Published for SISSA by

Springer

Received: January 9, 2024
Revised: May 26, 2024
Accepted: June 5, 2024
Published: July 4, 2024

A double copy from twisted (co)homology at genus one,a Andrzej Pokraka,a Lecheng Ren

a and Carlos RodriguezbaDepartment of Physics, Brown University,
Providence, RI 02912, U.S.A.
b
Department of Physics and Astronomy, Uppsala University,
Box 516, 75120 Uppsala, Sweden

E-mail: rishabh_bhardwaj@brown.edu, andrzej_pokraka@brown.edu,
lecheng_ren@brown.edu, carlos.rodriguez@physics.uu.se
Abstract: We study the twisted (co)homology of a family of genus-one integrals — the socalled Riemann-Wirtinger integrals. These integrals are closely related to one-loop string
amplitudes in chiral splitting where one leaves the loop-momentum, modulus and all but one
puncture un-integrated. While not actual one-loop string integrals, they share many properties
and are simple enough that the associated twisted (co)homologies have been completely
characterized [1]. Using intersection numbers — an inner product on the vector space ofallowed differential forms — we derive the Gauss-Manin connection for two bases of thetwisted cohomology providing an independent check of [2]. We also use the intersection index— an inner product on the vector space of allowed contours — to derive a double-copy formulafor the closed-string analogues of Riemann-Wirtinger integrals (one-dimensional integrals over
the torus). Similar to the celebrated KLT formula between open- and closed-string tree-level
amplitudes, these intersection indices form a genus-one KLT-like kernel defining bilinears inmeromorphic Riemann-Wirtinger integrals that are equal to their complex counterparts.
Keywords: Differential and Algebraic Geometry, Gauge-Gravity Correspondence,
Scattering Amplitudes
ArXiv ePrint: 2312.02148

Open Access, © The Authors.
Article funded by SCOAP3 .

https://doi.org/10.1007/JHEP07(2024)040

JHEP07(2024)040

Rishabh BhardwajContents
2

2 Fundamentals of twisted (co)homology and the double copy
2.1 A simple example
2.2 Lightning review of twisted (co)homology
2.3 Pairings
2.4 Quadratic identities and the double copy

5
5
6
10
12

3 One-loop string integrals and the Riemann-Wirtinger integral
3.1 One-loop string integrals
3.2 The Riemann-Wirtinger integral

13
13
16

4 Twisted cohomology of the Riemann-Wirtinger integral
4.1 Basis of cohomology
4.2 Intersection numbers: an inner product on cohomology

17
18
19

5 Twisted homology of the Riemann-Wirtinger integral
5.1 Basis of homology
5.2 The intersection index: an inner product on homology

22
22
24

6 The double copy of Riemann-Wirtinger integrals
6.1 Complex Riemann-Wirtinger integrals
6.2 The double copy for real Mandelstam variables
6.3 The double copy for complex Mandelstam variables
6.4 The double copy and modularity

28
28
30
33
35

7 Towards integrating two punctures

38

8 Future directions and discussion

42

A Local systems

44

B Computing intersection indices

44

C Solving for s1A

48

D The modular double copy: subleading terms

49

E Towards the analytic Riemann-Wirtinger integral in terms of eMPLs
E.1 Differential equations for the φ-basis
E.2 The O(α′ ) solution to the differential equations
E.3 Boundary values
E.4 The η → 0 limit

50
51
53
56
61

–1–

JHEP07(2024)040

1 Introduction

1

Introduction

The first instance of a double copy in the physics literature was discovered by KawaiLewellen-Tye (KLT) [3]. These authors factorized tree-level closed string scattering amplitudes
into bilinears of tree-level open string amplitudes, their complex conjugates and the so-called
KLT kernel. In parallel, similar relations appeared in the mathematics literature, more
specifically in Aomoto’s work on complex Selberg integrals [4]. In the decades since theseminal work [3], there has been a search for the analogous statement at genus-one andhigher [5–14].
While its origin may be in string theory, the double copy is perhaps best understoodin the field theory limit where there are several formulations [15]. In the field theory limit,
the matrix elements of the KLT kernel can be understood as color ordered amplitudes ofbi-adjoint scalar theory [16] or generalizations thereof [17–21].
In the scattering equation approach by Cachazo, He and Yuan (CHY) [16, 22, 23], with aloop-level extension in [24–29], n-point color-ordered amplitudes at L loops are represented byan integral over the (n + 2L)-punctured Riemann sphere. Here, the integrand is a numerator
specific to the scattering process divided by a product of n differences between the punctures.
In particular, the integrand only ever has simple poles when two punctures collide. Moreover,this integral representation of the amplitude localizes onto the locus of the scattering equations
turning the integral into a discrete sum. To obtain gravity amplitudes in this formalism,
one simply replaces the denominator of a color-ordered CHY integrand with the numerator
factor corresponding to another color-ordered CHY integrand.
Additionally, the color-kinematics duality pioneered by Bern, Carrasco, and Johansson(BCJ) [15, 30–32] has been tested and used at loop-level in a wide range of theories and up
to four-loops in N ≥ 4 supergravities [33–35]. Schematically, the numerators of tree-level
amplitudes or loop-level integrands are organized into the product of a color factor and akinematic factor such that both the color and kinematic factors obey the Jacobi identity
satisfied by the color algebra. Then, the corresponding gravity amplitude/integrand is
obtained by replacing all color factors in the color-ordered amplitude/integrand with anothercopy of the kinematic factors, which can even belong to a different theory.
In the CHY and string theory formulation of the double copy one inevitably encounters
integrals of the generalized Euler-Mellin type. Twisted (co)homology provides a generalmathematical framework for studying such integrals. In particular, it provides the necessary
machinery to construct the double copy for any family of Euler-Mellin integrals [36, 37]. In
this language, the double copy of gravitational CHY and tree-level closed string amplitudes–2–

JHEP07(2024)040

The double copy is a framework for computing closed-string/gravitational amplitudes using
simpler open-string/gauge theory amplitudes as input. This framework has revealed importantmathematical structures in field theory and string theory as well as providing an efficientmethod for computing scattering amplitudes that would otherwise be intractable. In particular,
the double copy is a bilinear map on open-string/gauge theory amplitudes that makes theslogan gravity = (gauge theory)2 precise. This is extremely surprising and useful because
gravitational amplitudes are orders of magnitude harder to compute than their gauge theory
analogues.

Double copy teaser. The simplest example of this double copy or genus-one RW-KLT
formula is for n = 3 punctures where the twisted (co)homology is 2-dimensional. Explicitly,Z

d2 z1 e2πis1A (z1 −z̄1 )Mϑ1 (z12 |τ ) 2s12
F (z1a , η|τ ) F (z1b , η|τ )ϑ1 (z13 |τ )

(1.1)
!


i sin πs12 0e−iπ(s1A −s12 )[γ2A |φa ⟩·=[γ2A |φb ⟩ [γ23 |φb ⟩ ·
iπ(s
−s
)
12
1A
2 sin πs1A−e2i sin π(s1A − s12 )[γ23 |φa ⟩

!

where
[γ|φa ⟩ =Z

dz1 e2πis1A z1γϑ1 (z12 |τ )
ϑ1 (z13 |τ )s12

F (z1a , η|τ ) ,(1.2)

a, b ∈ {2, 3} and the integration contours are γ2A = {z1 ∈ [z2 , z2 +1]} and γ23 = {z1 ∈ [z2 , z3 ]}.
We also set zij = zi − zj and often fix z2 as the origin: z2 = 0. In the complex integralof (1.1), M is the fundamental parallelogram with the punctures removed, and the measure isRchosen such that P d2 z1 = 1. Here, ϑ1 (z|τ ) is the odd Jacobi-theta function, F (z, η|τ ) is the–3–

JHEP07(2024)040

follows from a generalization of the well-known twisted Riemann bilinear relations [38](reviewed in section 2).
The integrand of a one-loop string amplitude also takes the form of a generalized EulerMellin integral on the torus. Thus, it is tempting to try using twisted (co)homology to build
the corresponding double copy. In fact, this line of research was initiated in [9] and [10].
Using twisted (co)homology, the authors derive the monodromy relations of [5, 8] for string
integrals at genus-one and higher. While the authors of [9] comment on the prospect ofderiving the one-loop KLT kernel, the presence of unexpected, unphysical cycles in the basisof homology and B-cycle monodromies that depend on the other punctures complicate theanalysis. The genus-one KLT formulae of [13, 14] derived from contour deformations callfor an understanding in terms of twisted (co)homology.
Generalized Euler-Mellin integrals also make a prominent appearance in the computationof Feynman integrals/field theory amplitudes. In particular, twisted (co)homology has foundinteresting applications to the study of CHY amplitudes [39–42], cosmological correlators [43]and Feynman integrals (in particular the coaction [44, 45] and integral reduction [46–56]).
Moreover, the study of elliptic Feynman integrals has boomed over the last decade (see thereview [57] and [58–62] for some recent works). Since both the α′ -expansion of one-loop
string integrands (pre τ and loop-momentum integration) and elliptic Feynman integrals
evaluate to the same class of functions — elliptic multiple polylogarithms (eMPLs) [63, 64]— it is interesting to explore one-loop string integrands and their connection to field theory
objects using tools from twisted (co)homology.
Motivated by the possibility of a KLT kernel for one-loop string integrands from twisted(co)homology and aiming for a better understanding the function space of genus-one integrals,
we study the family of so-called Riemann-Wirtinger integrals. This family of integrals is
closely related to one-loop string integrands and serves as a toy model for their twisted(co)homology, which has been fully characterized [1, 2, 65, 66]. This allows us to present an
explicit double copy formula for Riemann-Wirtinger (RW) integrals with a (RW-) KLT kernelbuilt out of intersection indices an inner product on the space of homology!

Kronecker-Eisenstein series and {s12 , s1A } are real numbers to be thought of as Mandelstam
invariants and η ∈ C. The genus-one RW-KLT formula (1.1) is contingent on the combination1−η + s1A τ − s12 z3 = s1B .

