One-loop open-string integrals from differential equations: all-order alpha'-expansions at n points

Carlos R. Mafra; Oliver Schlotterer

REVIEW 3 major objections 6 minor 2 cited by

One-loop open-string integrals from differential equations: all-order alpha'-expansions at n points

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper shows that every order in the α′ expansion of one-loop open-string integrals follows from a single first-order differential equation in τ, with uniform transcendentality and no twisted elliptic multiple zeta values.

desk verdict A genuinely new differential-equation framework for one-loop open-string α'-expansions, rigorous through n=5, with the all-order n-point claim resting on clearly labeled n≥6 conjectures and an open non-planar loophole. read the letter →

arxiv 1908.10830 v2 pith:NYLTUCO2 submitted 2019-08-28 hep-th

Carlos R. Mafra , Oliver Schlotterer This is my paper · ORCID

classification hep-th

keywords one-loopopen-stringamplitudesA-cycleintegralsellipticmultiplezetavaluesiteratedEisensteinalpha-primeexpansionTsunogaiderivationsKZBassociatorParke-Taylor

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that the generating functions of genus-one open-string integrals—the A-cycle integrals behind planar and non-planar one-loop superstring and bosonic-string amplitudes—obey a linear, homogeneous first-order differential equation in the modular parameter τ, with a universal form for every integration cycle. If true, every order in the inverse string tension α′ follows from elementary operations: matrix multiplication, η-differentiation, and known tree-level disk integrals at the cusp. The expansion is uniformly transcendental and expressible through iterated Eisenstein integrals, so the elliptic multiple zeta values that previously appeared are automatically presented in minimal form. A notable consequence is that non-planar one-loop integrals need no twisted elliptic multiple zeta values at any α′ order. The n-point differential operator is proven through five points and conjectured for all n.

What carries the argument

The machinery is a first-order differential equation of KZB type, $2\pi i\partial_{\tau}Z = D^{\tau}Z$, with $D^{\tau} = \sum_m (1-2m)G_{2m}(\tau)\, r_{\vec{\eta}}(\epsilon_{2m})$. The matrix entries $r_{\vec{\eta}}(\epsilon_{2m})$ are linear in $s_{ij}$ and contain $\eta$-derivatives; the Weierstrass function $\wp(\eta,\tau)$ generates the Eisenstein series. The companion ingredient is the degeneration at $\tau\to i\infty$, where the A-cycle pinches and the integrand reduces to Parke–Taylor disk integrands with two extra punctures; the contour deformation produces phases encoded in $H_{\alpha'}$ and $K_{\vec{\eta}}$ matrices. The S-map (4.31) organizes the combinatorics of the $\tau$-derivative at n points.

What would settle it

Compute $2\pi i\partial_{\tau}Z^{\tau}_{\vec{\eta}}$ at n=6 directly from (4.30), resolve the six-cycle of Kronecker–Eisenstein series by Fay identities, and compare the result with the S-map proposal (4.35). Any mismatch falsifies the all-order expansion; likewise, an n=6 contour-deformation check of the twisted-cycle degeneration (5.12) would test the cusp dictionary.

Watch

Extended reading notes

Core claim

The central discovery is that the (n−1)!-family of A-cycle integrals $Z^{\tau}_{\vec{\eta}}(\ast|1,A)$ closes under $2\pi i\partial_{\tau}$: the derivative of any member is a linear combination of the same family with coefficients linear in the Mandelstam variables $s_{ij}$. The coefficients assemble into the S-map and Weierstrass functions, giving the closed n-point formula (4.35), and their expansion in holomorphic Eisenstein series yields $(n-1)!\times(n-1)!$ matrix representations $r_{\vec{\eta}}(\epsilon_{2m})$ of Tsunogai's derivations. Picard iteration then expresses $Z^{\tau}_{\vec{\eta}}$ as a sum over words in these derivations acting on the cusp value $Z^{i\infty}_{\vec{\eta}}$, and the cusp values are identified with $(n+2)$-point Parke–Taylor disk integrals. This replaces step-by-step elliptic integration with a direct all-order expansion in iterated Eisenstein integrals $\gamma(k_1,\ldots,k_r|\tau)$.

Load-bearing premise

The load-bearing premise is that the closed n-point formula (4.35) for the $\tau$-derivative, proven only for $n\leq 5$ and conjectured for $n\geq 6$, is correct for every number of punctures; the all-order expansions and the non-planar no-twisted-eMZV result would not follow otherwise.

Editorial extensions

If this is right

Any desired order in $\alpha'$ of an n-point one-loop open-string integral is obtained by matrix multiplication and $\eta$-differentiation, starting from known disk integrals; no per-puncture elliptic integration or "z-removal" is needed.
Planar and non-planar cylinder integrals share one universal differential equation, so non-planar amplitudes are expressed in iterated Eisenstein integrals of $SL_2(\mathbb{Z})$ with no twisted elliptic multiple zeta values at any order.
Each $\alpha'$ order is uniformly transcendental: weight $k+n-1$ at order $\alpha'^{k}$, matching the pattern familiar from tree-level string integrals.
The A-cycle integrals are preserved by the coaction of iterated Eisenstein integrals, generalizing the motivic-coaction stability of disk integrals.
The $\tau\to i\infty$ degeneration ties genus-one integrals to $(n+2)$-point Parke–Taylor disk integrals, giving compact formulas for cylinder- and Möbius-strip boundary integrals.

Reading between the lines

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

If the conjectural n-point formula (4.35) survives a direct n=6 check, the same differential-equation architecture could yield recursions in loop order: genus-two integrals may satisfy similar equations in the off-diagonal period, with separating degenerations supplying initial data—a direction the authors flag but do not carry out.
The manifest absence of twisted elliptic multiple zeta values suggests a structural theorem: for any distribution of punctures on the two cylinder boundaries, the non-planar integral lies in the untwisted elliptic MZV algebra; proving this would require closing the Section 4.5 loophole about whether the representations annihilate non-planar initial values.
The method provides a practical algorithm for high-order $\alpha'$ expansions, and the natural next testbed is the n=6 differential operator, which the paper indicates is accessible by direct computation from (4.30).
One could try to import the same S-map and differential-equation structure into elliptic Feynman-integral computations, since the paper notes the equation is the string-theory analogue of an $\varepsilon$-form.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, this paper is a real advance in method: Mafra and Schlotterer show that the genus-one A-cycle integrals generating one-loop open-string amplitudes obey a first-order differential equation in τ of KZB type, and that Picard iteration turns the α'-expansion into iterated Eisenstein integrals acting on cusp values provided by (n+2)-point disk integrals. Second, the headline 'at n points' is conditional: the n-point differential operator (4.35) and the cusp relations (5.12), (5.17) are explicitly labeled conjectural for n≥6, and the non-planar case retains an openly stated loophole in Section 4.5.

