher-derivative couplings; the failure of $D^6R^4$ to fit the modular scheme makes it the natural next test case.
The equality of type IIA and type IIB amplitudes through two loops, if it survives higher-loop scrutiny, would suggest a non-renormalization pattern constraining which superspace integrals can contribute to these couplings.
A mismatch in the D-instanton coefficients would indicate that the eleven-dimensional supergravity calculation is missing non-perturbative states that the full M-theory supplies.

Original abstract

The two-loop Feynman diagram contribution to the four-graviton amplitude of eleven-dimensional supergravity compactified on a two-torus, T^2, is analyzed in detail. The Schwinger parameter integrations are re-expressed as integration over the moduli space of a second torus, \hat T^2, which enables the leading low-momentum contribution to be evaluated in terms of maps of \hat T^2 into T^2. The ultraviolet divergences associated with boundaries of moduli space are regularized in a manner that is consistent with the expected duality symmetries of string theory. This leads to an exact expression for terms of order D^4 R^4 in the effective M theory action (where R^4 denotes a contraction of four Weyl tensors), thereby extending earlier results for the R^4 term that were based on the one-loop eleven-dimensional amplitude. Precise agreement is found with terms in type IIA and IIB superstring theory that arise from the low energy expansion of the tree-level and one-loop string amplitudes and predictions are made for the coefficients of certain two-loop string theory terms as well as for an infinite set of D-instanton contributions. The contribution at the next order in the derivative expansion, D^6 R^4, is problematic, which may indicate that it mixes with higher-loop effects in eleven-dimensional supergravity.

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arXiv listing abstract v3. Results and the main claim are underlined; show every role.

The two-loop Feynman diagram contribution to the four-graviton amplitude of eleven-dimensional supergravity compactified on a two-torus, T^2, is analyzed in detail. The Schwinger parameter integrations are re-expressed as integration over the moduli space of a second torus, \hat T^2, which enables the leading low-momentum contribution to be evaluated in terms of maps of \hat T^2 into T^2. The ultraviolet divergences associated with boundaries of moduli space are regularized in a manner that is consistent with the expected duality symmetries of string theory. This leads to an exact expression for terms of order D^4 R^4 in the effective M theory action (where R^4 denotes a contraction of four Weyl tensors), thereby extending earlier results for the R^4 term that were based on the one-loop eleven-dimensional amplitude. Precise agreement is found with terms in type IIA and IIB superstring theory that arise from the low energy expansion of the tree-level and one-loop string amplitudes and predictions are made for the coefficients of certain two-loop string theory terms as well as for an infinite set of D-instanton contributions. The contribution at the next order in the derivative expansion, D^6 R^4, is problematic, which may indicate that it mixes with higher-loop effects in eleven-dimensional supergravity.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The two-loop Feynman diagram contribution to the four-graviton amplitude of eleven-dimensional supergravity compactified on a two-torus, T^2, is analyzed in detail.	contribution 0.99	0.90	0.05	asserted 0.98
s1	The Schwinger parameter integrations are re-expressed as integration over the moduli space of a second torus, \hat T^2, which enables the leading low-momentum contribution to be evaluated in terms of maps of \hat T^2 into T^2.	method 0.98	0.92	0.06	asserted 1.00
s2	The ultraviolet divergences associated with boundaries of moduli space are regularized in a manner that is consistent with the expected duality symmetries of string theory.	method 0.98	0.89	0.04	asserted 0.89
s3	This leads to an exact expression for terms of order D^4 R^4 in the effective M theory action (where R^4 denotes a contraction of four Weyl tensors), thereby extending earlier results for the R^4 term that were based on the one-loop eleven-dimensional amplitude.	result 0.98	0.93	0.49	asserted 1.00
s4	Precise agreement is found with terms in type IIA and IIB superstring theory that arise from the low energy expansion of the tree-level and one-loop string amplitudes and predictions are made for the coefficients of certain two-loop string theory terms as well as for an infinite set of D-instanton contributions.	result 0.98	0.92	0.33	asserted 1.00
s5	The contribution at the next order in the derivative expansion, D^6 R^4, is problematic, which may indicate that it mixes with higher-loop effects in eleven-dimensional supergravity.	limitation 0.95	0.82	0.05	hedged possibility 0.93

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

What was examined, and what was not

sources read

arXiv listing abstract v3

not examined

full text
figures and tables
references and citing sentences
relations to other papers

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P abstract

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** (a) The scalar field theory one-loop diagram contributing to the fourgraviton amplitude of compactified eleven-dimensional supergravity. (b) The oneloop R4 counterterm that cancels the cubic ultraviolet divergence. 2.1. One-loop contributions Some of the systematics of the correspondence between the loop calculations of eleven-dimensional supergravity compactified on a circle or on a two-torus can be understood from dimensional arguments. For example, the one-loop four-graviton diagram of fig. 1 has dimension (momentum) 11 but it actually only diverges cubically with momentum.…