(1.3)

Outline of paper. In section 2, we review the formalism of twisted (co)homology andthe associated construction of a double copy. Then, in section 3, we review one-loop string
integrands and introduce the family of Riemann-Wirtinger integrals. The twisted cohomology
of Riemann-Wirtinger integrals is reviewed in section 4. In particular, we provide a basisof differential forms compatible with the twisted (co)homology double copy in section 5.1.
While not needed for the practical construction of the twisted (co)homology double copy, awell-defined intersection number — an inner product on cohomology — is essential for the
existence of a double copy from twisted (co)homology. Therefore, we demonstrate how theintersection number is computed at genus-one in section 5.2 but relegate most of the details
and results to appendix E to not distract from the flow of the RW double copy narrative. We
review the twisted homology of Riemann-Wirtinger integrals in section 5 and the intersection
indices — an inner product on homology — for a basis of homology in section 5.2. These
intersection indices make up the double copy kernel. The double copy of Riemann-Wirtinger
integrals is constructed in section 6 and verified numerically for both real (section 6.2) andcomplex (section 6.3) Mandelstams. In 6.4 we write a version of the double copy of RiemannWirtinger integrals that has nice modular properties, which looks like a closed-integral after
integrating out the loop momentum. Currently, it is not known how to define RiemannWirtinger integrals where two (or more) punctures are integrated. Since understanding this is
an important step towards making the twisted (co)homology double copy applicable to actual
string integrands, we provide a discussion of the current problems preventing the definition ofmulti-puncture Riemann-Wirtinger integrals and conjecture how one might overcome these
problems in section 7. We conclude in section 8 with a discussion and future directions.
For the interested reader, we have collected additional novel results on Riemann-Wirtinger
integrals that would benefit any future work aimed at an analytic double copy in appendix E.
In appendix E.1, we verify the Gauss-Manin connection satisfied by the RW family of
1Mathematically, this condition ensures that the integrand in the l.h.s. of (1.1) is be doubly periodic in
z1 . We give a physical interpretation to this condition in connection to the integration of loop-momentum instring theory around (6.6).

–4–

JHEP07(2024)040

conspiring to a real constant s1B (we also generalize to complex Mandelstams s• ∈ C). To
our knowledge, this is the first genus-one KLT-like formula where the Mandelstam variables
are generic and the modulus can have arbitrary real part: Re(τ ) ̸= 0. The closest examples
to this formula in the literature are a pair of double-copy formulas exhibited in [14] for
2-point, 1-loop string amplitudes with integer Mandelstam variable s12 and arbitrary realpart of the modulus τ .
A similar double copy formula will be given for the generalization of the complex
z1 -integral (1.1) with additional unintegrated punctures z4 , z5 , · · · , zn . At n-points, the
RW-KLT kernel is an (n − 1) × (n − 1) matrix whose entries are known rational functions of
the monodromies e2πis• . Such matrices have been constructed and studied before in [1, 66].

integrals that was derived through independent means [2]. While not directly used to verify
the numeric double copy in the main text, the RW-DEQs are essential for obtaining analytic
expressions for the RW integrals and their double copy in terms of eMPLs. In particular, we
obtain the leading order solution in the α′ expansion to the RW-DEQs in terms of eMPLs
in appendix E.2. We also provide boundary values for the RW-DEQs in appendix E.3.
Moreover, since only genus-zero intersection numbers have appeared in the physics literature,these genus-one intersection number calculations are interesting in their own right and havepotential application to the integral reduction of hyperelliptic Feynman integrals [58].

Fundamentals of twisted (co)homology and the double copy

In this section, we give an overview of twisted (co)homology in the context of a simple
example: the tree-level four-point open string amplitude (or the Euler beta function). We
slowly build up the twisted (co)homology formalism ending with the twisted Riemann bilinear
relations. Then, we describe how these relations can be generalized to produce the double
copy and express the tree-level four-point closed string amplitude as a square of the open
string amplitudes.
Historically, twisted (co)homology has been primarily developed to study properties
of hypergeometric functions; see the textbooks [67, 68] or [37, 38, 69–75, 75–81] for avery incomplete list of research papers. Fortunately for physicists, these authors are often
concerned with explicit calculations and many worked examples can be found in their writtings.
Interestingly, the twisted (co)homology version of the double copy was first discovered bymathematicians studying Selberg integrals that appear in conformal field theory correlation
functions [37, 38, 75, 75, 76]. More recently, the twisted (co)homology double copy wasproved from a motivic perspective [82, 83]. In particular, we want to test whether the double
copy prescription of [37, 38, 75, 75, 76, 82, 83] holds at higher genus and access the possible
application to one-loop closed string amplitudes.
2.1

A simple example

To understand the construction of twisted (co)homology and its relation to the double copy,
it is best to have a simple example in mind. To this end, consider the tree-level masslessfour-point open string integral (Euler beta function)A

open(s, t) ∝Z 1

dz z s−1 (1 − z)t−1

(2.1)

0

where we have used the SL(2, C) gauge symmetry to fix three of the punctures to 0, 1 and∞ and are not concerned with the overall constant kinematic factors and the momentum
conserving delta function. It satisfies interesting linear relations such as
Aopen (s + 1, t) =s
Aopen (s, t) ,s+t

Aopen (s, t + 1) =andt
Aopen (s, t) ,
s+t

(2.2)

as well as, the quadratic identity
Aopen (s, t)Aopen (−s, −t) = −π



1 1
+
s
t

–5–



1
1
+
.
tan(πs) tan(πt)


(2.3)

JHEP07(2024)040

2

The analogue of (2.2) in quantum field theory corresponds to the dimension shifting identities
and differential equations satisfied by dimensionally regulated Feynman integrals. While
Feynman integrals also satisfy quadratic identities like (2.2) they are not particularly wellstudied in the literature. The corresponding closed string amplitude is [18]A

closed(s, t) ∝Z|z|2s |1 − z|2tCdz ∧ dz̄−π(s + t)2 Γ(s)Γ(t)Γ(−s − t)=.
|z|2 |1 − z|2
Γ(1 − s)Γ(1 − t)Γ(1 + s + t)

(2.4)

Again, we have dropped the proportionality factors and gauge fixed three of the punctures.
Of course it is well known that the integral (2.4) admits a double copy(s, t) =11+tan(πs) tan(πt)

−1(Aopen (s, t))2 .

(2.5)

That is, gravity (closed string) amplitudes are the square of gauge (open string) amplitudes.
2.2

Lightning review of twisted (co)homology

Twisted (co)homology provides a systematic formalism for understanding relations suchas (2.2), (2.3), (2.5) and more! Roughly speaking, twisted (co)homology is the (co)homology
theory for integrals whose integrands have prescribed multiplicative monodromies. Moreexplicitly, the theory of twisted (co)homology applies to integrals of the form
Z

uφ

(2.6)

γ

where u is a universal multi-valued function called the twist, φ is a single-valued differential
form on the space M = E \ {u = 0, ∞} and γ is a contour on M . In this language,equation (2.1) becomes
Aopen (s, t) =

Z 1

u φ,

(2.7)

0

with
u = z s (1 − z)t ,φ = d log

z
1−z

and

M = CP \ {0, 1, ∞} .

(2.8)

Note that the monodromies of the twist in this example are e2πis and e2πit as z winds aroundthe origin and z = 1 (there is also the monodromy e−2πi(s+t) as z winds about the point∞). The so-called dual 2 local system,Ľ(s, t) = Cu,(2.9)

keeps track of the branch choices of u. For this reason, twisted (co)homology is also known
as (co)homology with coefficients in a local system.
Importantly, the integral (2.6) is not well defined until a branch choice for u is made.
Thus, twisted contours must store information associated to the topological integration
2

The adjective “dual” is a matter of convention and its significance will become clear once we haveintroduced the intersection pairings.

–6–

JHEP07(2024)040

A

closedcontour γ as well as a branch choice uγ on this contour. Explicitly, a twisted contour isthe tensor product γ ⊗ uγ where ⊗uγ is the aforementioned coefficient in the dual local
system. Returning to our example, we can now write the open-string four-point amplitude
as an integral over a twisted contour:
Aopen (s, t) =Z

φ.

(2.10)(0,1)⊗uZ∇ω φ =γ⊗uγZu γ ∇ω φ =
γ

Z

Z

Z

d(uγ φ) =γuγ φ =∂γ

φ.

(2.11)∂ω (γ⊗uγ )Here, we have introduced the covariant derivative ∇ω := d + ω∧ where ω := d log(u) is a flat
connection ω ∧ ω = 0 (this guarantees that the covariant derivative is nilpotent: ∇ω ◦ ∇ω = 0).
Since the total derivative of the multi-valued integrand d(u φ) is equivalent to the covariant
derivative of the single-valued differential form ∇ω φ, we can effectively “forget” about u andwork only with single-valued differential forms by replacing the normal exterior derivativewith the covariant derivative.
The boundary of a twisted contour is defined to be the topological boundary of thecontour with each boundary component tensored with the corresponding branch induced
from the original branch choice. This is most easily illustrated on a one-dimensional contourthat has been smoothly triangulated. Using the standard notation for m-simplicies, the(triangulated) path from point a to b is denoted by ⟨ab⟩ and has boundary ∂⟨ab⟩ = ⟨b⟩ − ⟨a⟩.
Then, the boundary of the corresponding twisted contour ⟨ab⟩ ⊗ u⟨ab⟩ is∂ω ⟨ab⟩ ⊗ u⟨ab⟩ (z) := ⟨b⟩ ⊗ u⟨b⟩ (b) − ⟨a⟩ ⊗ u⟨a⟩ (a) ,(2.12)

where the branch choices u⟨b⟩ and u⟨b⟩ are induced from u⟨ab⟩ by restriction. While thegeneralization to higher dimensional simplices is straightforward, it will not be needed in thiswork. The interested reader is encouraged to consult [67, 68] as well as [9, 10, 18, 76, 84–86].
In our running example, both the twisted contour and twisted differential form are closed(i.e., their respective image under ∂ω or ∇ω vanishes). The twisted contour in the definitionof the beta function is closed because u vanishes at the end points
h

i

h

i

∂ω (0, 1) ⊗ u(0,1) (z) = ∂ω ⟨01⟩ ⊗ u⟨01⟩ (z) = ⟨1⟩ ⊗ u⟨1⟩ (1) − ⟨0⟩ ⊗ u⟨0⟩ (0) = 0 .

(2.13)On the other hand, the twisted differential form in (2.10) is closed because it is closed in theusual sense (has no image under d) and has vanishing wedge product with ω:


∇ω

z
d log
1−z



= d2 logzz+ ω ∧ d log
= 0.
1−z
1−z

3

(2.14)

Technically, one should work with L-valued differential forms. However, since the local system is a trivial
line bundle, there exists a global trivialization and we can work with ordinary complex differential forms
without loss of information.

–7–

JHEP07(2024)040

While the differential form φ is an ordinary differential form on M , we will also call it atwisted differential form.3
Even in the twisted setting, there exists a version of Stokes’ theorem [67, 68]

Z(φ + ∇ω ψ) =

Z

Z

γ⊗uγ

γ⊗uγ

Z

∂ω (γ⊗uγ )

Z

Z

φ=
γ⊗uγ +∂ω (δ⊗uδ )

Z

ψ=

φ+

∇ω φ =

φ+
γ⊗uγ

δ⊗uδ

φ,

(2.15)

φ,(2.16)

γ⊗uγ

Z
γ⊗uγ

since ∂ω (γ ⊗ uγ ) = 0 = ∇ω φ by definition. To remove the above redundancy in defining
twisted (co)cycles, it is useful to define the twisted (co)homology and work with equivalenceclasses of (co)cycles.
The p-th twisted homology Hp group is the space of closed twisted p-contours modulo
exact twisted (p + 1)-contours
Hp (M, Ľω ) =







 ,

ker ∂ω : Cp (M, Ľω ) → Cp−1 (M, Ľω )im ∂ω : Cp+1 (M, Ľω ) → Cp (M, Ľω )(2.17)

where Cp (M, Ľω ) is the space of twisted p-contours. Its elements (twisted homology classes)
are equivalence classes of twisted cycles and are denoted by a square bra. Explicitly, given a
twisted p-cycle γ ⊗ uγ ∈ Cp (M, Ľω ), its twisted homology class is[γ ⊗ uγ | = {γ ⊗ uγ + ∂ω (β ⊗ uβ ) | ∀ (β ⊗ uβ ) ∈ Cp+1 (M, Ľω )} .