What is genuinely new is the differential-equation organization itself, the S-map structure of the τ-derivative, and the concrete (n−1)!×(n−1)! matrix representations of Tsunogai's derivations. The paper does the work well: up to five points the derivations are rigorous, the cusp dictionary to disk integrals is carried out explicitly, and the outputs reproduce known eMZV and A-cycle graph function results from the literature. That agreement is genuine verification — the disk-integral inputs are independent of the eMZV data being matched — not fitting. The citation pattern is sound: the relevant math literature is present, and the self-citations trace a legitimate lineage through the authors' own earlier eMZV program.

The soft spot is exactly the n≥6 conjecture and the Section 4.5 loophole. If (4.35) fails at some multiplicity, the all-order expansion, the uniform-transcendentality argument, and the coaction result would need restriction. The authors are upfront about this — the abstract itself flags the representations as conjectural — so this is not a hidden flaw, but a referee should insist the conjecture be either proved or explicitly carved out of the claims. A direct n=6 check of (4.35) would go a long way.

Who this is for: string amplitude practitioners and anyone working on elliptic polylogarithms or period structures at genus one. It deserves a serious referee; the verified n≤5 core already earns publication, and the conjectural scaffold is concrete and testable rather than speculative. Send it to review, with the condition that the conjectural status and the non-planar loophole stay prominent.

Referee Report

3 major / 6 minor

Summary. The paper develops a differential-equation method for genus-one A-cycle integrals Z^τ_η(∗|1,2,...,n) of the type that enters one-loop open-string amplitudes. The main results are: (i) a linear, homogeneous first-order differential equation in the modular parameter τ for these integrals, rigorously derived for n ≤ 5 in Sections 4.1–4.3 and conjectured for all n in Eq. (4.35); (ii) a solution via Picard iteration, Eq. (1.7), expressing the α′-expansion at order α′^r in terms of iterated Eisenstein integrals γ(k1,...,kr|τ) and matrix representations r⃗η(ϵ2m) of Tsunogai derivations that are linear in the Mandelstam variables; (iii) a reduction of the cusp initial values at τ→i∞ to (n+2)-point Parke–Taylor disk integrals, proven through n = 5 and conjectural beyond, Eqs. (5.12) and (5.17); and (iv) structural consequences in Section 7: uniform transcendentality of the α′-expansion and invariance under the coaction of iterated Eisenstein integrals. The non-planar sector is claimed to be expressible without twisted eMZVs, with a residual caveat analyzed explicitly in Section 4.5. Extensive explicit expansions are given for n = 2,3,4,5 and cross-checked against known A-cycle graph functions from the literature.

Significance. The conjectural n ≥ 6 extension (4.35) and the companion cusp relations (5.12)/(5.17) are load-bearing, and the stress-test concern lands: as written, the title-level claim of all-order α′-expansions 'at n points' exceeds the proven range. If these conjectures hold, the paper is a substantial advance: it reduces all-order α′ expansions of genus-one A-cycle integrals to elementary matrix operations with no free parameters, and it makes uniform transcendentality and the coaction structure manifest. Independent of the conjecture, the n ≤ 5 results are rigorous, concrete, and falsifiable: the explicit expansions (5.38), (5.52), (5.55), (E.1), and the non-planar examples (6.12)–(6.24) are given in closed, machine-verifiable form and are benchmarked against previously published A-cycle graph functions and degeneration data. The paper is unusually transparent about its conjectural steps, provides explicit matrix representations in Appendix C, and documents detailed checks of the derivation-algebra relations; these are genuine strengths that make the core n ≤ 5 content reliable.

major comments (3)

[§4.4, Eq. (4.35)] The n-point differential equation (4.35) is the load-bearing input for the paper's central claims, and it is explicitly labelled conjectural for n ≥ 6. The rigorous derivations in §§4.1–4.3 cover n ≤ 5 only; the extension to arbitrary n is an extrapolation of the S-map pattern, and the cycle-resolution lemma (A.12) used in its derivation is itself tested only through n = 5 and conjectural beyond. Every subsequent all-order statement—the Picard-iteration expansion (1.7), the uniform-transcendentality claim (7.2) in §7.1, and the coaction formula (7.13) in §7.2—uses (4.35), so the title-level claim 'all-order α′-expansions at n points' and the abstract's unconditional phrasing exceed what is proven at n ≥ 6. Please either prove (4.35) (or at minimum verify it in detail at n = 6 and n = 7 as supporting evidence), or re-scope the abstract, title, and Section 7 statements so that the conjectural status is explicit.
[§5.2, Eqs. (5.12), (5.17); (A.11)] The cusp reduction of the initial values is a second load-bearing conjecture with the same n ≥ 6 gap. Equations (5.12) and (5.17), which express the planar degeneration of the A-cycle integrals in terms of (n+2)-point disk integrals, are declared conjectural at n ≥ 6, and the same holds for the identity (A.11)/(A.12) that resolves cycles of Kronecker–Eisenstein series. Since the expansion (1.7) requires both the differential operator and the initial values, the all-order results at n ≥ 6 rest on three independent conjectural inputs. The explicit n ≤ 5 checks against known graph functions and genus-zero data are commendable, but the statement in Section 8 that the method applies 'at n points' should be qualified in the same places where (4.35) is qualified.
[§4.5] Section 4.5 leaves open whether the matrices r⃗η(ϵ2m) preserve the Tsunogai derivation relations when acting on non-planar initial values, and the paper states this explicitly. This matters because the presentation of the non-planar results—'minimal form' with 'all relations among eMZVs already incorporated' in §1.1 and twisted eMZVs 'completely bypassed' in Section 8—is only fully justified if the iterated Eisenstein integrals in the non-planar expansion (1.7) conspire to eMZVs, which is exactly the §4.5 loophole. The planar case is protected by the contradiction argument in §4.5, but no analogous argument is given for non-planar cycles. Please either close this loophole, for example by showing that the non-planar initial values lie in the same class of functions under the r⃗η action as planar ones, or explicitly carry the §4.5 caveat into the abstract and the conclusions.

minor comments (6)

