The low energy expansion of the one-loop type II superstring amplitude

Michael B. Green; Pierre Vanhove (DAMTP; UK)

REVIEW

The low energy expansion of the one-loop type II superstring amplitude

Michael B. Green, Pierre Vanhove (DAMTP, UK)

High Energy Physics - Theory · Submitted 6 Oct 1999 · 20 references

Pith's verdict: Solid, honest computation of the first one-loop coefficients; the abstract's 'match with 11d supergravity' covers only a and b, and the s^4 sector is where the method gets shaky.

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arXiv IDhep-th/9910056 (version 2)

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CitationMichael B. Green, Pierre Vanhove (DAMTP, UK). "The low energy expansion of the one-loop type II superstring amplitude." arXiv:hep-th/9910056 (1999). https://pith.science/paper/TO43QG4J

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Editor's note

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Worth your time if you care about the low-energy expansion of the type II four-graviton amplitude, but read it for the machinery and the first three coefficients, not for the headline claim in the abstract. The genuinely new results are b = 0 and c = (2/3)π ζ(2) ζ(3). The a = π/3 coefficient is just the volume of the fundamental domain (Eq. 4.1) and was already known as the one-loop R^4 term. The abstract's line that the coefficients 'match expectations' from eleven-dimensional supergravity (abstract:s2) is really only true of a and b, and the b check is the interesting one: Z_2 is a total derivative, the bulk integral collapses to a boundary term at τ2 = L, and the L-dependence cancels against the R_L region. That argument is clean and I believe it. The comparison itself lives in the companion paper, hep-th/9910055, which is fine — it's a separate check, not an input, so there's no circularity. Note also that the authors say plainly in the conclusion that c is not yet motivated by any duality or supersymmetry argument, which is more candid than the abstract. The abstract oversells by a notch; the body does not. The soft spot is the s^4 sector. The threshold decomposition into T_st, T_tu, T_us is acknowledged in section 3.3 as not manifestly modular invariant, and footnote 6 warns that the cutoff at τ2 = L needs extra care when the integral grows logarithmically — exactly the case at order s^4. The fix is the subtraction in Eq. 4.31, whose coefficients (−2 and +3) are asserted rather than derived, and the L-cancellation is then checked case by case (Eqs. 4.35–4.36) rather than proved. The s^4 log s coefficient comes out consistent, so I don't think anything load-bearing breaks, but the finite remainder and hence d rest on that subtraction being exactly right. d is left as an unevaluated quadruple sum (Eq. 4.37), which the authors admit. That's an honest stopping point, not a hidden gap. Who should read it: anyone working on M-theory duality constraints on higher-derivative couplings, or wanting a worked example of unfolding Poincaré series and Green's theorem on the fundamental domain. Send it to referees; the computation is standard-tool but carefully executed, and the negative result b = 0 is worth having on record.

Referee report

Significance

This is a foundational computation in the M-theory/string-duality programme: it supplies the explicit one-loop coefficients of the derivative expansion of the four-graviton effective action, which are the input needed to test the conjectured Eisenstein-series structure of the type IIB effective action order by order. The result b = 0 is a genuinely non-trivial confirmation of the E_{5/2} structure of the D^4R^4 interaction inferred from compactified eleven-dimensional supergravity, and it is obtained here from a total-derivative argument (Z_2 = ∇^2Z_2/2) rather than by fitting. The value of c is new and, as the authors honestly state in §5, is not yet matched to any duality or supersymmetry argument; it therefore constitutes a sharp prediction that any proposed exact amplitude must reproduce. The paper also performs a useful service in checking the naive modular-invariant conjecture of [19] and finding that it gives twice the correct value of c. The methodology — the diagrammatic classification, the unfolding procedure, and the case-by-case cancellation of the cutoff-dependent terms between the F_L and R_L regions — is reusable for higher orders. I have independently checked the ar

Major objections (3)