(2.18)

When two twisted p-cycles, γ ⊗ uγ and γ ′ ⊗ uγ ′ , differ by the boundary of a twisted (p + 1)contour β ⊗ uβ ,γ ⊗ uγ ≃ γ ′ ⊗ uγ ′ ⇐⇒ γ ⊗ uγ − γ ′ ⊗ uγ ′ = ∂ω (β ⊗ uβ ) ,(2.19)

we say that they are homologous or equivalent in homology.
Similarly, the p-th twisted cohomology H p group is the space of closed twisted p-forms
modulo exact twisted (p − 1)-forms
ker ∇ω : Ωp (M, ∇ω ) → Ωp+1 (M, ∇ω )H (M, ∇ω ) =,im (∇ω : Ωp−1 (M, ∇ω ) → Ωp (M, ∇ω ))

p(2.20)

where Ωp (M, ∇ω ) is the space of twisted p-forms. Its elements (twisted cohomology classes)
are equivalence classes of twisted cocycles and are denoted by an angle ket. Explicitly, given a
twisted p-cocycle φ ∈ ker(∇ω : Ωp (M, ∇ω ) → Ωp+1 (M, ∇ω )), its twisted cohomology class is|φ⟩ = {φ + ∇ω ψ | ∀ ψ ∈ Ωp−1 (M, ∇ω )} .

–8–

(2.21)JHEP07(2024)040

Closed twisted contours and differential forms are called twisted cycles and cocycles respectively. Intuitively, cycles are contours with no boundary. Unfortunately, cocycles do
not have a simple intuitive interpretation — they are simply differential forms that vanish
when (covariantly) differentiated.
While twisted cycles and cocycles are the “interesting” twisted contours/differential
forms on M , they are not uniquely defined. The integral of a twisted cocycle (φ) over a
twisted cycle (γ ⊗ uγ ) does not change when the twisted cycle (cocycle) is shifted by anexact twisted contour (differential form). Here, an exact twisted contour is the image of theboundary operator (∂ω (δ ⊗ uδ )) and an exact twisted differential form is the image of thecovariant derivative (∇ω ψ). Using Stokes theorem, it is easy to verify that integrals overtwisted (co)cycles are indeed invariant under such shifts

pHdR(M, ∇ω )•|•

Hcn−p (M, ∇∨ω)

Period

Po
in

car
é

[•|•

Hpc (M, L∨
ω,η )Intersection
[•|•]

•|•]

lf
Hn−p
(M, Lω,η )When two twisted p-cocycles, φ and φ′ , differ by the covariant derivative of a twisted(p − 1)-form ∇ω ψ,φ ≃ φ′ ⇐⇒ φ − φ′ = ∇ω ψ ,(2.22)

we say that φ and φ′ are cohomologous or equivalent in cohomology. In the physics literature,
shifting a differential form by a total derivative is commonly known as integration-by-parts
and is the standard method for reducing integrals that appear in the calculation of multiloop perturbative amplitudes to a minimal basis of so-called master integrals. Developing
fast integration-by-parts computer algorithms and researching alternative integral reduction
methods is currently an active area of research [87–91] since integral reduction is essential
to precision phenomenology. The intersection number, introduced below, is one of the mostpromising alternatives to traditional integration-by-parts algorithms being developed [46–56].
An important fact about twisted (co)homology is that only the middle degree(co)homology is non-trivial
Hp (M, Ľω ) = 0 = H p (M, ∇ω )ifp ̸= n ,(2.23)

where n is the complex dimension of M (n = dimC M ) or equivalently half the real dimensionof M (n = 12 dimR M ). As a consequence, the dimension of the middle (co)homology groups
is given by the Euler characteristic of M [67]4
dim Hn (M, Ľω ) = dim H n (M, ∇ω ) = |χ(M )| .

(2.24)

Another important fact is that twisted (co)homology groups are actually vector spaces(existence of an inner product structure)!

–9–

JHEP07(2024)040

Figure 1. Diagram detailing the connections between twisted (co)homology groups and their duals.
The period pairings (integration) are denoted by the vertical arrows, while the intersection pairings
are denoted by the horizontal arrows. Poincaré duality is indicated by the diagonal arrows. We havealso included some subscripts and superscripts for the experts and can be safely ignored by readers
unfamiliar with twisted (co)homology.

2.3

Pairings

Now that we have defined the twisted (co)homology, we can interpret integration as a pairing
between homology and cohomologyp[•|•⟩ : Hp (M, Ľω ) × H (M, ∇ω ) → C

where[γ ⊗ uγ |φ⟩ :=Z

φ.

(2.25)γ⊗uγ

Formally, this pairing is know as the period pairing. In particular, we can now think of the
four-point open-string amplitude (2.1) as a twisted period
d log

z
1−z



.

(2.26)

However, it turns out that we can construct other pairings that involve only homology or only
cohomology: the intersection index 5 and intersection number. While it may seem unnatural
to try and pair cycles with cycles and cocycles with cocycles, these pairings define innerproducts on the vector space of homology and cohomology. Using the intersection pairings we
can project any twisted (co)cycle onto a basis of (co)homology. In particular, this provides asystematic way of discovering linear relations between integrals. Moreover, both intersection
pairings are essential for constructing a double copy.
The intersection pairing pairs the twisted (co)homology with an associated dual twisted(co)homology. The dual twisted (co)homology is simply defined to be the twisted (co)homology
associated to the inverse twist: ǔ := u−1 :[Hp (M, Ľω )]∨ := Hp (M, Lω )ˇ ω)[H p (M, ∇ω )]∨ := H p (M, ∇and(2.27)ˇ ω = ∇−ω = d−ω∧ is the dual covariant derivative.
where Lω = Cu−1 is the local system and ∇Dual integrals or dual periods correspond to the following pairingˇ ω ) × Hp (M, Lω ) → C⟨•|•] : H (M, ∇
p

where

⟨φ̌|γ̌ ⊗ ǔ] :=

Z

φ̌ ,

(2.28)γ̌⊗ǔγ̌

where |γ̌ ⊗ ǔγ̌ ] and ⟨φ̌| denote dual homology and cohomology classes. For example, theintegral Aopen (−s, −t) corresponds to the dual period
A

open

#

z
(−s, −t) = d log(0, 1) ⊗ z −s (1 − z)−t .
1−z

(2.29)

From Poincaré duality [67, 94] (see figure 1), the (normal) twisted cohomology is isomorphic
to the dual homology H p (M, ∇ω ) ∼= Hn−p (M, Lω ) and the (normal) twisted homology isˇ ω ). Moreover, the dual local
isomorphic to the dual cohomology Hp (M, Ľω ) ∼= H n−p (M, ∇system Ľω corresponds to the functions that are annihilated by the dual covariant derivativeˇ(∇(Cu)= 0) while the local system Lω corresponds to the functions that are annihilated
4

In the context of Feynman integrals, it is well known that the Euler characteristic gives the number of
independent integrals (dimension of the cohomology) [92].
5Here, we are following the naming convention of Pham [93, 94]. Often the intersection index is called thehomological intersection number. We prefer intersection index because it is more succinct.

– 10 –JHEP07(2024)040



Aopen (s, t) = (0, 1) ⊗ z a (1 − z)b

by the covariant derivative (∇(Cu−1 ) = 0). This is why the dual local system appears inthe (normal) twisted homology.
Now that the dual (co)homology has been defined, we can introduce the intersection
pairings[•|•] : Hp (M, Ľω ) × Hn−p (M, Lω ) → C
byX[γ ⊗ uγ |γ̌ ⊗ ǔγ̌ ] :=

top

(2.30)

Reg[φ̌] ∧ φ .

(2.31)

(uγ ǔγ̌ |x ) Reg[γ]|γ̌ x ,x∈γ∩γ̌andˇ ω ) × H p (M, ∇ω ) → C
(M, ∇

by

⟨φ̌|φ⟩ :=Z
M

In (2.30) and (2.31), the dual cycle or dual cocycle must be regularized. In particular, theymust be replaced by a (co)homologous (co)cycle that has compact support. For cycles, thisregularization procedure introduces rational functions of the monodromies (i.e., e2πis ande2πit ). On the other hand, the regularization of dual cocycles introduces rational functionsof the exponents in the twist (i.e., s and t). We will postpone the explicit recipe for the
regularization procedure to sections 4 and 5.
In definition (2.30), the set γ ∩ γ̌ corresponds to the set of all points where γ and γ̌
intersect. Here, [•|•]top
is the topological intersection number at x that evaluates to ±1xdepending on the relative orientation of γ and γ̌. Additionally, the factor (uγ ǔγ̌ |x ) simply
evaluates to a phase since ǔ = u−1 . Thus, we see that the intersection index (2.30) counts thenumber of times a (regularized) twisted cycle and dual cycle intersects (with sign) weighted
by rational functions of the monodromies. Similarly, the intersection number (2.31) counts
the number of overlapping singularities between twisted forms and dual forms weighted byrational functions of the exponents in the twist. While not obvious without knowing theregularization procedure, the integral in (2.31) is simple to compute and always evaluates
to a sequence of residues!
Using the intersection pairings as inner products on (co)homology, we have the following
decompositions of identityh

1 = |γ̌i ⊗ ǔγ̌i ] Hij−1 γj ⊗ uγjand−1
1 = |φi ⟩ Cij⟨φ̌j | .

(2.32)

Here, Hij = [γi ⊗ uγi |γ̌j ⊗ uγ̌j ] is the homology intersection matrix corresponding to thebases {[γi ⊗ uγi |} and {|γ̌i ⊗ uγ̌i ]}. Similarly, Cij = ⟨φ̌i |φj ⟩ is the cohomology intersection
matrix corresponding to the bases {|φi ⟩} and {⟨φ̌i |}. Using these formula, any (co)cycle
can be projected onto a basis.
In our example of the beta function, only the first (co)homology is non-trivial. Moreover,
it is a one dimensional vector space since |χ(M )| = 1. Therefore, we can choose the basesof (co)homology and dual (co)homology to be
H1 (M, Ľω ) = Spannh

H1 (M, Lω ) = Spann(0, 1) ⊗ z s (1 − z)to(0, 1) ⊗ z −s (1 − z)−tioz
, (2.33)1−zˇ ω ) = Span d log z
H 1 (M, ∇. (2.34)1−z
H 1 (M, ∇ω ) = Span,,

– 11 –





d log

JHEP07(2024)040⟨•|•⟩ : Hn−pNote that one can choose the same topological cycles and differential forms for the dual bases
since the underlying topology of M is the same. With these choices, equation (2.32) becomesih

1 = (0, 1) ⊗ z −s (1 − z)−t H −1 (0, 1) ⊗ z s (1 − z)t(2.35)andz
1 = d log
1−z



1
1
+
tan(πs) tan(πt)



C

−1



d log

z
,
1−z

(2.36)where,

C = (2πi)

and

s+t
.
st

(2.37)

Now, we can understand the origin of the first equation of (2.2) by acting with the decomz
position of identity on |z d log 1−z⟩:zz d log
1−z



s
z
=
d log
s+t
1−z



,

(2.38)zzswhere ⟨z d log 1−z|d log 1−z⟩ = s+t
C. We also remark that the leading term in the α′
expansion of H corresponds to a four-point bi-adjoint scalar amplitude (s, t ∝ α′ ).