[Abstract and §1.1] The claims 'These integrals are shown to satisfy... a differential equation', 'a universal form for the integration cycles', and 'any desired order is accessible' are stated unconditionally; please add an explicit sentence distinguishing the rigorously established n ≤ 5 range from the conjectural n ≥ 6 status of (4.35), (5.12), and (5.17).
[Footnote 8] The stated derivative 2πi∂τ(iπτ/4) = −3ζ2/2 appears inconsistent with the paper's conventions: from ζ2 = π²/6 one obtains 2πi∂τ(iπτ/4) = −π²/2 = −3ζ2, which is also the value needed for the non-planar case of (2.16) to hold. Please verify and correct the footnoted value.
[§5.4 near (5.45) and App. E.1] The phrase 'irreducible iterated Einstein integrals' should read 'irreducible iterated Eisenstein integrals' in the two occurrences next to Eq. (5.45) and in Appendix E.1.
[Notation (1.1), (3.33), (5.40)] The integration cycle is denoted by '*' in (1.1) and (3.33) but by A in (5.40), (5.54), and (6.19) without an explicit statement that the two notations denote the same object; please add a remark at the first such switch.
[§4.3, Eq. (4.28)] Identity (4.28) is used to reduce the n! Fay-expanded terms to the (n−1)! basis on which (4.35) acts, but it is imported from reference [96] rather than derived in the genus-one setting; a short derivation, or at least an explicit statement of its status at n ≥ 6, would make the conjectural chain easier to audit.
[§5.2, after Eq. (5.17)] The expression 'Sα′/2(n,...,3,2|n,...,3,2)1' appears to contain a stray '1' in the notation for the diagonal KLT entry; please clarify or remove it.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the all-order alpha'-expansions are obtained by solving explicit differential equations with initial values reduced to independent disk integrals; the paper's own conjectural n>=6 caveats are proof gaps, not circularity.

full rationale

The derivation chain is non-circular. The A-cycle integrals are defined in (1.1), and their tau-derivatives are computed directly from the mixed heat equation (2.7), Fay identities, and integration by parts; for n<=5 the resulting linear system is proven and then solved by Picard iteration in (1.7). The initial values at tau -> i infinity in sections 5 and 6 are reduced to (n+2)-point disk integrals whose alpha'-expansions come from prior tree-level results [4,5,23,24], i.e. independent genus-zero inputs rather than quantities fitted to the one-loop target. The r_eta(epsilon) matrices are read off as coefficients of Eisenstein series in the differential operator, not adjusted to reproduce the claimed expansions. Validation against earlier eMZV and graph-function results (e.g. [20,21,81]) is an external consistency check. The only load-bearing caveats are explicit conjectures, not circular definitions: Eq. (4.35) for the n-point tau-derivative is 'conjectural at n>=6'; the twisted-cycle cusp relations (5.12) and (5.17) carry the same caveat; and Section 4.5 leaves open whether non-planar initial values are annihilated by the derivation-algebra relations, which would affect whether the iterated Eisenstein integrals conspire to eMZVs. These are gaps in proof at n>=6, which the paper states transparently, rather than cases of a result being equivalent to its input by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

No numerical parameters are fitted to data. The computations rely on standard identities for Kronecker-Eisenstein series and on several explicitly labeled conjectures for n>=6, which are the true cost of the all-order claim.

assumptions (6)

standard math Matthes' linear independence of iterated Eisenstein integrals over eMZVs (cited as [27]).
Used to state that the alpha-prime expansions in (1.7) are minimal and all eMZV relations are built in; Section 2.4 and footnote 3.
ad hoc to paper Identity (A.11) and (A.12) for resolving cycles of Kronecker-Eisenstein series at n>=6.
The authors state it is tested up to n=5 and conjectural at higher n; the n-point differential equation depends on it.
ad hoc to paper Conjectural n-point differential equation (4.35).
Section 4.4 explicitly labels (4.35) conjectural at n>=6; it is the basis for solving (1.7).
ad hoc to paper Conjectural twisted-cycle relation (5.12) and symmetric cycle (5.17) at n>=6.
Section 5.2 states these are conjectural at n>=6; they determine the initial values in terms of disk integrals.
domain assumption The (n-1)!-dimensionality of the genus-one twisted cohomology underlying the basis (1.1).
Proposed in Section 1.1 based on Schwarz and Fuchs discussions, not proven; the closure under tau-derivatives is supporting evidence.
ad hoc to paper The matrix representations r_eta(epsilon_{2m}) preserve Tsunogai derivation relations on relevant functions.
Section 4.5 verifies only low-weight commutators and argues planar annihilation; the non-planar case has a stated loophole.

invented entities (2)

Matrix representations r_eta(epsilon_{2m}) of Tsunogai derivations independent evidence
purpose: Generate the alpha-prime expansion of A-cycle integrals via (1.7) and encode the derivation algebra in (n-1)! times (n-1)! matrices.
They are conjectural but checked against commutator relations for low weights and against known A-cycle graph functions; hence they yield falsifiable predictions for higher orders.
(n-1)!-family of A-cycle integrands as representatives of genus-one twisted cohomology
purpose: Provides the closed basis in which the differential equations and alpha-prime expansions are stated.
Conjectured in Section 1.1; supported by low-point counts and closure under tau-derivatives, but no proof is supplied.

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Pith. "Pith review of One-loop open-string integrals from differential equations: all-order alpha'-expansions at n points." pith.science (2026). https://pith.science/paper/NYLTUCO2

@misc{pith2026190810830,
  author       = {Pith},
  title        = {Pith review of: One-loop open-string integrals from differential equations: all-order alpha'-expansions at n points},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NYLTUCO2}},
  note         = {Machine review of arXiv:1908.10830}
}

abstract

We study generating functions of moduli-space integrals at genus one that are expected to form a basis for massless $n$-point one-loop amplitudes of open superstrings and open bosonic strings. These integrals are shown to satisfy the same type of linear and homogeneous first-order differential equation w.r.t. the modular parameter $\tau$ which is known from the A-elliptic Knizhnik--Zamolodchikov--Bernard associator. The expressions for their $\tau$-derivatives take a universal form for the integration cycles in planar and non-planar one-loop open-string amplitudes. These differential equations manifest the uniformly transcendental appearance of iterated integrals over holomorphic Eisenstein series in the low-energy expansion w.r.t. the inverse string tension $\alpha'$. In fact, we are led to matrix representations of certain derivations dual to Eisenstein series. Like this, also the $\alpha'$-expansion of non-planar integrals is manifestly expressible in terms of iterated Eisenstein integrals without referring to twisted elliptic multiple zeta values. The degeneration of the moduli-space integrals at $\tau \rightarrow i\infty$ is expressed in terms of their genus-zero analogues -- $(n{+}2)$-point Parke--Taylor integrals over disk boundaries. Our results yield a compact formula for $\alpha'$-expansions of $n$-point integrals over boundaries of cylinder- or Moebius-strip worldsheets, where any desired order is accessible from elementary operations.