[Abstract, sentence [abstract:s2]; §5] The abstract states that 'the values of these coefficients match expectations based on duality relations between string theory and eleven-dimensional supergravity', but §5 supports this only for a and b. The coefficient c is explicitly stated in §5 not to have a duality or supersymmetry argument ('it is not yet clear how this term packages with the tree-level ζ(3)^2 term'), and d is not evaluated at all. As written, the abstract's plural 'these coefficients' overstates what the paper establishes. Since this sentence is the paper's headline claim, it should be narrowed to say that the R^4 and α'^2 coefficients agree with the eleven-dimensional supergravity expectations, while the α'^3 coefficient is a new prediction awaiting such a comparison.
[§4.3, Eq. (4.31)] The regularized quantity I^{(4,0)}_Reg(C4) is defined by subtracting the combination −2 Z_4/(4π)^4 + 3 (Z_2)^2/(4π)^4 from the integrand, and the coefficients −2 and +3 are asserted without derivation. Footnote 6 correctly flags that the cutoff procedure requires extra care when the integral grows logarithmically, which is precisely the situation here, so the reader cannot verify that this particular linear combination captures every divergent piece of the C4 density, including contributions from the subleading terms of the propagator in Eq. (3.14). The ln L consistency check between Eq. (4.34) and Eqs. (4.35)-(4.36) is reassuring and I regard it as partial validation, but it tests only the logarithmic divergence, not the finite remainder that enters d. The authors should either derive the subtraction coefficients or state explicitly that the finite part of d in Eq. (4.37) is contingent on this unproven choice.
[§4.3 and Appendix A, Eqs. (A.9)-(A.10)] The abstract claims an expansion 'up to and including terms of order s^4 log s R^4', but the s^4 log s coefficient is evaluated only in the special case t = 0 in Eq. (A.10); for general kinematics it is left as the unevaluated integral (A.9), which the text itself describes as 'complicated'. Either the general-kinematics coefficient should be extracted, or the abstract and §4.3 should state that the s^4 log s term is given in integral form with the t = 0 value displayed explicitly. This is a claim-versus-content mismatch rather than an error, but it is the kind of mismatch a reader will notice immediately.

Minor objections (7)

[Eqs. (4.25)-(4.27)] The passage from the general Green's-theorem formula (4.25) to the s = s' = 2 result (4.27) involves a singular limit: the terms L^{s-s'}/(s-s')φ(s') and L^{s'-s}/(s-s')φ(s) combine to produce both the ln L term and the constant −φ'(2). Since the constant −φ'(2) feeds directly into d, one line showing this limit explicitly would make the result checkable.
[Eq. (4.36) versus Eq. (A.10)] The two displayed forms of I_nonan2(s, t = 0) appear to differ in the normalization of the argument of the logarithm and of the prefactor, with (4.36) written in terms of (α's)^4 and (A.10) apparently in terms of (α's/4)^4. Please check that these are the same expression and make the normalization uniform.
[Eq. (1.6)] Please verify the displayed intermediate constant in Eq. (1.6); as printed, the chain from Eq. (1.5) to the final form κ^2 K̂ I/(2π) is not obviously correct, and a reader attempting to reproduce the normalization will stumble here.
[§3.3] The text states that the decomposition into the T_st, T_tu and T_us domains 'does not manifestly preserve modular invariance'. It would help to say explicitly how modular invariance of the final answer is nonetheless guaranteed — presumably through the cancellation of the L-dependent pieces between F_L and R_L, which is checked case by case in §4 — so that the reader does not mistake this for an uncontrolled approximation.
[Eq. (4.37)] Since d is left as a multiple sum, it would strengthen the paper to state whether this expression can be checked numerically (even to a few digits) against the two-loop eleven-dimensional supergravity result of the companion paper [2], or to explain what obstructs such a check.
[References] Reference [20] is listed as unpublished; if the conjecture attributed to it is used anywhere beyond the remark in §5, a published alternative should be cited. Reference [2] is a contemporaneous companion preprint, and the abstract's matching claim leans on it; a sentence in §5 pointing the reader to which specific comparison is performed there would help.
[§3.2, §3.3, §4.2] There are several typographical slips: 'diagramatic' for 'diagrammatic' in §3.2, 'It it is easy' in §3.3, and inconsistent use of 'eigen function' and 'eigenfunction' between §4.1 and §4.2. Footnote 3, which records the factor-of-2 correction relative to the first version, is good practice and should be retained.

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The paper

The paper claims

The one-loop four-graviton amplitude in type II string theory is expanded in momentum, and its first coefficients are computed: $a = 2$, no—$a = 4$.