2.4

Quadratic identities and the double copy

Having introduced all the pieces featured in (2.2) (equations (2.26), (2.29) and (2.37)), weturn our attention to the quadratic identity (2.3). This identity follows from the twisted
Riemann bilinear relations (also known as twisted period relations)C = P̌ · H −1 · P⊤or

H = P · C −1 · P̌ .

(2.39)

Here, P and P̌ are the twisted period matrices for a given basis choice of the twisted(co)homology and dual (co)homology,

Pij = γi ⊗ uγi φjP̌ij = φ̌i γ̌j ⊗ uγ̌j .

and(2.40)

Now, while (2.3) and (2.39) are not quite what we mean by a double copy they are only one
step away. In the remainder of this section we will describe how to generalize equation (2.39)
and obtain the double copy (2.5).
The closed string amplitude (2.4) already looks like a double copy since its integrandis simply two copies of the same thing (with one complex conjugated)
Aclosed (s, t) =Z(u φ) ∧ (u φ) =C

Z

|u|2 φ̄ ∧ φ(2.41)

Cz
where |u|2 = |z|2s |1 − z|2t and φ = d log 1−z. It also looks like the intersection number (2.31)
where we use the complex conjugate of φ instead of the regulated dual form. That is, we
can interpret Aclosed as the pairing⟨¯•|•⟩ : H n (M̄ , ∇ω̄ ) × H n (M, ∇ω ) → Cby– 12 –⟨φ̄a |φb ⟩ :=Z
M

|u|2 φ̄a ∧ φb .

(2.42)JHEP07(2024)040i
H=2

Note that the complex conjugated dual local system has the same monodromies as the local
system for real exponents (s, t ∈ R). For example, ū → e2πis ū = e−2πis ū and u−1 → e−2πis u−1
as z winds around the origin. Thus, the local system is isomorphic to the complex conjugateof the dual local system:
Cu−1 =: Lω (s, t) ∼= Ľω (s, t) := Cū

(2.43)

for real exponents s and t. Replacing dual (co)cycles by complex conjugated (co)cycles in
the Riemann bilinear relations yields the double copyM|u|2 φ̄a ∧ φb =

P · H −1 · P


ba

.

(2.44)

Note that since the Ľω ∼= Lω , we do not need to recompute the intersection matrix H.
Moreover, since the intersection matrix H is meromorphic in the exponents of the twist,
we can take these exponents to be complex. We can check that the above prescription
indeed reproduces equation (2.5)−11
1+(Aopen (s, t))2 ,tan(πs) tan(πt)−π(s + t)2 Γ(s)Γ(t)Γ(−s − t)=,Γ(1 − s)Γ(1 − t)Γ(1 + s + t)Z
dz ∧ dz̄
=|z|2s |1 − z|2t 2= Aclosed (s, t) .
2
|z|
|1
−
z|
C

P̄ H −1 P =(2.45)

Note that the double copy (2.44) (and (2.45)) are manifestly single-valued. Thus, the double
copy procedure is a mechanism to generate interesting single-valued integrals.
The main goal of this work is to explore the structure of twisted cohomology at higher
genus and test if the double copy prescription of [83] holds. For one-loop string amplitudes,
the KLT kernel has been long sought after [5, 7–11, 13, 95, 96] culminating in [13, 14]. Our
goal is to asses whether twisted (co)homology is indeed the correct formalism to compute
KLT kernels for higher genus string amplitudes. Our hope is that twisted (co)homology could
simplify the procedure of determining the KLT kernel and turn the complicated contour
deformation arguments in [13, 14] into more linear algebra like statements.

3

One-loop string integrals and the Riemann-Wirtinger integral

In this section, we introduce one-loop string integrals after chiral splitting (section 3.1)
and explain how the Riemann-Wirtinger family of integrals (the main focus of this work)
is related (section 3.2).
3.1

One-loop string integrals

In the chiral splitting formalism [97], n-point one-loop open-string integrals take the form
Z

dD ℓ

Z

′

dτ eπiα τ ℓ2Zdn z e2πiα′ ℓ·

Pn
j=1

kj zj

Y

e

α′ kj ·kk log

ϑ1 (zj −zk )ϑ′ (0)
1

fstring (ℓµ , zj , τ )1≤j<k<n

∝

Z

dD ℓ

Z

′

dτ eπiα τ ℓ2Zdn z KNτn fstring (ℓµ , zj , τ )– 13 –(3.1)

JHEP07(2024)040⟨φ̄a |φb ⟩ :=Z

where D is the space-time dimension. Here, KNτn is the Koba-Nielsen for n-punctures is amulti-valued function universal to all one-loop string integrals
KNτn =nY!e2πisiA zi 

i=1s
ϑ1jk (zj − zk )

Y

(3.2)

1≤j<k≤nwhere ϑ1 is the Jacobi theta function and we have hidden most of the loop-momentum
dependence in the si ’s(3.3)

The sij ’s are the familiar planar Mandelstam invariants sij = sji = 12 (ki + kj )2 = ki · kj
where kiµ is the momentum associated to the ith particle. Moreover, these variables satisfyP
the momentum conservation constraint j̸=i sij = 0.
One-loop string integrals are integrals over the moduli space of the n-punctured torus:
M1,n . The modulus τ (ratio of the A- and B-cycle periods) controls the shape of the torus,
while the zi are the position of punctures on the torus. Meromorphic functions on the torus arecalled elliptic functions and must be doubly periodic under integer shifts of the A- and B-cycle
periods. Another way of saying this is that elliptic functions transform covariantly undermodular transformations (SL(2, Z) transformation of the periods). In particular, the modulus
can always be mapped to the upper half plane using a modular transformation. Thus, without
loss of generality, we normalize the A-cycle period to one and the B-cycle period to τ ∈ H.
These periods define a lattice Λτ = Z + τ Z that tiles C and the torus is the corresponding
quotient E = C/Λτ . We define the fundamental parallelogram to be P = {z = a + bτ |a, b ∈[0, 1]} and assume all zi ∈ P (one can always preform a modular transformation to move azi outside of the fundamental parallelogram into P ). While functions on the torus should
be doubly periodic it is often convenient to work with quasi-periodic functions like the odd
Jacobi-theta function ϑ1 . We also use conventions for the Jacobi-theta function such that
ϑ1 (z + 1|τ ) = −ϑ1 (z|τ ),ϑ1 (z + τ |τ ) = −e−iπ(τ +2z) ϑ1 (z|τ ).

(3.4)

In later sections, we will often omit the explicit dependence on τ since we always workat fixed τ .
In this work, we concentrate on the dτ dD ℓ integrands of string integrals, which arethemselves integrals over the configuration space of the n-punctured torus. Under A- andB-cycle shifts of the punctures, the Koba-Nielsen factor becomes
A-cycle :KNτn |zi →zi +1= exp [2πisiA ] ,KNτn

B-cycle :

(3.5)





n



X
KNτn |zi →zi +τ



=
exp
2πi
s
τ
+
s
z


ij j  ,

 iA

KNτn
j=1
j̸=i

where we have used momentum conservationP– 14 –

j̸=i sij = 0.

(3.6)

JHEP07(2024)040

siA = ℓ · ki .

The function fstring in (3.1) is a meromorphic and almost periodic function of punctures
zi=1,...,n , modulus τ and the loop momentum ℓµ . At genus-one, products of the KroneckerEisenstein function,
F (z, η|τ ) =ϑ′1 (0|τ )ϑ1 (z + η|τ ),ϑ1 (z|τ )ϑ1 (η|τ )(3.7)

at z → zi − zj form generating series for one-loop integrands fstring . Here, η is treatedas a formal parameter and the coefficients of the η-expansion of F appear in the one-loop
integrands:Xg (k) (z|τ ) η k−1 .

(3.8)

k≥0

With the exception of g (0) (z|τ ) = 1, all other Kronecker-Eisenstein coefficients g (k≥1) (z|τ )are meromorphic functions of z with at most simple poles. In particular, g (k≥1) (z)dz shouldbe thought of as the genus-one analogue of d log-forms. These functions are the integration
kernels that define elliptic multiple polylogarithms (eMPLs), which are becoming increasingly
well studied in the physics literature (see [58–62, 98–103] for an incomplete survey of thelast few years).
Elliptic functions cannot have only a simple pole.6 Therefore, the g (k≥1) are not elliptic
functions and have non-trivial B-cycle transformations
A-cycle :g (k) (z + 1|τ ) = g (k) (z|τ ),B-cycle :g (k) (z + τ |τ ) =kX(−2πi)j
j=0

j!

(3.9)
g (k−j) (z, τ ).

(3.10)

While the g (k≥1) ’s are not elliptic, one can construct doubly periodic linear combinations. On
the other hand, the generating series F has comparatively simple B-cycle monodromies
A-cycle :F (z + 1, η|τ ) = F (z|τ ),B-cycle :F (z + τ, η|τ ) = F (z, η|τ )e(3.11)−2πiη

.

(3.12)

Thus, it is often more convenient to consider a generating series of one-loop string integrands
built from Kronecker-Eisenstein functions.
Moreover, the Kronecker-Eisenstein functions satisfy a genus-one analogue of partial
fractions called the Fay identity
F (zik , ζi )F (zjk , ζj ) = F (zik , ζij )F (zji , ζj ) + F (zjk , ζij )F (zij , ζi )(3.13)

zij = zi − zj

(3.14)

ζij = ζi + ζj .

(3.15)

Since the Fay identity holds order by order in η, the coefficient functions g (k) also satisfy
partial fraction like identities. However, these g (k) identities are much more complicated —another reason to prefer working with the generating functions F over the g (k) ’s.
6

Double periodicity forces elliptic functions to have a double pole or more than one simple pole.