Figures

Figures reproduced from arXiv: 1908.10830 by the authors.

**Figure 1.** The cylinder worldsheet for one-loop open-string amplitudes seen in the left panel is parameterized by the gray rectangle in the right panel, where the periodic direction is reflected by the identification of horizontal lines. The cylinder boundaries drawn in red are the integration domain C(∗) of the integrals in (1.1) where a cyclic ordering of the punctures is prescribed within both boundaries. The asterisk is a… view at source ↗

**Figure 2.** The torus will be parameterized through a complex coordinate z, where the A-cycle and B-cycle translate into periodicities z ∼= z+1 and z ∼= z+τ , respectively. 2.1 Kronecker–Eisenstein series The integrands under investigation in (1.1) are built from a non-holomorphic version of the Kronecker–Eisenstein series [78, 79], Ω(z, η, τ ) ≡ exp 2πiη Im z Im τ θ 0 1 (0, τ )θ1(z + η, τ ) θ1(z, τ )θ1(η, τ ) , (2.2) where … view at source ↗

**Figure 3.** The degeneration of the torus at τ → i∞ pinches the A-cycle and yields the topology of a Riemann sphere. In particular, the pinched A-cycle introduces a pair of identified punctures σ+ = 0 and σ− = ∞ on the Riemann sphere. 3.3 The initial value at the cusp: degenerating the integrand Given that the pattern of iterated Eisenstein integrals in the two-point integrals is fixed by (3.14), the next step is to find the ex… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: As depicted in the left panel, the integration contour C(1, 2) in the σ2 = e 2πiz2 variable is the unit circle |σ2| = 1. When replacing C(1, 2) by the homotopy-equivalent combination of paths visualized the right panel, the multivalued part of the integrand σ s12/2 2…

**Figure 5.** Figure 5: In the σj = e 2πizj variables, the integration contour C(1, 2, 3) is the unit circle in the left panel, where the phases of σ2 and σ3 are ordered according to z2 < z3. Similar to figure 4, we replace C(1, 2, 3) by the homotopy-equivalent combination of paths visualized…

**Figure 6.** Figure 6: Genus-zero contributions Dijk to the degenerate genus-one integral over the fourpoint cycle C1234 = {0 < z2 < z3 < z4 < 1}. + 2isin π 2 (s14+s24+s34+s45) e iπ 2 (−s12−s13+s15−s23+s25−s35)D4352 + 2isin π 2 (s14+s24+s34+s45) e iπ 2 (−s12−s13+s15−s23+s25+s35)D4532 …

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Cited by 2 Pith papers

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All-order alpha'-expansion of one-loop open-string integrals
hep-th 2019-08 conditional novelty 8.0 of 10

A KZB-type differential equation and Picard iteration yield the all-order alpha-prime expansion of one-loop open-string integrals in terms of iterated Eisenstein integrals.
Associators for AdS string amplitude building blocks
hep-th 2025-05 conditional novelty 6.0 of 10

Open-string AdS building blocks can be generated by Drinfeld associator recursions and closed-string ones by Deligne associator recursions, yielding all-order zeta-valued expansions.

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Works this paper leans on

126 extracted references · 8 canonical work pages · cited by 2 Pith papers

[1]

C. R. Mafra and O. Schlotterer, All-order α′-expansion of one-loop open-string integrals , 1908.09848

arXiv 1908
[2]

Terasoma, Selberg integrals and multiple zeta values , Compositio Mathematica 133 (2002) 1

T. Terasoma, Selberg integrals and multiple zeta values , Compositio Mathematica 133 (2002) 1

2002
[3]

Brown, Multiple zeta values and periods of moduli spaces M0,n(R), Annales Sci

F. Brown, Multiple zeta values and periods of moduli spaces M0,n(R), Annales Sci. Ecole Norm. Sup. 42 (2009) 371, [ math/0606419]

arXiv 2009
[4]

Schlotterer and S

O. Schlotterer and S. Stieberger, Motivic Multiple Zeta Values and Superstring Amplitudes , J. Phys. A46 (2013) 475401, [ 1205.1516]

arXiv 2013
[5]

Broedel, O

J. Broedel, O. Schlotterer, S. Stieberger and T. Terasoma, All order α′-expansion of superstring trees from the Drinfeld associator , Phys. Rev. D89 (2014) 066014, [ 1304.7304]

arXiv 2014
[6]

Schnetz, Graphical functions and single-valued multiple polylogarithms , Commun

O. Schnetz, Graphical functions and single-valued multiple polylogarithms , Commun. Num. Theor. Phys. 08 (2014) 589–675, [ 1302.6445]

arXiv 2014
[7]

Brown, Single-valued Motivic Periods and Multiple Zeta Values , SIGMA 2 (2014) e25, [1309.5309]

F. Brown, Single-valued Motivic Periods and Multiple Zeta Values , SIGMA 2 (2014) e25, [1309.5309]

arXiv 2014
[8]

Stieberger, Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator, J

S. Stieberger, Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator, J. Phys. A47 (2014) 155401, [ 1310.3259]

arXiv 2014

Show all 126 references

[9]

Stieberger and T

S. Stieberger and T. R. Taylor, Closed String Amplitudes as Single-Valued Open String Amplitudes, Nucl. Phys. B881 (2014) 269–287, [ 1401.1218]

2014 arXiv
[10]

Schlotterer and O

O. Schlotterer and O. Schnetz, Closed strings as single-valued open strings: A genus-zero derivation, J. Phys. A52 (2019) 045401, [ 1808.00713]

2019 arXiv
[11]

Brown and C

F. Brown and C. Dupont, Single-valued integration and double copy , 1810.07682

arXiv
[12]

Vanhove and F

P. Vanhove and F. Zerbini, Closed string amplitudes from single-valued correlation functions , 1812.03018

arXiv
[13]

Brown and C

F. Brown and C. Dupont, Single-valued integration and superstring amplitudes in genus zero , 1910.01107

1910 arXiv
[14]

L. A. Barreiro and R. Medina, 5-ﬁeld terms in the open superstring eﬀective action , JHEP 03 (2005) 055, [ hep-th/0503182]

2005 arXiv
[15]

Oprisa and S

D. Oprisa and S. Stieberger, Six gluon open superstring disk amplitude, multiple hypergeometric series and Euler-Zagier sums , hep-th/0509042

arXiv
[16]

Stieberger and T

S. Stieberger and T. R. Taylor, Multi-Gluon Scattering in Open Superstring Theory , Phys. Rev. D74 (2006) 126007, [ hep-th/0609175]