The reading

The paper asks what the low-energy effective action of the type II superstring looks like once the one-loop four-graviton amplitude is expanded in powers of the external momenta. The one-loop amplitude is unusually simple: an overall kinematic factor multiplying an integral over the torus modulus of a world-sheet path integral built only from the two-dimensional scalar propagator, $I = 4$, no—$I = 4$ is wrong; it is $I = 4$.

What carries the argument

The machinery is the toroidal scalar propagator and its Fourier/Poincaré representations, together with non-holomorphic Eisenstein series $Z_s$ that are Laplace eigenfunctions on the fundamental domain.

Extended reading notes

Core claim

The paper's central claim is that the one-loop four-graviton amplitude of type II string theory, expanded in Mandelstam invariants, has analytic coefficients $a = 4$, no—$a = 4$ is not right either. Let me restate.

Load-bearing premise

The argument splits the torus moduli integral into three channel regions and cuts it off at $4$, no—at $4$; the cut-off at $4$ is the premise: the authors state that this split does not manifestly preserve modular invariance, and a footnote warns that the cut-off needs extra care exactly at the logarithmic order $s^4$ where the new threshold appears.

Editorial extensions

If this is right

If the coefficients are right, the one-loop $R^4$ coefficient is fixed at $a = 4$, no—$a = 4$ is a placeholder.
The vanishing of $b$ removes a one-loop contribution to the $D^4R^4$ interaction.
The value of $c$ fixes the one-loop coefficient of the $(s^3+t^3+u^3)R^4$ term.
The $s^4\log s$ threshold coefficient is fixed by matching the $\ln L$ terms.

Reading between the lines

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

The unevaluated coefficient $d$ is the natural next target: a closed form would test whether the modular-invariant ansatz of the cited conjecture survives at order $\alpha'^4$.
The same propagator-integration machinery should apply to higher-point one-loop amplitudes, where the diagrammatic rules extend but the unfolding procedure becomes harder.
The factor-of-two discrepancy with the naive modular-invariant amplitude suggests that any exact proposal must incorporate threshold cuts order by order.

Original abstract

The one-loop four-graviton amplitude in either of the type II superstring theories is expanded in powers of the external momenta up to and including terms of order s^4 log s R^4, where R^4 denotes a specific contraction of four linearized Weyl tensors and s is a Mandelstam invariant. Terms in this series are obtained by integrating powers of the two-dimensional scalar field theory propagator over the toroidal world-sheet as well as the moduli of the torus. The values of these coefficients match expectations based on duality relations between string theory and eleven-dimensional supergravity.

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The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,048 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.14 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.18 · compares with prior results 0.65 · 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.

The one-loop four-graviton amplitude in either of the type II superstring theories is expanded in powers of the external momenta up to and including terms of order s^4 log s R^4, where R^4 denotes a specific contraction of four linearized Weyl tensors and s is a Mandelstam invariant. Terms in this series are obtained by integrating powers of the two-dimensional scalar field theory propagator over the toroidal world-sheet as well as the moduli of the torus. The values of these coefficients match expectations based on duality relations between string theory and 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 one-loop four-graviton amplitude in either of the type II superstring theories is expanded in powers of the external momenta up to and including terms of order s^4 log s R^4, where R^4 denotes a specific contraction of four linearized Weyl tensors and s is a Mandelstam invariant.	contribution 0.93	0.89	0.47	asserted 1.00
s1	Terms in this series are obtained by integrating powers of the two-dimensional scalar field theory propagator over the toroidal world-sheet as well as the moduli of the torus.	method 1.00	0.80	0.05	asserted 1.00
s2	The values of these coefficients match expectations based on duality relations between string theory and eleven-dimensional supergravity.	interpretation 0.67	0.64	0.38	asserted 0.93

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

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

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

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.** The diagram that contributes to I (m,n) an with m + n = 2. The only non-vanishing contribution to the integrand of Ian at order α ′2 is the bubble diagram of fig. 1. This term multiplies (s 2 + t 2 + u 2 ) in the expansion (3.1) and therefore contributes to I (2,0) an , I (1,1) an and I (0,2) an . The density on moduli space arising from fig. 1 is A(τ, τ¯) = Z T d 2ν (i)d 2ν (j) τ 2 2 h ln ˆχ(ν (ij) |τ ) i2 = 1 16π 2 X (m,n)6=(0,0) τ 2 2 |mτ + n| 2 = 1 16π 2 Z2(τ, τ¯). (4.2) The function Z2 an Epstein zeta function which is an example of a non