– 15 –JHEP07(2024)040

F (z, η|τ ) =In section 6.4, we will also use the non-meromorphic but doubly-periodic version of
the Kronecker Eisenstein series:Im z
F (z, η|τ ) .
Im τΩ(z, η|τ ) = exp 2πiη(3.16)

Like its meromorphic cousin, the doubly-periodic Kronecker-Eisenstein series can be ηexpandedXΩ(z, η|τ ) =f (k) (z|τ ) η k−1 ,(3.17)k≥0

3.2

The Riemann-Wirtinger integral

The one-loop string integrals (3.1) are closely related to a family of genus-one integrals —the so-called Riemann-Wirtinger integral — recently studied in the mathematics literature [1,2, 65, 66, 104]. The integrals are essentially the dz1 -part of (3.1)
Z

u(z1 ) = e2πis1A z1

u(z1 ) F (z1 − zj , η|τ ) dz1 ,γinYϑs11i (z1 − zi ),(3.18)

i=2

with n ≥ 3 punctures and an extra conditions1B = s1A τ +nX

s1j zj − η = const. ,(3.19)

j=2

where s1B ∈ C is a generic complex number.7 Crucially, this condition ensures that both theA- and B-cycle monodromies of the integrand in (3.18) are treated symmetrically:
A-cycle :B-cycle :KNτn F (z, η|τ )|z1 →z1 +1= e2πis1A ,KNτn F (z|τ )KNτn F (z, η|τ )|z1 →z1 +τ= e2πis1B .
KNτn F (z|τ )(3.20)(3.21)The twist also has monodromies as z1 loops around another puncture zj
z1 loop around zj :KNτn |⟲1j= e2πis1j .
KNτn(3.22)To understand the linear and quadratic relations among Riemann-Wirtinger integrals, weconstruct the corresponding twisted (co)homology. In particular, we need to understandhow the local system is modified at genus-one.
In the following, we review the construction of the local system since there are newfeatures at genus-one. Specifically, the “total” local system is the tensor product of two
local systems: one from the multi-valuedness of the twist and one from the quasi-periodicityof the Kronecker-Eisenstein function. While the local system will be important, much of7

To convert from the notation of [1] to our notation one makes the replacement c∞ → −s1B , ci>0 → s1i ,c0 → s1A .

– 16 –JHEP07(2024)040

to obtain doubly-periodic coefficients f (k) .

the underlying mathematical formalism (see appendix A and references therein) can beignored by first-time readers.
We begin with the local system and dual local system associated to the multi-valuednessof the twist
Lω := Lω (s1A , s12 , . . . , s1n ) = Cu−1 ,(3.23)

Ľω := L−ω (−s1A , −s12 , . . . , −s1n ) = Cu ,(3.24)

ω := d log(u) .

(3.25)

Lω,η (s1A , s12 , . . . , s1n , s1B ) = Lω ⊗ Lη = Cu−1 ⊗ Lη ,(3.26)

Ľω,η = Lω (−s1A , −s12 , . . . , −s1n , −s1B ) ⊗ L−η = Cu ⊗ L−η .

(3.27)As usual, these local systems are connected by the involution ∨ : s• 7→ −s• . From this,we construct the twisted cohomology and homology of the Riemann-Wirtinger integral(H p (M, Lω,η ) and Hp (M, Ľω,η )) as well as their duals (H p (M, Ľω,η ) and Hp (M, Lω,η )) in
sections 4 and 5.

4

Twisted cohomology of the Riemann-Wirtinger integral

Since one cannot put any differential form into the double copy formula, the space of allowed
differential forms needs to be understood. To this end, we review the construction of
the twisted cohomology of the Riemann-Wirtinger integral family. We define the twisted
cohomology for η ̸= 0 in section 4.1 and give two different bases one of which has a smoothη = 0 limit (see appendix E.4 and [1] for more). Then, in section 4.2, we define the associateddual twisted cohomology and an inner product called the intersection number. While only
8

By flat, we mean that ω ∧ ω = 0 =⇒ ∇2 = 0.

– 17 –JHEP07(2024)040

These local systems are line bundles that keep track of the branch choices of the twist anddual (inverse) twist. More precisely, the universal covering M̃ of M is a principle π1 (M )bundle where π1 (M ) is the first homotopy group. By exponentiating the monodromies of u
we define a one-dimensional representation of π1 (M ). Then, the local system is the associatedline bundle. More succinctly, the (dual) local system is the locally constant sheaf of solutions
ˇ = 0) with the flat8 (dual) covariant derivative ∇ = d + ω∧to the equation ∇f = 0 (∇f
ˇ = d − ω∧). In other words, the (dual) local system is the set of all possible branch(∇choices for the dual twist ǔ = u−1 (twist u). These covariant derivatives replace the normal
exterior derivative in the twisted setting.
So far the genus-one construction mirrors the genus-zero construction in section 2.
However, at genus-one, we have an additional local system associated to the multi-valued-nessof the Kronecker-Eisensten functions (3.12). Following [1, 2], we define a 1-dimensional
representation of the fundamental group π1 (E) for η ∈ C. We denote this by eη : π1 (E) ≃Λτ ∋ γ 7→ eη (γ) ∈ C∗ such that eη (1) = 1 and eη (τ ) = e−2πiη . This representation corresponds
to the allowed phases of our integrand modulo the twist (recall equation (3.11) and (3.12)). We
denote the rank-1 local system on E determined by the above representation of π1 (E) by Lη .
Thus, the combined local system of the integrand of (3.18) and its dual arethe existence of a well-defined intersection number is needed to have a well-defined double
copy, we illustrate how to compute intersection numbers at genus-one in order to provide
a complete presentation of the twisted (co)homology of Riemann-Wirtinger integrals. As asimple application of the intersection number, we verify the over-completeness of the second
basis of cohomology by deriving the linear relation satisfied by its elements.
The interested reader can find a derivation of the DEQs satisfied by the RW family of
integrals for both η ̸= 0 and η = 0 via intersection numbers in appendices E.1 and E.4.
4.1

Basis of cohomologyZZd(u φ) =u(∇φ) = 0.

(4.1)

This means that the Riemann-Wirtinger integrand is not unique — we can always add a
total covariant derivative to φ without changing the value of the integral. For this reason,
we would like to work with equivalence classes of forms that are unique and independentof such shifts. We would also like any representative of such a class to be closed: ∇φ = 0.
This ensures that there are no boundary terms in the twisted version of Stokes’ theorem.
Thus, we are lead to define the twisted cohomology
H p (M, Lω,η ) =









(4.2)

∇φ = 0} .

(4.3)

ker ∇ : Ωpη (M ) → Ωp+1
η (M )pim ∇ : Ωp−1
η (M ) → Ωη (M )Explicitly, the twisted cohomology class represented by φ is|φ⟩ = {φ + ∇ψ|ψ ∈ Ωdegφ−1
(M )
η

and

Here, the local system appears in the second argument of H p instead of ∇ because ∇ only
knows about the Lω half of the local system Lω,η .
For each p = 0, . . . , dimR M , the twisted cohomology is a finite dimensional vector space.
Moreover, from equations (2.23), only the middle-dimensional cohomology p = 12 dimR M is
non-trivial. Thus, there are no non-trivial closed twisted 0- or 2-forms
dimH p=0,2 (M, Lω,η ) = 0.

(4.4)

By a theorem from [1, 2], we also know that the dimension of the twisted cohomologyof 1-forms is
dimH 1 (M, Lω,η ) = n − 1– 18 –(4.5)

JHEP07(2024)040

Recall that the twisted cohomology is essentially the cohomology with respect to the covariant
derivative ∇ = d + ω∧ where ω = d log u and ∇2 = 0. This covariant derivative keeps track of
the monodromies generated by the twist u. It also means that we can trade the multi-valued
integrand uφ for simpler integrands (Lη -valued differential forms) since d(uφ) = u(∇φ). We
denote the space of twisted differential p-forms on M by Ωpη (M ) = Ωp (M ) ⊗ Lη .
Because the coefficients s• are generic, the total derivative always vanisheswhere n is the number of punctures zi . Note that in [1, 2] they effectively have one additionalpuncture (the variable u in [1, 2] should be thought of as the puncture zn+1 ) and therefore,
the dimension of their cohomology is n instead of n − 1.
We close this section by introducing two useful bases for the twisted cohomology. The
first basis contains forms that have at most a simple poles at one of the punctures zi≥2
φa = F (z1 − za+1 , η) dz1

fora = 1, 2, · · · , n − 1.

(4.6)

The second is a spanning set of forms

ξp(p) = ∂1 F (z1 − zp , η) dz1

(4.7)

(p)

ξa≥2 = [F (z1 − za , η) − F (z1 − zp , η)] dz1

fora ̸= psubject to a single relation








0 ≃ 2πis1A + s1p g (1) (η) +− (s1,p − 1)η|ξp(p) ⟩ + η

n
X







(p)

s1j F (zj − zp , η) − g (1) (zj − zp )  |ξ1 ⟩

j=2
j̸=p
n
X

(p)

s1j F (zj − zp , η) |ξj ⟩.

(4.8)

j=2
j̸=p
(p)

This identity follows from the fact that ∇ξp ≃ 0 in cohomology (and can be shown using
the Fay identity or intersection numbers). This basis has a well defined η → 0 limit andspans the twisted cohomology in this limit [1] (see appendix E.4). In the next section, we
will verify that the intersection matrix of these sets has rank n − 1 confirming that the set(p){φa } is indeed a basis and that the set {ξa } is over-complete.
4.2

Intersection numbers: an inner product on cohomology

Recall from section 2.3 that the dual cohomology isomorphic to the twisted cohomology
H p (M, Lω,η ) is defined by changing the sign of all s• and consequently η[H p (M, Lω,η )]∨ := H p (M, Ľω,η ).

(4.9)

ˇ = d + ω̌∧ where ω̌ = d log u−1 = −ω. As a consequence
The dual covariant derivative is ∇of changing the signs of the s• ’s, the dual Riemann-Wirtinger integral comes with a dual
twist ǔ = u−1 . We can also think of ∨ as a map from H p (M, Lω,η ) to the dual cohomology:φ̌ = [φ]∨ := φ|s• →−s• ,η→−η . In particular, the set {φ̌a } is basis for the dual cohomology and(p){ξˇa } spans the dual cohomology subject to the dualized version of (4.8).
The intersection number pairs the cohomology and its dual to form an inner product⟨•|•⟩ : H p (M, Ľω,η ) × H p (M, Lω,η ) → C(s• )– 19 –(4.10)JHEP07(2024)040(p)ξ1 = ηF (z1 − zp , η) dz1

by⟨φ̌|φ⟩ =Z

Reg[φ̌] ∧ φ(4.11)

M

Reg[φ̌] := φ̌ −


n
X





∇ Θ(|z1i | < ϵ) ψ̌i

i=2
n
X

= 1−



Θ(|z1i | < ϵ) φ̌ −

i=2

n
X

ψ̌i dΘ(|z1i | < ϵ),(4.12)

i=2

where ϵ is arbitrarily small enough such that the Heaviside-Θ functions do not overlap. Here,ˇ ψ̌i = φ̌ for z1 ∼ zi . While any
the 0-form ψ̌i is a local primitive for φ̌ near z1 = zi : ∇global primitive is indeed multi-valued, the local primitive ψ̌i is single-valued and can be
expressed as a Laurent polynomial in z1 − zi . Clearly, Reg[φ̌] has no support for z1 ∼ zi .
Moreover, it differs from φ̌ by a total derivative and is thus equivalent in cohomology.10
It is essential that we use the compactly supported version of φ̌ otherwise the intersection
number would not be finite.11 Moreover, since only the anti-holomorphic part of Reg[φ̌]survives the wedge product in (4.11), the intersection number of Riemann-Wirtinger 1-forms
is given by the following simple residue formula⟨φ̌|φ⟩ = −n ZXψ̌i dΘ(|z1i | < ϵ) ∧ φ = −

i=2 M

n
X

Resz1 =zi [ψ̌i φ].