2006 arXiv
[17]

Stieberger, Constraints on Tree-Level Higher Order Gravitational Couplings in Superstring Theory, Phys

S. Stieberger, Constraints on Tree-Level Higher Order Gravitational Couplings in Superstring Theory, Phys. Rev. Lett. 106 (2011) 111601, [ 0910.0180]

2011 arXiv
[18]

R. H. Boels, On the ﬁeld theory expansion of superstring ﬁve point amplitudes , Nucl. Phys. B876 (2013) 215–233, [ 1304.7918]

2013 arXiv
[19]

Puhlfuerst and S

G. Puhlfuerst and S. Stieberger, Diﬀerential Equations, Associators, and Recurrences for Amplitudes, Nucl. Phys. B902 (2016) 186–245, [ 1507.01582]. – 74 –

2016 arXiv
[20]

Broedel, C

J. Broedel, C. R. Mafra, N. Matthes and O. Schlotterer, Elliptic multiple zeta values and one-loop superstring amplitudes, JHEP 07 (2015) 112, [ 1412.5535]

2015 arXiv
[21]

Broedel, N

J. Broedel, N. Matthes, G. Richter and O. Schlotterer, Twisted elliptic multiple zeta values and non-planar one-loop open-string amplitudes , J. Phys. A51 (2018) 285401, [ 1704.03449]

2018 arXiv
[22]

Enriquez, Analogues elliptiques des nombres multiz´ etas, Bull

B. Enriquez, Analogues elliptiques des nombres multiz´ etas, Bull. Soc. Math. France 144 (2016) 395–427, [1301.3042]

2016 arXiv
[23]

Broedel, O

J. Broedel, O. Schlotterer and S. Stieberger. http://mzv.mpp.mpg.de
[24]

C. R. Mafra and O. Schlotterer, Non-abelianZ-theory: Berends-Giele recursion for the α′-expansion of disk integrals , JHEP 01 (2017) 031, [ 1609.07078]

2017 arXiv
[25]

Broedel, N

J. Broedel, N. Matthes and O. Schlotterer, Relations between elliptic multiple zeta values and a special derivation algebra , J. Phys. A49 (2016) 155203, [ 1507.02254]

2016 arXiv
[26]

Broedel and O

J. Broedel and O. Schlotterer, One-Loop String Scattering Amplitudes as Iterated Eisenstein Integrals, in Proceedings, KMPB Conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory: Zeuthen, Germany, October 23-26, 2017 , pp. 133–159, 2019. DOI

2017
[27]

Matthes, On the algebraic structure of iterated integrals of quasimodular forms , Algebra & Number Theory 11-9 (2017) 2113–2130, [ 1708.04561]

N. Matthes, On the algebraic structure of iterated integrals of quasimodular forms , Algebra & Number Theory 11-9 (2017) 2113–2130, [ 1708.04561]

2017 arXiv
[28]

F. C. S. Brown, On the decomposition of motivic multiple zeta values , in Galois-Teichm¨ uller theory and arithmetic geometry , vol. 63 of Adv. Stud. Pure Math. , pp. 31–58. Math. Soc. Japan, Tokyo, 2012. 1102.1310

2012 arXiv
[29]

Brown, Mixed Tate motives over Z, Ann

F. Brown, Mixed Tate motives over Z, Ann. Math. 175 (2012) 949–976

2012
[30]

J. M. Drummond and E. Ragoucy, Superstring amplitudes and the associator , JHEP 08 (2013) 135, [ 1301.0794]

2013 arXiv
[31]

A. B. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry , Duke Math. J. 128 (2005) 209–284, [ math/0208144]

2005 arXiv
[32]

Brown, Motivic periods and the projective line minus three points , 1407.5165

F. Brown, Motivic periods and the projective line minus three points , 1407.5165

arXiv
[33]

Brown, Notes on motivic periods , Communications in Number Theory and Physics 11 (2015) 557–655, [ 1512.06410]

F. Brown, Notes on motivic periods , Communications in Number Theory and Physics 11 (2015) 557–655, [ 1512.06410]

2015 arXiv
[34]

Broedel, C

J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series , JHEP 08 (2018) 014, [1803.10256]

2018 arXiv
[35]

Mizera, Aspects of Scattering Amplitudes and Moduli Space Localization

S. Mizera, Aspects of Scattering Amplitudes and Moduli Space Localization . PhD thesis, Perimeter Inst. Theor. Phys., 2019. 1906.02099

2019 arXiv
[36]

Calaque, B

D. Calaque, B. Enriquez and P. Etingof, Universal KZB equations: the elliptic case , in Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I , vol. 269 of Progr. Math., pp. 165–266. Birkh¨ auser Boston, Inc., Boston, MA, 2009

2009
[37]

Enriquez, Elliptic associators, Selecta Math

B. Enriquez, Elliptic associators, Selecta Math. (N.S.) 20 (2014) 491–584

2014
[38]

Hain, Notes on the universal elliptic KZB equation , 1309.0580

R. Hain, Notes on the universal elliptic KZB equation , 1309.0580. – 75 –

arXiv
[39]

Tsunogai, On some derivations of Lie algebras related to Galois representations , Publ

H. Tsunogai, On some derivations of Lie algebras related to Galois representations , Publ. Res. Inst. Math. Sci. 31 (1995) 113–134

1995
[40]

Aomoto, Gauss-Manin connection of integral of diﬀerence products , J

K. Aomoto, Gauss-Manin connection of integral of diﬀerence products , J. Math. Soc. Japan 39 (1987) 191–208

1987
[41]

C. R. Mafra, O. Schlotterer and S. Stieberger, Complete N-Point Superstring Disk Amplitude I. Pure Spinor Computation , Nucl. Phys. B873 (2013) 419–460, [ 1106.2645]

2013 arXiv
[42]

Broedel, O

J. Broedel, O. Schlotterer and S. Stieberger, Polylogarithms, Multiple Zeta Values and Superstring Amplitudes, Fortsch. Phys. 61 (2013) 812–870, [ 1304.7267]

2013 arXiv
[43]

Huang, O

Y.-t. Huang, O. Schlotterer and C. Wen, Universality in string interactions , JHEP 09 (2016) 155, [1602.01674]

2016 arXiv
[44]

Azevedo, M

T. Azevedo, M. Chiodaroli, H. Johansson and O. Schlotterer, Heterotic and bosonic string amplitudes via ﬁeld theory , JHEP 10 (2018) 012, [ 1803.05452]

2018 arXiv
[45]

A. V. Kotikov, Diﬀerential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B254 (1991) 158–164

1991
[46]