**Figure 2.** (a) A scalar field theory two-loop diagram that contributes to four-graviton scattering. (b) One-loop diagrams in which one vertex is the R4 counterterm cancel the sub-divergences of the two-loop diagrams. (c) A two-loop counterterm proportional to S 2 R4 . New primitive divergences arise at each order in perturbation theory. For example, the two-loop Feynman diagrams contributing to the four-graviton amplitude in elevendimensional supergravity have the superficial degree of divergence Λ20. However, the amplitude has an expansion in powers of derivatives acting on R4 so there are a

**Figure 3.** The ‘S-channel’ scalar field theory diagrams that contribute to the twoloop four-graviton amplitude of eleven-dimensional supergravity. (a) The (S, T) planar diagram, I P (S, T); (b) The (S, T) non-planar diagram, I NP (S, T). 19

**Figure 4.** The domain of integration over the parameters τ1 and τ2, bounded by the thick line, is the fundamental domain of Γ0(2). A more complete analysis of the divergences is greatly facilitated by the observation that the integrand possesses a secret SL(2, Z) symmetry that is not at all apparent in the λ, ˆ ρ, ˆ σˆ parameterization. This symmetry is made manifest by redefining the integration variables in (4.15) so that the parameters, ˆρ, λˆ and ˆσ, are replaced by the dimensionless volume, V , and complex structure, τ = τ1 + iτ2, of a two-torus, Tˆ 2 , defined by τ1 = ρˆ ρˆ+ λˆ , τ2 =

Figures reproduced from arXiv:hep-th/9910055 by the authors.

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hep-th 2019-12 conditional novelty 7.0 of 10

The large-N expansion of a four-point correlator in N=4 super-Yang-Mills reproduces the SL(2,Z)-invariant Eisenstein series coefficients of the R^4 and D^4R^4 interactions in type IIB string theory, including D-instan...
Genus-2 Holographic Correlator on $AdS_5 \times S^5$ from Localization
hep-th 2019-08 conditional novelty 7.0 of 10

The genus-two D4R4 coefficient in the strongly coupled N=4 SYM stress-tensor correlator is fixed to 7/3072 by combining supersymmetric localization with the flat-space IIB S-matrix.
AdS$_4$/CFT$_3$ from Weak to Strong String Coupling
hep-th 2019-06 accept novelty 7.0 of 10

The R^4 correction to the stress-tensor four-point function in ABJM theory is fixed at finite type IIA string coupling and matches the known genus-zero and genus-one string S-matrix.
$\mathcal{N}=4$ Super-Yang-Mills Correlators at Strong Coupling from String Theory and Localization
hep-th 2019-02 conditional novelty 7.0 of 10

Localization and string-theory computations of 1/λ corrections to half-BPS four-point functions in N=4 SYM agree at R^4 order, confirming AdS/CFT beyond supergravity and fixing the D^4R^4 correction.
Genus-One String Amplitudes from Conformal Field Theory
hep-th 2018-09 conditional novelty 7.0 of 10

The double discontinuity of the one-loop, four-point AdS5 x S5 amplitude is computed from CFT data and matched to the genus-one type IIB string amplitude, fixing two subleading effective-action coefficients.
Duality and higher derivative terms in M theory
hep-th 2005-10 accept novelty 7.0 of 10

The coupling dependence of the D^6R^4 term in the IIA and IIB effective actions is fixed by a two-loop eleven-dimensional supergravity computation, yielding an SL(2,Z) modular function that satisfies an inhomogeneous ...
Modular Features of Superstring Scattering Amplitudes: Generalised Eisenstein Series and Theta Lifts
hep-th 2025-01 conditional novelty 6.0 of 10

A single theta-lift construction generates both two-loop modular graph functions and S-dual modular functions as rational combinations of generalised Eisenstein series with all cusp-form L-values cancelling.
Higher-derivative corrections in M-theory from precision numerical bootstrap
hep-th 2024-12 conditional novelty 6.0 of 10

A numerical bootstrap with localization inputs extracts a tentative D^8R^4 higher-derivative coefficient in M-theory from ABJM correlators.
Species Scale in Diverse Dimensions
hep-th 2023-10 conditional novelty 6.0 of 10

Using exactly known R4 and R2 coefficients in 5 to 10 dimensions, the authors locate desert points, compute moduli-space diameters, and argue the species scale obeys |grad Lambda_s|^2/Lambda_s^2 <= 1/(d-2) everywhere ...
(Non)-projective bounds on gravitational EFT
hep-th 2022-01 accept novelty 6.0 of 10