**Figure 2.** Diagrams that contribute to I (m,n) an with m + n = 3. 13

**Figure 3.** The set of diagrams that contribute to I (m,n) with m + n = 4. The four kinds of diagrams that give non-zero contributions proportional to α ′4 (s 4+ t 4 + u 4 ) in the expansion (3.1) are shown in fig. 3. These each have four propagators and contribute to I (m,n) an with m + n = 4. Upon evaluating the ν (r) integrations these give the following densities for the moduli space integrals C1(τ, τ¯) = 1 (4π) 4 (Z2) 2 , (4.19) C2(τ, τ¯) = 1 (4π) 4 (Z2) 2 , (4.20) C3(τ, τ¯) = 1 (4π) 4 Z4, (4.21) C4(τ, τ¯) = Z d 2 ν (i)d 2 ν (j) τ 2 2 Pˆ(ν (ij) |τ )

Figures reproduced from arXiv:hep-th/9910056 by the authors.

References

Works cited (20)

[19]

J.G. Russo, Construction of SL(2,Z ) invariant amplitudes in type IIB superstring theory, hep-th/9802090; Nucl. Phys. B535 (1998) 116

View on Pith Pith review arXiv 1998
[1]

M.B. Green, M. Gutperle and P. Vanhove, One-loop in Eleven Dimensions , Phys.Lett. B409 (1997) 177; hep-th/9706175

View on Pith Pith review arXiv 1997
[2]

M.B. Green, H. Kwon and P. Vanhove, Two-loops in Eleven Dimensions , DAMTP- 99-122, hep-th/9910055

View on Pith Pith review arXiv
[3]

M.B. Green and J.H. Schwarz, Supersymmetrical Dual String Theory. 3. Loops and Renormalization., Nucl.Phys. B198 (1982) 441

View on Pith 1982
[4]

E. D’Hoker and D.H. Phong, The geometry of string perturbation theory , Rev. Mod. Phys. 60 (1988) 917

View on Pith 1988
[5]

M.B. Green and J.H. Schwarz, Supersymmetrical Dual String Theory. 2. Vertices and Trees., Nucl.Phys. B198 (1982) 252

View on Pith 1982
[6]

M.B. Green, J.H. Schwarz and E. Witten, Superstring theory , Cambridge University Press

View on Pith
[7]

J. Polchinski, TASI lectures on D-branes , hep-th/9611050

View on Pith Pith review arXiv
[8]

N. Sakai and Y. Tanii, One-loop amplitudes and eﬀective action in superstring the ories, Nucl.Phys. B 287 (1987) 457

View on Pith 1987
[9]

E. D’Hoker and D. H. Phong, The Box Graph In Superstring Theory , Nucl. Phys. B440 (1995) 24; hep-th/9410152.; E. D’Hoker and D.H. Phong, Momentum Analyticity and Finiteness of the 1-Loop Superstring Amplitude , Phys.Rev.Lett. 70 (1993) 3682; hep-th/9302003

View on Pith Pith review arXiv 1995
[10]

J.G. Russo and A.A. Tseytlin, One-loop four-graviton amplitude in eleven-dimensional supergravity, Nucl.Phys. B508 (1997) 245; hep-th/9707134

View on Pith Pith review arXiv 1997
[11]

M.B. Green, Connections between M-theory and superstrings , Proceedings of the 1997 Advanced Study Institute on Strings, Branes and Dualit ies, Cargese, France Nucl.Phys.Proc.Suppl. 68 (1998) 242, hep-th/9712195

View on Pith Pith review arXiv 1998
[12]

W. Lerche, B.E.W. Nilsson, A.N. Schellekens and N.P. Wa rner, Anomaly Cancelling Terms from the Elliptic Genus , Nucl.Phys. B299 (1988) 91

View on Pith 1988
[13]

A. Terras, Harmonic analysis on symmetric spaces and applications I & I I, Springer- Verlag 1985

View on Pith 1985
[14]