Arkani-Hamed, J

N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo and J. Trnka, Local Integrals for Planar Scattering Amplitudes, JHEP 06 (2012) 125, [ 1012.6032]

2012 arXiv
[47]

J. M. Henn, Multiloop integrals in dimensional regularization made simple , Phys. Rev. Lett. 110 (2013) 251601, [ 1304.1806]

2013 arXiv
[48]

Adams and S

L. Adams and S. Weinzierl, The ε-form of the diﬀerential equations for Feynman integrals in the elliptic case , Phys. Lett. B781 (2018) 270–278, [ 1802.05020]

2018 arXiv
[49]

Broedel, C

J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Elliptic Feynman integrals and pure functions, JHEP 01 (2019) 023, [ 1809.10698]

2019 arXiv
[50]

J. E. Gerken, A. Kleinschmidt and O. Schlotterer, Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings , JHEP 01 (2019) 052, [ 1811.02548]

2019 arXiv
[51]

D’Hoker and M

E. D’Hoker and M. B. Green, Exploring transcendentality in superstring amplitudes , JHEP 07 (2019) 149, [ 1906.01652]

2019 arXiv
[52]

M. B. Green and J. H. Schwarz, Inﬁnity Cancellations in SO(32) Superstring Theory , Phys. Lett. 151B (1985) 21–25

1985
[53]

Tsuchiya, More on One Loop Massless Amplitudes of Superstring Theories , Phys

A. Tsuchiya, More on One Loop Massless Amplitudes of Superstring Theories , Phys. Rev. D39 (1989) 1626

1989
[54]

Bianchi and D

M. Bianchi and D. Consoli, Simplifying one-loop amplitudes in superstring theory , JHEP 01 (2016) 043, [ 1508.00421]

2016 arXiv
[55]

M. Berg, I. Buchberger and O. Schlotterer, From maximal to minimal supersymmetry in string loop amplitudes, JHEP 04 (2017) 163, [ 1603.05262]

2017 arXiv
[56]

Dolan and P

L. Dolan and P. Goddard, Current Algebra on the Torus , Commun. Math. Phys. 285 (2009) 219–264, [0710.3743]

2009 arXiv
[57]

Mizera, Combinatorics and Topology of Kawai-Lewellen-Tye Relations , JHEP 08 (2017) 097, [1706.08527]

S. Mizera, Combinatorics and Topology of Kawai-Lewellen-Tye Relations , JHEP 08 (2017) 097, [1706.08527]. – 76 –

2017 arXiv
[58]

Luque, J.-C

J.-G. Luque, J.-C. Novelli and J.-Y. Thibon, Period polynomials and Ihara brackets , J. Lie Theory 17 (2007) 229–239, [ math/0606301]

2007 arXiv
[59]

Relations between derivations arising from modular forms

A. Pollack, “Relations between derivations arising from modular forms.” 2009

2009
[60]

Bloch and P

S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph , J. Number Theor. 148 (2015) 328–364, [ 1309.5865]

2015 arXiv
[61]

Bloch, M

S. Bloch, M. Kerr and P. Vanhove, A Feynman integral via higher normal functions , Compos. Math. 151 (2015) 2329–2375, [ 1406.2664]

2015 arXiv
[62]

Adams and S

L. Adams and S. Weinzierl, Feynman integrals and iterated integrals of modular forms , Commun. Num. Theor. Phys. 12 (2018) 193–251, [ 1704.08895]

2018 arXiv
[63]

Ablinger, J

J. Ablinger, J. Bl¨ umlein, A. De Freitas, M. van Hoeij, E. Imamoglu, C. G. Raab et al., Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams , J. Math. Phys. 59 (2018) 062305, [1706.01299]

2018 arXiv
[64]

Remiddi and L

E. Remiddi and L. Tancredi, An Elliptic Generalization of Multiple Polylogarithms , Nucl. Phys. B925 (2017) 212–251, [ 1709.03622]

2017 arXiv
[65]

J. L. Bourjaily, A. J. McLeod, M. Spradlin, M. von Hippel and M. Wilhelm, Elliptic Double-Box Integrals: Massless Scattering Amplitudes beyond Polylogarithms , Phys. Rev. Lett. 120 (2018) 121603, [ 1712.02785]

2018 arXiv
[66]

Broedel, C

J. Broedel, C. Duhr, F. Dulat and L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism , JHEP 05 (2018) 093, [ 1712.07089]

2018 arXiv
[67]

Broedel, C

J. Broedel, C. Duhr, F. Dulat and L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral , Phys. Rev. D97 (2018) 116009, [1712.07095]

2018 arXiv
[68]

Adams, E

L. Adams, E. Chaubey and S. Weinzierl, Planar Double Box Integral for Top Pair Production with a Closed Top Loop to all orders in the Dimensional Regularization Parameter , Phys. Rev. Lett. 121 (2018) 142001, [ 1804.11144]

2018 arXiv
[69]

Adams, E

L. Adams, E. Chaubey and S. Weinzierl, Analytic results for the planar double box integral relevant to top-pair production with a closed top loop , JHEP 10 (2018) 206, [ 1806.04981]

2018 arXiv
[70]

Broedel, C

J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, From modular forms to diﬀerential equations for Feynman integrals, in KMPB Conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory Zeuthen, Germany, October 23-26, 2017 , 2018. 1807.00842

2017 arXiv
[71]

Bl¨ umlein, A

J. Bl¨ umlein, A. De Freitas, M. Van Hoeij, E. Imamoglu, P. Marquard and C. Schneider, The ρ parameter at three loops and elliptic integrals , PoS LL2018 (2018) 017, [ 1807.05287]

2018 arXiv
[72]

Broedel, C

J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Elliptic polylogarithms and Feynman parameter integrals, JHEP 05 (2019) 120, [ 1902.09971]

2019 arXiv
[73]

Bogner, S

C. Bogner, S. M¨ uller-Stach and S. Weinzierl,The unequal mass sunrise integral expressed through iterated integrals onM1,3, 1907.01251

1907 arXiv
[74]

Broedel, C

J. Broedel, C. Duhr, F. Dulat, R. Marzucca, B. Penante and L. Tancredi, An analytic solution for the equal-mass banana graph , JHEP 09 (2019) 112, [ 1907.03787]. – 77 –

2019 arXiv
[75]

J. A. Minahan, One Loop Amplitudes on Orbifolds and the Renormalization of Coupling Constants, Nucl. Phys. B298 (1988) 36–74

1988
[76]

M. Berg, I. Buchberger and O. Schlotterer, String-motivated one-loop amplitudes in gauge theories with half-maximal supersymmetry , JHEP 07 (2017) 138, [ 1611.03459]