Mapping the gravitational EFThedron onto the mathematical L-moment problem yields two-sided bounds on individual Wilson coefficients, including 0 <= b0,0/(4 pi)^2 <= 0.337 M^8 for the R^4 coupling.
New Modular Invariants in $\mathcal{N}=4$ Super-Yang-Mills Theory
hep-th 2020-08 conditional novelty 6.0 of 10

Fourth derivatives of the N=2* squashed-sphere partition function, expanded at large N, produce a new class of modular functions (generalized Eisenstein series) whose coefficients match the R4, D4R4 and D6R4 interacti...
Asymptotics of the $D^8 R^4$ genus-two string invariant
hep-th 2018-06 conditional novelty 6.0 of 10

The genus-two D^8 R^4 string invariant is expanded near non-separating, separating, and tropical degeneration limits, and its tropical limit reproduces two-loop supergravity at leading order plus subleading odd-zeta s...
M-Theory Reconstruction from (2,0) CFT and the Chiral Algebra Conjecture
hep-th 2018-05 accept novelty 6.0 of 10

The R4 coefficient in the 11d M-theory four-graviton amplitude is derived from 6d CFT data via the W_N chiral algebra, matching the independently known string theory value.
The M-Theory S-Matrix From ABJM: Beyond 11D Supergravity
hep-th 2018-04 accept novelty 6.0 of 10

The known R^4 coefficient of the M-theory four-graviton amplitude is rederived from ABJM localization data via the flat-space limit of a Mellin amplitude.
From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop
hep-th 2018-03 conditional novelty 6.0 of 10

A conjectured elliptic single-valued projection converts symmetrized one-loop open-string integrals into modular graph functions, verified through weight seven and seventh order in the low-energy expansion.
On the modular structure of the genus-one Type II superstring low energy expansion
hep-th 2015-02 conditional novelty 6.0 of 10

An infinite class of modular functions from genus-one string amplitudes satisfies exact Laplace eigenvalue equations, and conjectured algebraic relations determine one-loop D^8 R^4 and D^10 R^4 coefficients.
Matching the $D^6 {\cal R}^4$ interaction at two-loops
hep-th 2014-05 accept novelty 6.0 of 10

The Zhang-Kawazumi invariant is an eigenfunction of the Laplace-Beltrami operator with eigenvalue 5 on genus-two moduli space, and its integral over that space equals 2π³/45, matching the S-duality prediction for the ...
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...
Low energy expansion of the four-particle genus-one amplitude in type II superstring theory
hep-th 2008-01 accept novelty 6.0 of 10

The one-loop four-graviton amplitude in type II string theory is expanded through order s^6 R^4, yielding explicit coefficients that are rational multiples of Riemann zeta values and that constrain non-perturbative co...
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.
Inflationary constraints on the moduli-dependent species scale in modular invariant theories
hep-th 2024-11 conditional novelty 5.0 of 10

For SL(2,Z) modular inflation models, the CMB spectral index and tensor-to-scalar ratio imply a gravitational species scale of about 10^16 GeV.
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.

How later papers used it

78 later papers cite it

Evidence: substantial. 63 of the citing papers share no author with it, and 13 of those carry a confident label from Pith's citation classifier.

Pith read 16 of the 198 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

3 later papers by other authors are listed below: 2 used its method or tool and 1 used its data or values. 1 more paper by the same authors is listed: 1 used its method or tool.

Uses its method or tool 3 of 10 papers with this label · 1 by the same authors

Followed this paper's approach in using the results in another work to analyze the SL(2,Z)-invariant theta integral.

Some finite terms from ladder diagrams in three and four loop maximal supergravity (2015) · 0.99 · paraphrased: the licence does not allow quoting
Isolated the boundary contribution along the lines of this paper and mentioned only the results.

The D^6 R^4 term from three loop maximal supergravity (2014) · 0.99 · paraphrased: the licence does not allow quoting

By the same authors

Used this paper's method of orbits to split the integral into three kinds of pieces.

Automorphic properties of low energy string amplitudes in various dimensions (2010) (same authors) · 0.99 · paraphrased: the licence does not allow quoting

Uses its data or values 1 of 5 papers with this label

Fixed its overall normalization using the answer from this paper's two-loop supergravity.

Perturbative type II amplitudes for BPS interactions (2015) · 0.87 · paraphrased: the licence does not allow quoting

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72 arXiv papers cite this in 196 sentences

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Builds on it 1 paper

The method used to evaluate these integrals is an extension of that used for the $(0,1)$ case in [ref] , which we will review in the appendix [ref] ($I_(1,0)^(d=10)$ was evaluated in [ref] ).