(4.13)

i=2

One important feature of this formula is that the intersection number vanishes whenever theforms φ̌ and φ do not have overlapping singularities. Moreover, the intersection numbers
satisfy the following relation[⟨φ̌|ϕ⟩]∨ = −⟨ϕ̌|φ⟩ .

(4.14)

Thus, one only has to compute the upper/lower triangular part of the intersection matrix.
9

Formally, Reg is a map from the dual twisted cohomology to the compactly supported dual twisted
ˇ → Hcp (M, ∇).
ˇ
cohomology Reg : H p (M, ∇)10
Forms differing by a covariant derivative are in the same cohomology class and are said to be cohomologous.
The replacement of φ̌ with Reg[φ̌] does not change the intersection number because it only depends on theclass not the representative.
11
More precisely, for generic s• , Reg is an isomorphism between H p (M, Ľω,η ) and its compactly supported
version Hcp (M, Ľω,η ). Alternatively, one could also regulate Riemann-Wirtinger cohomology instead of thedual cohomology.

– 20 –JHEP07(2024)040

where Reg maps the representative φ̌ to a new representative that has no support in theneighbourhood of each puncture zi≥2 .9 Note that our definition for the intersection number
differs from [1] by a sign because we choose to put the dual forms on the left hand side
because it is more natural in the bra-ket notation of quantum mechanics.
Since the Riemann-Wirtinger integral is one-dimensional we only need to know the map
Reg in this case. Explicitly, for any 1-form φ̌, its image under Reg is

In particular, we can already predict that the intersection matrix ⟨φ̌a |φb ⟩ is diagonal. To
compute the proportionality constant, we need to construct the primitives of φ̌a"1
2πis1Ag (1) (−η)ψ̌a,i=a = −
+
−
+
s1a
(s1a −1)s1a
s1a −1ψ̌a,i̸=a = −P

i̸=a s1i g(1) (z1a )#(s1a −1)s1a

2z1a + O z1a, 
F (zia , −η)2z1i + O z1i,
s1i −1(4.15)(4.16)ˇ ψ̌a,i = φ̌a near z1 = zi . Using (4.13), we obtain
where ∇
δab.
s1a(4.17)

In particular, this intersection matrix has full rank verifying that the set {φa } forms a basis.
Equations (4.15) and (4.16) are also enough to determine almost all of the intersection(p)numbers for the set {ξa }:
(p)

 2
η


 s1p


(p)

⟨ξˇ1 |ξa ⟩η
= − s1p−2πi 2iπs −s
1A



(p)

(p)

⟨ξˇa≥2,a̸=p |ξb≥2 ⟩−2πi=

b = 1,
a = 1, b ≥ 2 and b ̸= p,(4.18)Pn(1) (η)+s g (1) (zpi )1p gi=2;i̸=p 1i(s1p −1)s1p

b = p,

 s +s− 1a 1p s1a s1p

b=a

− 1

s1p
Pn


(1)
(1)

 −s1p F (zap ,η)−s1p g (η)+ i=2;i̸=p s1i g (zpi )+2iπs1A(s1p −1)s1pb ̸= a, p(4.19)

b = p.

For a = b = p, one needs to go beyond the local primitives (4.15) and (4.16)
(p)

2



(p)

n
n
X
X
⟨ξˇp |ξp ⟩= (2πs1A )2 + 2s21p g (2) (η) +s1p s1i g (1) (zpi ) − 
s1i g (1) (zpi )
−2πi
i=2,i̸=p
i=2,i̸=p− 4iπs1AnX

s1i g (1) (zpi ) − 2s1pi=2,i̸=p(p)nXs1i ℘(zpi ).

(4.20)

i=2,i̸=p(p)

All remaining ⟨ξa |ξb ⟩ intersection numbers can be obtained from the above using the relation(p)(p)[⟨φ̌|ϕ⟩]∨ = −⟨ϕ̌|φ⟩. One can then verify that the n × n intersection matrix ⟨ξa=1,...,n |ξb=1,...,n ⟩has rank n − 1 confirming that the set {ξ (p) }a=1,...,n is over-complete. In particular, theidentity (4.8) follows from a straightforward application of the resolution of identity (2.32).
(p)(p)
Choosing {ξa }a̸=p and {ξˇa }a̸=p for our basis of cohomology and dual cohomology, we set(p) (p)(p)
Cab = ⟨ξˇa |ξb ⟩ and apply (2.32) to |ξp ⟩:|ξp(p) ⟩ ≃r.h.s.(4.8)−1 ˇ(p) (p)|ξa(p) ⟩ Cab⟨ξb |ξp ⟩ =.
η(s12 − 1)a,b̸=pX(p)

While more complicated than the forms φa , the spanning set ξaspans the cohomology in the η → 0 limit (see appendix (E.4)).

– 21 –(4.21)is still useful because it

JHEP07(2024)040(Cφ )ab = ⟨φ̌a |φb ⟩ = 2πi5Twisted homology of the Riemann-Wirtinger integral

After reviewing the construction of the twisted homology for Riemann-Wirtinger integrals,
we give a basis for homology (section 5.1). Then, in section 5.2, we compute the intersection
matrix that becomes the double copy kernel in section 6.
For the interested reader, we describe how to take the τ → i∞ limit of the Riemann
Wirtinger integral and obtain boundary values for the twisted contours in section E.3.
5.1

Basis of homology

γ = γ̃ ⊗ uγ̃ .

(5.1)

Technically, the contour should also be loaded with a branch choice eγ̃ on E (section ofĽη ). However, since we can regard the Kronecker-Eisenstein function F (z, η) as a global
meromorphic section of Ľη , such a loading will not play an important role in the computationof intersection indices.
The twisted homology is defined in the usual way
Hm (M, Ľω,η ) =









ker ∂ω,η : Cm (M, Ľω,η ) → Cm−1 (M, Ľω,η )im ∂ω,η : Cm+1 (M, Ľω,η ) → Cm (M, Ľω,η )(5.2)

where Cm (M, Ľω,η ) is the space of twisted m-contours. Elements of the twisted homology
are equivalence classes of closed twisted cycles (twisted contours with no boundary) modulo
exact cycles (twisted contours that are boundaries) and denoted by the square bra. Explicitly,
the homology class of an m-cycle γ ⊗ uγ ∈ ker(∂ω,η : Cm (, Ľω ) → Cm−1 (, Ľω )) is[γ ⊗ uγ | = {γ ⊗ uγ + ∂ω (β ⊗ uβ ) | (β ⊗ uβ ) ∈ Cm+1 (, Ľω )} .

(5.3)

That is, a representative of a twisted homology class, γ ⊗ uγ , is a twisted contour that has
no boundary and is not a boundary itself. We also define the dual twisted homology byexchanging the dual local system Ľω,η in (5.2) for the local system Lω,η[Hm (M, Ľω,η )]∨ := Hm (M, Lω,η ) =









ker ∂ˇω,η : Cm (M, Lω,η ) → Cm−1 (M, Lω,η )im ∂ˇω,η : Cm+1 (M, Lω,η ) → Cm (M, Lω,η )(5.4)

where Lω,η = [Ľω,η ]∨ = Ľω,η (−s1A , −s12 , . . . , −s1n , −s1B ) = Cu−1 , Cm (M, Lω,η ) is the spaceof dual twisted m-contours and ∂ˇω,η = ∂−ω,−η is the dual boundary operator. Representativesof dual homology classes are topological cycles loaded with a branch choice of the inverse
twist: γ̌ = γ̌ ′ ⊗ u−1γ̌ ′ . Given a representative of the dual homology, its class is denoted by′a square ket |γ̌ ⊗ u−1γ̌ ′ ].

– 22 –JHEP07(2024)040

As reviewed in section 2, twisted or loaded contours are represented by the tensor product of
a “normal” topological contour with a branch choice. That is, given a topological contourγ̃ on the manifold M , we “load” it with a branch choice of the twist uγ̃ (section of Ľω,η )to form a twisted contour γ

z2 +τ
γ̌6B

z6
γ̌6A
z5

γ
z4

z3

z2 +1

γ2A

Figure 2. An over-complete set of cycles in blue and an over-complete set of dual cycles in green
for n = 6 punctures. The (dual) cycles depicted by solid lines form a basis and the dashed red line
denotes our choice of branch cut. All of the cycles start at the point z2 while all of the dual cycles
start at the point zn = z6 . The solid n − 1 = 5 (dual) cycles define a basis for the (dual) twisted
homology. The dashed cycles are naturally defined but linearly depend on the cycles denoted by solidlines. The contours drawn here are technically cycles because the end points do not belong to M .
Such cycles are known as locally finite or Borel-Moore cycles in the mathematics literature.

Since only the middle dimensional twisted cohomology is non-trivial, dim H0 = dim H2 =0. We also know from [1, 2] that dim H1 = n − 1 where n is the number of punctures.12 Of
course, this counting is the same for the cohomology (section 4).
The remaining task is to provide a basis for the twisted homology and its dual. A
particularly convenient choice of topological cycles is depicted in figure 2 with cycles in blueand dual cycles in green. All of the cycles start at the puncture z2 while the dual cycles startat zn (where n = 6 in figure 2). These topological cycles are then loaded with a branch of u(ǔ) to form twisted (dual) cycles where the branch cut is denoted by the dashed red line infigure 2. We will always choose branches such that on the interval ∆k = [zk , zk+1 ]k=2,...,n−1
z 1 − zjarg
=π
z1 − z i

for 2 ≤ i ≤ k < j ≤ n .

(5.5)

Furthermore, note that the cycles in figure 2 intersect only one dual cycle. This is enoughto guarantee that the corresponding intersection matrix is diagonal.
We have actually drawn an over complete set of n (dual) cycles in figure 2 since weknow that dim H1 = n − 1. To find a linear relation amongst these cycles, start with aclosed circular contour that does not encircle any singularities. Then, expand this contour
without crossing the branch cuts until the contour runs along the edge of the fundamental
parallelogram and the branch cut. Rewriting the deformed contour in terms of the set
12Note that in [1, 2], the variable u should be thought of as an additional puncture zn+1 and therefore, theirhomology has n elements.