2017 arXiv
[77]

S. He, O. Schlotterer and Y. Zhang, New BCJ representations for one-loop amplitudes in gauge theories and gravity , Nucl. Phys. B930 (2018) 328–383, [ 1706.00640]

2018 arXiv
[78]

Kronecker, Zur Theorie der elliptischen Funktionen , Mathematische Werke IV (1881) 313–318

L. Kronecker, Zur Theorie der elliptischen Funktionen , Mathematische Werke IV (1881) 313–318
[79]

Brown and A

F. Brown and A. Levin, Multiple elliptic polylogarithms , 1110.6917

arXiv
[80]

Mumford, M

D. Mumford, M. Nori and P. Norman, Tata Lectures on Theta I, II . No. Bd. 2. Birkh¨ auser, 1983, 1984

1983
[81]

Broedel, O

J. Broedel, O. Schlotterer and F. Zerbini, From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop , JHEP 01 (2019) 155, [ 1803.00527]

2019 arXiv
[82]

Zerbini, Elliptic multiple zeta values, modular graph functions and genus 1 superstring scattering amplitudes

F. Zerbini, Elliptic multiple zeta values, modular graph functions and genus 1 superstring scattering amplitudes. PhD thesis, Bonn U., 2017. 1804.07989

2017 arXiv
[83]

F. Zerbini, Modular and holomorphic graph function from superstring amplitudes , in KMPB Conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory Zeuthen, Germany, October 23-26, 2017 , 2018. 1807.04506

2017 arXiv
[84]

Brown, Multiple modular values and the relative completion of the fundamental group of m1,1, 1407.5167

F. Brown, Multiple modular values and the relative completion of the fundamental group of m1,1, 1407.5167

arXiv
[85]

Broedel, N

J. Broedel, N. Matthes and O. Schlotterer. https://tools.aei.mpg.de/emzv
[86]

Matthes, The meta-abelian elliptic KZB associator and periods of Eisenstein series , N

N. Matthes, The meta-abelian elliptic KZB associator and periods of Eisenstein series , N. Sel. Math. New Ser. 24 (2018) 3217–3239, [ 1608.00740]

2018 arXiv
[87]

Kawai, D

H. Kawai, D. C. Lewellen and S. H. H. Tye, A Relation Between Tree Amplitudes of Closed and Open Strings , Nucl. Phys. B269 (1986) 1–23

1986
[88]

N. E. J. Bjerrum-Bohr, P. H. Damgaard and P. Vanhove, Minimal Basis for Gauge Theory Amplitudes, Phys. Rev. Lett. 103 (2009) 161602, [ 0907.1425]

2009 arXiv
[89]

Stieberger, Open & Closed vs

S. Stieberger, Open & Closed vs. Pure Open String Disk Amplitudes , 0907.2211

arXiv
[90]

Cho and K

K. Cho and K. Matsumoto, Intersection theory for twisted cohomologies and twisted Riemann’s period relations I, Nagoya Math. J. 139 (1995) 67–86

1995
[91]

M. B. Green, J. G. Russo and P. Vanhove, Low energy expansion of the four-particle genus-one amplitude in type II superstring theory , JHEP 02 (2008) 020, [ 0801.0322]

2008 arXiv
[92]

D’Hoker, M

E. D’Hoker, M. B. Green, ¨O. G¨ urdogan and P. Vanhove,Modular graph functions, Commun. Num. Theor. Phys. 11 (2017) 165–218, [ 1512.06779]

2017 arXiv
[93]

Zerbini, Single-valued multiple zeta values in genus 1 superstring amplitudes , Commun

F. Zerbini, Single-valued multiple zeta values in genus 1 superstring amplitudes , Commun. Num. Theor. Phys. 10 (2016) 703–737, [ 1512.05689]

2016 arXiv
[94]

Zagier and F

D. Zagier and F. Zerbini, Genus-zero and genus-one string amplitudes and special multiple zeta values, 1906.12339. – 78 –

1906 arXiv
[95]

D’Hoker and M

E. D’Hoker and M. B. Green, Absence of irreducible multiple zeta-values in melon modular graph functions, 1904.06603

1904 arXiv
[96]

Schocker, Lie elements and Knuth relations , Can.J.Math 56 (2004) 871, [ math/0209327]

M. Schocker, Lie elements and Knuth relations , Can.J.Math 56 (2004) 871, [ math/0209327]

2004 arXiv
[97]

C. R. Mafra and O. Schlotterer, Multiparticle SYM equations of motion and pure spinor BRST blocks, JHEP 07 (2014) 153, [ 1404.4986]

2014 arXiv
[98]

C. R. Mafra and O. Schlotterer, Cohomology foundations of one-loop amplitudes in pure spinor superspace, 1408.3605

arXiv
[99]

J. J. M. Carrasco, C. R. Mafra and O. Schlotterer, Semi-abelian Z-theory: NLSM+φ3 from the open string, JHEP 08 (2017) 135, [ 1612.06446]

2017 arXiv
[100]

Kleiss and H

R. Kleiss and H. Kuijf, Multi - Gluon Cross-sections and Five Jet Production at Hadron Colliders, Nucl. Phys. B312 (1989) 616–644

1989
[101]

Z. Bern, J. J. M. Carrasco and H. Johansson, New Relations for Gauge-Theory Amplitudes , Phys. Rev. D78 (2008) 085011, [ 0805.3993]

2008 arXiv
[102]

M. B. Green, J. H. Schwarz and L. Brink, N=4 Yang-Mills and N=8 Supergravity as Limits of String Theories, Nucl. Phys. B198 (1982) 474–492

1982
[103]

M. B. Green and J. H. Schwarz, Supersymmetrical Dual String Theory. 3. Loops and Renormalization, Nucl. Phys. B198 (1982) 441–460

1982
[104]

Hohenegger and S

S. Hohenegger and S. Stieberger, Monodromy Relations in Higher-Loop String Amplitudes , Nucl. Phys. B925 (2017) 63–134, [ 1702.04963]

2017 arXiv
[105]

Z. Bern, L. J. Dixon, M. Perelstein and J. S. Rozowsky, Multileg one loop gravity amplitudes from gauge theory, Nucl. Phys. B546 (1999) 423–479, [ hep-th/9811140]

1999 arXiv
[106]

N. E. J. Bjerrum-Bohr, P. H. Damgaard, T. Sondergaard and P. Vanhove, The Momentum Kernel of Gauge and Gravity Theories , JHEP 01 (2011) 001, [ 1010.3933]

2011 arXiv
[107]