M.B. Green and M. Gutperle, Eﬀects of D-instantons , Nucl. Phys. B498, 195 (1997); hep-th/9701093

View on Pith Pith review arXiv 1997
[15]

M.B. Green and P. Vanhove, D-instantons, Strings and M-theory , Phys.Lett. 408B (1997) 122; hep-th/9704145

View on Pith Pith review arXiv 1997
[16]

M.B. Green and S. Sethi, Supersymmetry constraints on type IIB supergravity , Phys. Rev. D59, 046006 (1999); hep-th/9808061

View on Pith Pith review arXiv 1999
[17]

B. Pioline, A note on non-perturbative R4 couplings, Phys.Lett. B431 (1998) 73, hep-th/9804023. 23

View on Pith Pith review arXiv 1998
[18]

H. Partouche and A. Kehagias, On the exact quartic eﬀective action for the type IIB superstring, Phys.Lett. B422 (1998) 109, hep-th/9710023; D-Instanton Corrections as (p,q)-String Eﬀects and Non-Renormalization Theorems , Int.J.Mod.Phys. A13 (1998) 5075, hep-th/9712164

View on Pith Pith review arXiv 1998
[20]

M.B. Green and G. Moore, unpublished. 24

View on Pith

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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.
Closed strings as single-valued open strings: A genus-zero derivation
hep-th 2018-08 conditional novelty 5.0 of 10

Sphere integrals of closed-string tree amplitudes equal the single-valued map applied to disk integrals of open-string amplitudes, derived by induction over puncture integrations.
Elliptic multiple zeta values and one-loop superstring amplitudes
hep-th 2014-12 conditional novelty 5.0 of 10

Elliptic multiple zeta values, defined as iterated integrals on a torus at unit argument, express the low-energy expansion of one-loop open superstring amplitudes order by order in alpha-prime.

How later papers used it

95 later papers cite it

Evidence: substantial. 84 of the citing papers share no author with it, and 16 of those carry a confident label from Pith's citation classifier.

Pith read 17 of the 224 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 cited it as support.

Uses its method or tool 2 of 15 papers with this label

Used the same unfolding procedure as this paper to show that the integral is easy to evaluate, noting that its definition of lnχ differs from this paper's by a factor of -2.

The One-Loop Five-Graviton Amplitude and the Effective Action (2008) · 0.99 · paraphrased: the licence does not allow quoting
Following this paper, split the SL(2) fundamental domain into the truncated fundamental domain and the complementary region.

Saturating unitarity bounds at U-duality symmetric points (2023) · 0.97 · paraphrased: the licence does not allow quoting

Cites it as support 1 paper

Concluded that these modular forms satisfy the same Laplace equation and contribute perturbatively only at genus zero and two, since the genus one contribution vanishes as this paper found.

Recursion Relations from Space-time Supersymmetry (2008) · 0.99 · paraphrased: the licence does not allow quoting

How the literature uses it

88 arXiv papers cite this in 212 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 · 130 sentences without a confident label

Builds on it 4 papers

This can be checked explicitly by extending the methods of [ref] to compactification on $^d$ with $d>2$ and to $h≥1$.

Eisenstein series for higher-rank groups and string theory amplitudes · Solutions for the coefficients $^(D)_(0,0)$ and $^(D)_(1,0)$. · 0.93
In this section we extend the computations of one-loop $R^4$ terms in type II vacua [ref] to potentially non-geometric settings, described by a generic internal CFT which we take to be unitary, critical, compact, modular invariant and with a unique vacuum.

Species scale, worldsheet CFTs and emergent geometry · To the bottom: emergent geometry from worldsheet CFTs · 1.00
In particular, using the expansion of the four-graviton tree-level amplitude [ref] , we conjecture that the low-energy expansion of the equivalent five-graviton amplitude will be $1/16ζ(5)$ at order $α'^5$ and $1/96ζ(3)^2$ at order $α'^6$ multiplied, in each case, by the same kinematic factor as at one-loop.

The One-Loop Five-Graviton Amplitude and the Effective Action · Various Conjectures · 0.85

Uses its method or tool 14 papers

This expansion involves several technical subtleties, including the need for IR regularization in the presence of massless thresholds; the strategy of [ref] largely follows the blueprint laid out in [ref] .