– 23 –JHEP07(2024)040

z2

{γ2A , γ2B } ∪ {γ2j }j=3,...,n yields the relation [1](e2πis1A − 1)[γ2B | + (1 − e2πis1B )[γ2A | −nXe−2πi(s12 +···+s1j ) (1 − e2πis1j )[γ2j | ≃ 0,

(5.6)

j=3

5.2

The intersection index: an inner product on homology

In this section, we regularize the cycles defined in the previous section in order to computethe intersection matrix that becomes the double copy kernel in section 6. The intersection
indices for Riemann-Wirtinger integrals were first derived in [66] for η = 0 and more recently
revisited in [1] for η ̸= 0.
As reviewed in section 2, the intersection index (2.30) is a pairing between the twisted
homology with local system Ľω,η and the twisted homology with local system Lω,η[•|•] : Hp (M, Ľω ) × Hn−p (M, Lω ) → C ,(5.7)

given by[γ ⊗ uγ |γ̌ ⊗ ǔγ̌ ] :=

X



top

(uγ ǔγ̌ |x ) Reg[γ]|γ̌ x ,(5.8)

x∈γ∩γ̌

where the set γ ∩ γ̌ corresponds to the set of all points where γ and γ̌ intersect and thefactor (uγ ǔγ̌ )|x , evaluates to a phase. Here, [•|•]top
is the topological intersection indexx
at x that evaluates to ±1 depending on the relative orientation of γ and γ̌. Following theconventions of [18, 105], we define

.

(5.9)

The only remaining piece is to specify the map Reg.
13

Note that the BCJ relations are linear relations among color-ordered amplitudes and should not be
confused with color-kinematics duality.

– 24 –JHEP07(2024)040(a similar identity exists for the set of dual cycles {γ̌nA , γ̌nB } ∪ {γ̌nj }j=2,...,n−1 ). We will oftenchoose the cycles drawn in solid lines as our basis for the (dual) homology. Moreover, we will
also rediscover the relation (5.6) using the intersection indices in the next section.
In string theory context, relations like (5.6) are often called monodromy relations because
one projects a contour with one color ordering onto another with different color ordering,which corresponds to the winding or exchanging of punctures. Such identities relate various
color orderings for open string amplitudes. The tree-level gauge theory equivalent of (5.6)
are the so-called Kleiss-Kuijf and BCJ relations. The Kleiss-Kuijf follow from the leading
α′ expansion of the string monodromy relations while the BCJ relations follow from thesubleading order in α′ .13

z2 +τ

γ̌6B

(3)

P2

s6
γ̌6A
ℓ1B

(2)

P2

(2)

P2

m2

z2

(2)

P0

(2) m3
P3

ℓ1A

z2 +1

Figure 3. The over-complete set of cycles (dual cycles) subject to (5.6) in the compact homology (dualhomology). Those depicted by solid lines form a basis. In particular, this choice has an orthogonalintersection matrix since each cycle only intersects one dual cycles!

In order for the definition (5.8) to make sense, at least one contour should have compact
support. Thus, Reg is a map from the normal14 twisted homology described in section 5.1
to the compactly supported twisted homologycReg : Hm (M, Ľω,η ) → Hm
(M, Ľω,η ) .

(5.10)

When all of the exponents s• are generic, this map is an isomorphism and we can use it tocompute intersections involving non-compact cycles.
Compact cycles cannot have endpoints like in figure 2 even if the endpoints do not belong
to the space (as in our case). Thus, compact cycles must wind around the punctures insteadof ending at the punctures. This necessarily means that the compact cycles will cross abranch cut. To be a twisted cycle with compact support, it must have no boundary. Becausesuch a cycle crosses branch cuts there will be relative phases that prevent ∂ω,η from vanishing.
Only by breaking up the compact cycle into pieces and combining these pieces with particularphases do we obtain a genuine compact twisted cycle.
This procedure is best illustrated through examples. Consider the cycles of figure 3 inblue. Clearly, these are compact. The compact cycles corresponding to the non-compact
γ2j=3...,n,A,B cycles have been broken up into three pieces: one encircling the puncture z2 ,one encircling the puncture zj and one connecting each of these circles below the branch cut.
When j = A or j = B, the compact cycle encircles z2 +1 or z2 +τ . To understand how to
combine the contours mj , Sj and ℓ2j into a genuine compact twisted cycle, we enumerate
14

To be precise, the homology described in section 5.1 is called locally finite or Borel-Moore [93].

– 25 –JHEP07(2024)040

(2)

P1
m

m1

their twisted boundaries:∂ω,η mj =P (2) − P (2)j+1

for i ∈ {1, 2, 3} ,j

e2πis12 P (2) − P (2)1(j)∂ω,η Sj = (e2πis1j − 1)P0
∂ω,η ℓ2j =


(j)
(2)


P − P0 0e2πisA P

for j = 0 ,

0

(2)

−P

for j ∈ {3, . . . , n},(5.11)

for i ∈ {1, 2, 3} ,
(2)

2
0



e−2πisB P (2) − P (2)31

for j = A ,for j = B .

2πis1A m
m0 + e2πis12 (m1 + m2 + m3 )1
2πis1A (m2 + m3 + m0 ) + e
+ℓ
−e
2A
2πis
2πis
12
12
e
−1
e
−1
2πis12
m1 + m2 + m3 + m0
(m
+
m
)
+
(m
+
m
3
0
1
2 )e
−2πis1B
Reg[γ2B ] :=
+
ℓ
−
e
2B
e2πis12 − 1e2πis12 − 1
m0 + e2πis12 (m1 + m2 + m3 )SjReg[γ2j ] :=+ ℓ2j − 2πis1j −1
for j = 3, . . . , ne2πis12 − 1eReg[γ2A ] :=(5.12)(5.13)(5.14)

have no twisted boundary. In practice, this regularisation procedure is equivalent to an
analytic continuation by using a Pochhammer contour [18]. The regularization of these cycles
almost identical to the genus-zero case and equivalent to [80].
Since the intersection matrix is a basis dependent object, there is no unique best choice.
Therefore, we will provide the intersection matrix for two choices of dual cycles in this section.
Choosing the dual basis in figure 2 and 3, yields the following diagonal intersection matrix
(2n)

Hab

:= [γ2a |γ̌nb ] 2πi(s(5.15)
+s

)

12
13
,e
= diag e 1−e2πis
132πi(s12 +s13 +s14 )
1−e2πis14,··· , e

2πi(s12 +s13 +···+s1,n−1 )1−e2πis1,n−1

, e−2πi(s1n +s1B ) , e−2πis1A ,where a = 3, . . . , n − 1, A, B and b = 3, . . . , n − 1, B, A. Note that we have to invert theordering of the A- and B-cycles in the dual basis to make the intersection matrix diagonal.
This form of the intersection matrix is most useful in the decomposition of identity (2.32)
since H (2n) is easy to invert. For example, we can prove (5.6) by applying (2.32) to [γ2n |[γ2n | ≃n−1
X[γ2n |γ̌na ][γ2n |γ̌nB ][γ2n |γ̌nA ]−l.h.s. (5.6)[γ2a | +[γ2A | +[γ2B | =.
[γ2a |γ̌na ][γ2A |γ̌nB ][γ2B |γ̌nA ]1 − e2πis1na=3(5.16)

Here, we have used (5.15) and[γ2n |γ̌nb ]b=3,...,n−1,B,A =no−1
2πis1n−2πis1B
2πis1A
1,
.
.
.
,
1,
e
(1
−
e
),
(1
−
e
)
. (5.17)
1 − e2iπs1nWhile the dual basis {γ̌nb }b=3,...,n−1,B,A is convenient for many computations, it is not thebest for the double copy because it requires evaluating two sets of twisted periods: [γ2a |φb ⟩and ⟨φ̌a |γ̌nb ]. On the other hand, useing the basis of topological cycles γ2a for the dual basis
as well (i.e., use γ̌2b = [γ2b ]∨ = γ2b |s• →−s• as our dual basis), one easily obtains the dual
periods from the “known” integrals: ⟨φ̌a |γ̌2b ] = [γ2b |φa ⟩|s• →−s• . The price one pays is that– 26 –JHEP07(2024)040

Using (5.11), the reader can verify that the regulated contours

the intersection matrix becomes incredibly dense and hard to invert. However, inverting thismatrix is still much easier than evaluating two sets of periods. Explicitly, we set
(22)

Hab

:= [γ2a |γ̌2b ] ,(5.18)

for a = 3, . . . , n − 1, A, B and b = 3, . . . , n − 1, A, B where[γ2j |γ̌2k ] = 2πis1j
 e 2πis1−e1j

for j < k[γ2j |γ̌2A ] =,1 − e2πi(s12 +s1j ),(1 − e2πis12 )(1 − e2πis1j )1 − e2πis1A,
1 − e2πis12
1 − e−2πis1B[γ2B |γ̌2j ] =,1 − e2πis12[γ2A |γ̌2j ] =1 − e2πis12
e2πis12 (1 − e2πis1B )[γ2j |γ̌2B ] =,1 − e2πis12(e2πis1A − 1)(e2πis1A − e2πis12 )[γ2A |γ̌2A ] = −,e2πis1A (1 − e2πis12 )(e2πis1B − 1)(e2πis1B − e2πis12 )[γ2B |γ̌2B ] = −,e2πis1B (1 − e2πis12 )(5.19)

e2πis12 − e2πi(s1A +s12 ) − e2πi(s1B +s12 ) + e2πi(s1B +s1A ),1 − e2πis12
1 − e−2πis1B − e−2πis1A + e2πi(s12 −s1B −s1A )[γ2B |γ̌2A ] =.
1 − e2πis12[γ2A |γ̌2B ] =Aside from a change in notation, the above intersection indices can be found in [1] and arequoted here for the convenience of the reader. Pedagogical examples of selected intersection
indices have also been provided in appendix B.
Even though it is not obvious that H (22) is invertible, its determinant is easily computed
1 − e−2πis1ndet(H (22) ) = Qn−1
̸= 0 ,
2πis1m )
m=2 (1 − e(5.20)and non-vanishing. Similarly, the determinant of H (2n) is also non-vanishing. However,notice that these determinants are finite only when s12 , . . . , s1n−1 ∈/ Z. Thus, the bases oftwisted cycles introduced in this section are only valid when the Mandelstams are generic:s12 , . . . , s1n ∈/ Z. Also note that when s1A = s1B[γ2A |γ̌2A ] [γ2A |γ̌2B ][γ2B |γ̌2A ] [γ2B |γ̌2B ]

!

=

01
−1 0

!

(5.21)

and we can identify γ2A and γ2B with the usual basis of the non-twisted homology H1 (E, Z).
Before closing this section, we note that intersection indices can be used to compute themonodromy matrices associated to zj A- and B-cycles as well as encircling zk≥2 with zj≥2
(i.e., the circuit matrix Mjk ) [1]. While we do not explicitly need these matrices in this work,
it is worth stressing that their computation is rather straightforward from the intersection
point of view. Moreover, these matrices may be the key to understanding Riemann-Wirtinger
integrals when more than one puncture is integrated (see section 7 for some speculation
on how this may work)! While the derivation of the circuit matrices is almost identical to– 27 –JHEP07(2024)040

1
j > k,
1−e2πis1j
e2πis12 (1 − e−2πis1A )[γ2j |γ̌2j ] =the analogous genus-zero case, something new and interesting happens when computing thezj≥2 A- and B-cycle monodromy matrices. In addition to a phase, the zj≥2 A- and B-cycles
induce a change in the local system wherezj≥2 A-cycle : s1B 7→ s1B − s1j ,(5.22)

zj≥2 B-cycle : s1A 7→ s1A + s1j .