Abreu, R

S. Abreu, R. Britto, C. Duhr and E. Gardi, Algebraic Structure of Cut Feynman Integrals and the Diagrammatic Coaction, Phys. Rev. Lett. 119 (2017) 051601, [ 1703.05064]

2017 arXiv
[108]

Abreu, R

S. Abreu, R. Britto, C. Duhr and E. Gardi, Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case , JHEP 12 (2017) 090, [ 1704.07931]

2017 arXiv
[109]

Abreu, R

S. Abreu, R. Britto, C. Duhr and E. Gardi, The diagrammatic coaction and the algebraic structure of cut Feynman integrals , PoS RADCOR2017 (2018) 002, [ 1803.05894]

2018 arXiv
[110]

Abreu, R

S. Abreu, R. Britto, C. Duhr, E. Gardi and J. Matthew, Coaction for Feynman integrals and diagrams, PoS LL2018 (2018) 047, [ 1808.00069]

2018 arXiv
[111]

Brown and C

F. Brown and C. Dupont, Lauricella hypergeometric functions, unipotent fundamental groups of the punctured Riemann sphere, and their motivic coactions , 1907.06603

1907 arXiv
[112]

Bl¨ umlein, D

J. Bl¨ umlein, D. J. Broadhurst and J. A. M. Vermaseren, The Multiple Zeta Value Data Mine , Comput. Phys. Commun. 181 (2010) 582–625, [ 0907.2557]

2010 arXiv
[113]

Multiple zeta values, from numbers to motives

J. I. Burgos Gil and F. J., “Multiple zeta values, from numbers to motives.” http://javier.fresan.perso.math.cnrs.fr/mzv.pdf
[114]

J. J. M. Carrasco, C. R. Mafra and O. Schlotterer, Abelian Z-theory: NLSM amplitudes and α’-corrections from the open string , JHEP 06 (2017) 093, [ 1608.02569]. – 79 –

2017 arXiv
[115]

J. E. Gerken, A. Kleinschmidt and O. Schlotterer, All-order diﬀerential equations for one-loop closed-string integrals and modular graph forms , 1911.03476

1911 arXiv
[116]

D’Hoker, M

E. D’Hoker, M. B. Green and P. Vanhove, On the modular structure of the genus-one Type II superstring low energy expansion , JHEP 1508 (2015) 041, [ 1502.06698]

2015 arXiv
[117]

D’Hoker and M

E. D’Hoker and M. B. Green, Identities between Modular Graph Forms, J. Number Theor. 189 (2018) 25–88, [ 1603.00839]

2018 arXiv
[118]

D’Hoker and J

E. D’Hoker and J. Kaidi, Hierarchy of Modular Graph Identities , JHEP 11 (2016) 051, [1608.04393]

2016 arXiv
[119]

Kleinschmidt and V

A. Kleinschmidt and V. Verschinin, Tetrahedral modular graph functions, JHEP 09 (2017) 155, [1706.01889]

2017 arXiv
[120]

Basu, Eigenvalue equation for the modular graph Ca,b,c,d, JHEP 07 (2019) 126, [1906.02674]

A. Basu, Eigenvalue equation for the modular graph Ca,b,c,d, JHEP 07 (2019) 126, [1906.02674]

2019 arXiv
[121]

Brown, A class of non-holomorphic modular forms I , Res

F. Brown, A class of non-holomorphic modular forms I , Res. Math. Sci. 5 (2018) 5:7, [1707.01230]

2018 arXiv
[122]

Brown, A class of non-holomorphic modular forms II : equivariant iterated Eisenstein integrals, 1708.03354

F. Brown, A class of non-holomorphic modular forms II : equivariant iterated Eisenstein integrals, 1708.03354

arXiv
[123]

Tourkine and P

P. Tourkine and P. Vanhove, Higher-loop amplitude monodromy relations in string and gauge theory, Phys. Rev. Lett. 117 (2016) 211601, [ 1608.01665]

2016 arXiv
[124]

D’Hoker, Integral of two-loop modular graph functions , JHEP 06 (2019) 092, [ 1905.06217]

E. D’Hoker, Integral of two-loop modular graph functions , JHEP 06 (2019) 092, [ 1905.06217]

2019 arXiv
[125]

D’Hoker, M

E. D’Hoker, M. B. Green and B. Pioline, Higher genus modular graph functions, string invariants, and their exact asymptotics , Commun. Math. Phys. 366 (2019) 927–979, [1712.06135]

2019 arXiv
[126]

D’Hoker, M

E. D’Hoker, M. B. Green and B. Pioline, Asymptotics of the D8R4 genus-two string invariant , Commun. Num. Theor. Phys. 13 (2019) 351–462, [ 1806.02691]. – 80 –

2019 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

C. R. Mafra and O. Schlotterer, All-order α′-expansion of one-loop open-string integrals , 1908.09848

arXiv 1908

[2] [2]

Terasoma, Selberg integrals and multiple zeta values , Compositio Mathematica 133 (2002) 1

T. Terasoma, Selberg integrals and multiple zeta values , Compositio Mathematica 133 (2002) 1

2002

[3] [3]

Brown, Multiple zeta values and periods of moduli spaces M0,n(R), Annales Sci

F. Brown, Multiple zeta values and periods of moduli spaces M0,n(R), Annales Sci. Ecole Norm. Sup. 42 (2009) 371, [ math/0606419]

arXiv 2009

[4] [4]

Schlotterer and S

O. Schlotterer and S. Stieberger, Motivic Multiple Zeta Values and Superstring Amplitudes , J. Phys. A46 (2013) 475401, [ 1205.1516]

arXiv 2013

[5] [5]

Broedel, O

J. Broedel, O. Schlotterer, S. Stieberger and T. Terasoma, All order α′-expansion of superstring trees from the Drinfeld associator , Phys. Rev. D89 (2014) 066014, [ 1304.7304]

arXiv 2014

[6] [6]

Schnetz, Graphical functions and single-valued multiple polylogarithms , Commun

O. Schnetz, Graphical functions and single-valued multiple polylogarithms , Commun. Num. Theor. Phys. 08 (2014) 589–675, [ 1302.6445]

arXiv 2014

[7] [7]

Brown, Single-valued Motivic Periods and Multiple Zeta Values , SIGMA 2 (2014) e25, [1309.5309]

F. Brown, Single-valued Motivic Periods and Multiple Zeta Values , SIGMA 2 (2014) e25, [1309.5309]

arXiv 2014

[8] [8]

Stieberger, Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator, J

S. Stieberger, Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator, J. Phys. A47 (2014) 155401, [ 1310.3259]

arXiv 2014