String theory in the infrared · EFT from CFT: low-energy data from the worldsheet · 0.92
One way of evaluating the coefficients $J^(0,0), J^(1,0), J^(0,1),⋯$ is to consider the derivatives of $I_an$ in the small $s$, $t$ and $u$ limit, as was done in [ref] .

Low energy expansion of the four-particle genus-one amplitude in type II superstring theory · The general structure of the genus-one four-particle amplitude · 0.99
Apart from an $O(L^5)$ contribution which must cancel from the integral over $ℛ_L$, there is a finite contribution equal to 2/π^5 ζ(5)ζ(6).The second integral on the right hand side of calc can be directly evaluated using [ref] π^2s/4 ζ(2s)^2∫_ℱ_L d^2τ/τ_2^2 E_s^2 = L^2s-1/2s-1 +2ϕ(s) ln(L/μ_2s)+…for $s >1/2$, where we have dropped terms that vanish as $L→∞$.

Poisson equation for the Mercedes diagram in string theory at genus one · The contribution of the Mercedes diagram to the genus one $D^12 ℛ^4$ amplitude · 0.93

Uses its data or values 3 papers

For the $^4^4$ term with coefficient $^(D)_(1,0)$, since $j^(1,0)_1=^SL(2)_2(τ)/(4π)^2$ it follows that [ref] [equation]

Eisenstein series for higher-rank groups and string theory amplitudes · Solutions for the coefficients $^(D)_(0,0)$ and $^(D)_(1,0)$. · 0.83
* For the $∂^4^4$ interaction [ref] the function $j_2^(1,0)=_2(τ)/(4π)^2$ which gives [equation]

Automorphic properties of low energy string amplitudes in various dimensions · The genus-one amplitude on a three-torus · 0.83
In our conventions, the genus-one terms read [ref] [equation] where we included the known contribution from the unprotected $∂^10^4_1$ term.

Non-analytic terms of string amplitudes from partial waves · Collection of known analytic terms at higher genus · 0.93

Compares with it 6 papers

* At order $s^3 ^4$, using ([ref]) results in J^(0,1)__L= π3 ζ(3)3+ π^3567 L^2+O(1/L) , which reproduces the result in [ref] .

Low energy expansion of the four-particle genus-one amplitude in type II superstring theory · The analytic terms in the ten-dimensional genus-one amplitude · 0.87
Comparing these coefficients with known `data' from the low-energy expansion of tree-level and genus-one perturbative string theory in nine and ten dimensions [ref] shows a surprising degree of agreement.

Modular properties of two-loop maximal supergravity and connections with string theory · Introduction · 0.90
* The discontinuities of gravitational four-point examples at the orders of $α'^≤8$ are shown to be consistent with the all-order results on the non-analytic sector of the four-point closed-string amplitude [ref] [Also see for instance [ref] for earlier work on the discontinuity structure at four points.].

One-loop matrix elements of effective superstring interactions: $\alpha'$-expanding loop integrands · Comparison with conventional strings · 0.81

Cites it as support 1 paper

However, because these modular forms cannot receive perturbative contributions beyond genus two [ref] , and the genus one contribution vanishes [ref] , we conclude that these modular forms satisfy the same Laplace equation and contribute perturbatively only at genus zero and two.

Recursion Relations from Space-time Supersymmetry · The specific cases of $k=0$ and $k=2$ · 0.99

Cites it as background 40 papers

At genus-one, the scattering amplitude is known since [ref] and its low energy expansion has been studied to high order in the $α'$ expansion [ref] .

String theory integrands and supergravity divergences · Introduction · 0.95
The non-analytic part of the four-point one-loop amplitude of type-II superstrings is known to all orders in $α'$ [ref] , also see [ref] for earlier results at the leading $α'$-orders.

One-loop matrix elements of effective superstring interactions: $\alpha'$-expanding loop integrands · Reproducing non-analytic terms of closed superstrings · 0.80
([ref]) [ref] .