Modular properties of two-loop maximal supergravity and connections with string theory · Interactions from circle compactification to ten dimensions · 0.92

Uses its method or tool 10 papers

To evaluate eval, we further define [ref] τ_1 = ρ̂/ρ̂+λ̂, τ_2 = √(Δ̂_2(,ł̂,̊̂))/ρ̂+λ̂, V_2 = l_11^2 √(Δ̂_2(,ł̂,̊̂)),to get that 𝒜_D^4ℛ^4^(2) = π^11κ_11^4/(2π)^D l_11^8 _2ℛ^4∑_m̂^I, n̂^I∫_0^∞dV_2 V_2^3 ∫_ℱ_2 d^2 τ/τ_2^2 e^-π^2 G_IJ (m̂+n̂τ)^I (m̂+n̂τ̅)^J V_2/τ_2,where $d^2 τ= dτ_1 dτ_2$ and $ℱ_2$ is the fundamental domain of $SL(2,ℤ)$ defined by ℱ_2 = { -1/2 ≤τ_1 ≤1/2, τ_2 ≥0, |τ|^2 ≥1}.Thus $V_2$ and $τ$ parametrize the volume and complex structure of an auxiliary $T^2$.

Non--analytic terms from nested divergences in maximal supergravity · The $ℛ^4$, $D^2ℛ^4$ and $D^4ℛ^4$ coefficient functions from supergravity amplitudes · 0.91
The integral over $τ$ in the $(m̂,n̂) (0,0)$ piece can be `unfolded' to the infinite strip as in [ref] , giving h_(p,q)^0 (1) = π^5 b_(p,q)^0 ∫_chi^dVV ( ∫_d^2ττ_2^2 +∫_0^∞dtt^2∑_p≠0exp(-πp^2 v V/τ_2)) = π^6 b_(p,q)^03 ∫_chi^dVV + π^6 b_(p,q)^03v ∫_chi^dVV^2 .

Modular properties of two-loop maximal supergravity and connections with string theory · $h_(3,0)^0$, $h_(0,2)^0$ · 0.96
Hence we only need to isolate the boundary contribution, which is done along the lines of [ref] and so we only mention the results.

The D^6 R^4 term from three loop maximal supergravity · The two loop four graviton amplitude · 0.99

Uses its data or values 5 papers

Using the ten dimensional values $λ^(10)_(0,0)=3/4$, $λ^(10)_(1,0)=15/4$ and $λ^(10)_(0,1)=12$ determined in [ref] , we deduce that the coefficients of the terms discussed in [ref] satisfy the following set of Laplace eigenvalue equations with source terms, [equation]

String theory dualities and supergravity divergences · Constraints on the coefficients · 0.99
The contribution to the coefficient of the $σ_2 ^4∼^4 ^4$ term was shown in [ref] to have the form I^(d=10)_(1,0) + I_▹ (1,0)^(d=10) + δ_2 I^(d=10)_(1,0) = ζ(5)4 R_11^5 ,

Modular properties of two-loop maximal supergravity and connections with string theory · $(p,q)=(1,0)$ · 0.81
For the $D^4ℛ^4$ interaction, from 1loop2 we get that ( λ^2 ^2/λ^2 +2 λ/λ-2)Î^(1)_D^4ℛ^4 =0,leading to I^(1)_D^4 ℛ^4 = 4/π^2ζ(3)ζ(4)(r^3 +1/r^3 ), where we have fixed the overall normalization using the answer from two loop supergravity [ref] .

Perturbative type II amplitudes for BPS interactions · The genus one amplitudes · 0.87

Compares with it 6 papers

The various contributions in [ref] agree with the expressions obtained by evaluating the sum of one-loop and two-loop Feynman diagram contributions to the amplitude in eleven-dimensional supergravity compactified on a $(d+1)$-torus [ref] .

Eisenstein series for higher-rank groups and string theory amplitudes · Solutions for the coefficients $^(D)_(0,0)$ and $^(D)_(1,0)$. · 0.98
In the absence of concrete answers at three and four loops, we consider in detail the analysis at two loops, where several amplitudes are known [ref] and we can use the results to compare with what we get based on our analysis in the main text.

Some finite terms from ladder diagrams in three and four loop maximal supergravity · Generalities and our analysis of the two loop amplitude · 0.91
These coefficients follow from the methods described in appendix [ref], and the results agree with the type IIA expressions derived by taking the one-loop amplitude of eleven dimensional supergravity on $T^2$ [ref] .

Low energy expansion of the four-particle genus-one amplitude in type II superstring theory · Summary of the expansion of the nine-dimensional genus-one amplitude · 1.00

Cites it as background 26 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
The terms in the low momentum expansion of the two loop four graviton amplitude up to the $D^12ℛ^4$ term have been obtained in [ref] .

Supergravity limit of genus two modular graph functions in the worldline formalism · Introduction · 0.90
We review various details of the four graviton amplitude in $N=1$ supergravity in 11 dimensions compactified on $ℝ^8,1 ×T^2$ at one and two loops [ref] .

The D^6 R^4 term from three loop maximal supergravity · The four graviton supergravity amplitude at one and two loops · 0.91

Discussion

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