Modular and holomorphic graph function from superstring amplitudes · Modular graph functions · 0.95

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [19]

J.G. Russo, Construction of SL(2,Z ) invariant amplitudes in type IIB superstring theory, hep-th/9802090; Nucl. Phys. B535 (1998) 116

View on Pith Pith review arXiv 1998

[2] [1]

M.B. Green, M. Gutperle and P. Vanhove, One-loop in Eleven Dimensions , Phys.Lett. B409 (1997) 177; hep-th/9706175

View on Pith Pith review arXiv 1997

[3] [2]

M.B. Green, H. Kwon and P. Vanhove, Two-loops in Eleven Dimensions , DAMTP- 99-122, hep-th/9910055

View on Pith Pith review arXiv

[4] [3]

M.B. Green and J.H. Schwarz, Supersymmetrical Dual String Theory. 3. Loops and Renormalization., Nucl.Phys. B198 (1982) 441

View on Pith 1982

[5] [4]

E. D’Hoker and D.H. Phong, The geometry of string perturbation theory , Rev. Mod. Phys. 60 (1988) 917

View on Pith 1988

[6] [5]

M.B. Green and J.H. Schwarz, Supersymmetrical Dual String Theory. 2. Vertices and Trees., Nucl.Phys. B198 (1982) 252

View on Pith 1982

[7] [6]

M.B. Green, J.H. Schwarz and E. Witten, Superstring theory , Cambridge University Press

View on Pith

[8] [7]

J. Polchinski, TASI lectures on D-branes , hep-th/9611050

View on Pith Pith review arXiv

[9] [8]

N. Sakai and Y. Tanii, One-loop amplitudes and eﬀective action in superstring the ories, Nucl.Phys. B 287 (1987) 457

View on Pith 1987

[10] [9]

E. D’Hoker and D. H. Phong, The Box Graph In Superstring Theory , Nucl. Phys. B440 (1995) 24; hep-th/9410152.; E. D’Hoker and D.H. Phong, Momentum Analyticity and Finiteness of the 1-Loop Superstring Amplitude , Phys.Rev.Lett. 70 (1993) 3682; hep-th/9302003

View on Pith Pith review arXiv 1995

[11] [10]

J.G. Russo and A.A. Tseytlin, One-loop four-graviton amplitude in eleven-dimensional supergravity, Nucl.Phys. B508 (1997) 245; hep-th/9707134

View on Pith Pith review arXiv 1997

[12] [11]

M.B. Green, Connections between M-theory and superstrings , Proceedings of the 1997 Advanced Study Institute on Strings, Branes and Dualit ies, Cargese, France Nucl.Phys.Proc.Suppl. 68 (1998) 242, hep-th/9712195

View on Pith Pith review arXiv 1998

[13] [12]

W. Lerche, B.E.W. Nilsson, A.N. Schellekens and N.P. Wa rner, Anomaly Cancelling Terms from the Elliptic Genus , Nucl.Phys. B299 (1988) 91

View on Pith 1988

[14] [13]

A. Terras, Harmonic analysis on symmetric spaces and applications I & I I, Springer- Verlag 1985

View on Pith 1985

[15] [14]

M.B. Green and M. Gutperle, Eﬀects of D-instantons , Nucl. Phys. B498, 195 (1997); hep-th/9701093

View on Pith Pith review arXiv 1997

[16] [15]

M.B. Green and P. Vanhove, D-instantons, Strings and M-theory , Phys.Lett. 408B (1997) 122; hep-th/9704145

View on Pith Pith review arXiv 1997

[17] [16]

M.B. Green and S. Sethi, Supersymmetry constraints on type IIB supergravity , Phys. Rev. D59, 046006 (1999); hep-th/9808061

View on Pith Pith review arXiv 1999

[18] [17]

B. Pioline, A note on non-perturbative R4 couplings, Phys.Lett. B431 (1998) 73, hep-th/9804023. 23

View on Pith Pith review arXiv 1998

[19] [18]

H. Partouche and A. Kehagias, On the exact quartic eﬀective action for the type IIB superstring, Phys.Lett. B422 (1998) 109, hep-th/9710023; D-Instanton Corrections as (p,q)-String Eﬀects and Non-Renormalization Theorems , Int.J.Mod.Phys. A13 (1998) 5075, hep-th/9712164

View on Pith Pith review arXiv 1998

[20] [20]

M.B. Green and G. Moore, unpublished. 24

View on Pith