(5.23)

The corresponding monodromy matrices realize a linear map from the homology with s1A
and s1B to the homology with s1B − s1j or s1A + s1j . For more details, see [1].

The double copy of Riemann-Wirtinger integrals

In this section, we introduce the closed-string analogues of the Riemann-Wirtinger integrals,
which we call complex or single-valued Riemann-Wirtinger integrals (the first example ofwhich was considered in section 4.4.4 of [66]). We define the complex Riemann-Wirtinger
integral and its double copy in section 6.1. Then, in sections 6.2 we study the double copy forreal Mandelstam variables {s1j }nj=2 , s1A , s1B ∈ R. However, since double copy formulas aremeromorphic functions of the Mandelstams, we can take them to be complex quantities in theend. Two natural ways of relaxing the reality condition are explored in sections 6.3 and 6.4.
6.1

Complex Riemann-Wirtinger integrals

In section 2, we defined the single-valued pairing between a twisted form and a complexconjugated twisted form as the intersection pairing⟨•|•⟩ : H 1 (M , Ľω,η ) × H 1 (M, Lω,η ) → C .

(6.1)

This pairing is single valued in the sense that the integrand insensitive to all z1 -monodromies.
Explicitly, we define⟨φ̄k |φl ⟩ :=Z
M|u(z1 )|2 F (z1 − zl , η|τ ) F (z1 − zk , η|τ ) d2 z1 ,(6.2)

1 ∧dz1
where d2 z1 = 2i dzIm
is the normalized volume form,τ|u(z1 )|2 := e2πis1A (z1 −z 1 )nY|ϑ1 (z1 − zj )|2s1j ,

(6.3)

j=2

and P is the fundamental parallelogram. Note that equation (6.3) is a definition when the
Mandelstams are complex since, in this case, uū ̸= |u|2 .
One can think of this pairing as an intersection number where the dual twist is ǔ = ū.
To see this, we note that there is secretly a hidden factor of ǔ u inside the integral defining
the intersection number (2.31). Usually (outside of this section) ǔ = u−1 and the productǔ u = 1 drops out of the formula; the resulting integrand is single-valued/doubly-periodic.
However, it is crucial to choose ǔ such that it cancels the multi-valuedness of the integrandwhen defining the intersection number. Equivalently, this conditions tells us what spaces weare allowed to pair in this manner. For example, the choice ǔ = ū also renders the integrand– 28 –JHEP07(2024)040

6

single-valued/doubly-periodic. This suggests that we can take the replacement ǔ → ū in thetwisted Riemann-bilinear relations seriously and define the double copy⟨φ̄a |φb ⟩ =

X
X
i n−1
i n−1
⟨φ̄a |δ̄i ] (Hij−1 )⊤ [γj |φb ⟩ =[γi |φb ⟩ Hij−1 [δj |φa ⟩ ,2 i,j=1
2 i,j=1(6.4)

B-cycle :|u(z1 )|2 F (z1 − zj , η|τ )F (z1 − zk , η|τ )|z1 →z1 +τ= e2πi(s1B −s1B ) .
|u(z1 )|2 F (z1 − zj |τ )F (z1 − zk , η|τ )(6.5)

Thus, to ensure that the z1 -integral over the fundamental parallelogram is indeed the integral
over the whole space, we further require the reality of the Mandelstam variable
s1B = s1B .

(6.6)

Since s1B is defined to satisfy (3.19), complex conjugation has a natural action on s1B via thecomplex conjugation on punctures zj , η and the modulus τ and thus (6.6) is non-trivial.15 One
possible way of satisfying (6.6) numerically is by first choosing a value of s1B and then using
the constraint (3.19) to find a value of η given τ and the rest of the Mandelstam variables.
Another important reason why we require the condition in (6.6) is because we will use
the isomorphism of local systems Lω,η ∼= Ľω,η whereĽω,η = Ľω,η̄ (−s1A , −s12 , . . . , −s1n , −s1B ) = Cu ⊗ Lη̄ ,(6.7)

is the complex conjugate of the dual local system (cf., equation (3.27)). However, thiscoincides with (3.26)
Ľω,η ∼= Lω,η (s1A , s12 , . . . , s1n , s1B )ifs• ∈ R ,(6.8)

only if all s• ’s are real — in this case, u−1 and ū have the same monodromies (we will show
this more explicitly in section 6.2).
Strictly speaking, the double copy formula (6.4) should include the complex-conjugate∼ Ľω,η 16 allows us totwisted-cycles γj ∈ Hm (M, Ľω,η ). However, the isomorphism Lω,η =recycle the intersection indices of section 5.2 without modification. In fact, we can use the15

A version of (6.6) is found in equation (63) of [66].

16

Explicitly, given a section, av̄, of Ľω where a ∈ C, this isomorphism is given byav̄ 7→ a

v̄
.
ūu

v̄
Since v̄/ū ∈ C is a phase, a ūu∈ Cu−1 = Lω . Furthermore, this induces an isomorphism between homologiesv̄ −1
via γ̄ = γ̃ ⊗ v̄ → γ̃ ⊗ ū u .

– 29 –JHEP07(2024)040

where H is the homology intersection matrix corresponding to a basis choice {γi } and {δi }.
The factor of i/2 in (6.4) comes from our particular normalization of d2 z1 .
However, as written, one should not expect (6.2) to have any cohomological meaningfor generic complex η since the integrand is not necessarily doubly-periodic, even if s1A
and the {s1j }nj=1 (all but s1B ) are real. In fact, the integrand of (6.2) precisely fails to be
doubly-periodic because it has a B-cycle monodromy:same intersection indices even when the s• ’s are complex since they enter the double copy
in a meromorphic way (sections 6.3 and 6.4). Similarly, we should also have the conjugatedperiod pairing ⟨φa |γj ] in (6.4) where the conjugated cohomology comes with the connection¯ or equivalently the local system Lω,η .
∇We start by building up our intuition about the double copy for real s• in section 6.2
and describe how this condition can be relaxed later in sections 6.3 and 6.4. In particular, wehave checked the identity (6.4) both for real and complex Mandelstams {s1A , s1B , s1j=2,...,n }whenever the l.h.s. of (6.4) converges. Moreover, one can interpret (6.4) as defining theanalytic continuation of ⟨φ̄a |φb ⟩ when (6.2) does not converge.
The double copy for real Mandelstam variables

In this section, we test the double copy formula (6.4) when all Mandelstams (including
s1B ) are real.
We start by understanding the local system Ľω,η . It is defined by the multi-valuednessof the integrands u F (z, η|τ ):A-cycle :u F (z, η|τ )|z1 →z1 +1= e−2πis1A ,u F (z|τ )(6.9)

B-cycle :u F (z, η|τ )|z1 →z1 +τ= e−2πis1B ,u F (z|τ )(6.10)

s1j zj − η = const ∈ R.

(6.11)

where (3.19) implies that we haves1B = s1A τ +

n
X
j=2

Then, the twisted cycles γj ∈ H1 (M , Ľω,η ) can be paired up with twisted forms φa ∈H 1 (M , Lω,η ) to form complex-conjugated Riemann-Wirtinger integrals⟨φa |γj ] = [γj |φa ⟩ .

(6.12)

These conjugated integrals also satisfy a version of the over-completeness relations (5.6):(e−2πis1A − 1)⟨φ|γ2B ] + (1 − e2πis1B )⟨φ|γ2A ]−nXe−2πi(s12 +···+s1j ) (1 − e−2πis1j )⟨φ|γ 2j ] = 0,

(6.13)

j=3

which is as a direct corollary of (6.12) and complex-conjugating (5.6).
In principle, as long as one uses the same topological cycles for a basis of both homologies
and the same complex differential forms for the basis of both cohomologies, the conjugatedintegrals do not need to be evaluated from scratch. However, for the purposes of numerics, thefollowing fact17 provides a practical definition of the conjugated Riemann-Wirtinger integral.
17This comes naturally from the contour integral as a Riemann integral, and linearity of complex integration(using a real parametrization of the contour), see equation (4) in Chapter 4 of [106] and the paragraphs below
equation (6) in the same chapter for a similar expression.

– 30 –JHEP07(2024)040

6.2

Given a meromorphic function f : C → C and a real-parameterized path γ : [0, 1] 7→ C
we have the factZZ

f (z)dz =γ

f (z)dz.

(6.14)

γ

Thanks to (6.14), the conjugated integral can be rewritten as⟨φa |γj ] =Zu(z1 ) F (z1 − za , η|τ ) dz1 =Z

γj(6.15)nYϑ1 (z1 − zi |τ )

s1i

n
Y
ϑs11i (z1 − zi | − τ ) ,(6.16)= e−2πis1A z1

i=2

i=2

which we contrast to (3.18). Note that we have used the following identity for theta functions,ϑ1 (z|τ ) = ϑ1 (z| − τ ) ,(6.17)

which serves to emphasize that the integrand is a meromorphic function of z1 , up to the
expected branch cuts.18
For most numerical checks of (6.4), the regularized twisted cycles (5.12)–(5.14) are overkill
since there are many choices of s1j , zj and τ for which the integrals over the un-regulated
cycles converge. For example,
Z 1

u(z1 ) F (z1 − za , η|τ ) dz1(6.18)

0is convergent for Re(s12 ) > 0. As a function of s12 , the regulated cycle γ2A = Reg(0, 1)(remember that we have gauged z2 = 0) analytically continues this integral to negative s12 .
A numeric implementation of these integrals and the resulting double copy is provided in
the ancillary Mathematica notebook in the supplementary material.
In all cases, we will choose {φa=2,...,n } as the basis for the cohomology and {φ̄a=2,...,n }as the basis for the complex-conjugated cohomology. In particular, the double copy of the“diagonal” pairings ⟨φ̄a |φa ⟩ are real and positive. Furthermore, we note that intersection
matrix (and its inverse transpose) must be anti-Hermitian in order for the r.h.s. of (6.4) to beconsistent with the reality of the l.h.s. (6.4). To see this, choose the same basis of topologicalcycles for the homology and its dual in (6.4). This is one way to rediscover the identity [80][γ|δ̌] = −[δ|γ̌]∨ ,(6.19)

where we recall that the operator ∨ implements the involution (see below (3.27))(s1A , s1B , s12 , . . . , s1n ) → (−s1A , −s1B , −s12 , . . . , −s1n ) .

(6.20)

Thus, the operator ∨ is equivalent to complex conjugation of the intersection number,[δ|γ̌]∨ = [δ|γ̌] whenever the Mandelstam variables are real.
In figures 4 and 5 we plot the numerical values for every term that appears in the double
copy formula 6.4 for n = 3 as s1A varies. Figure 4 corresponds to choosing a symmetric basis
18

This is because, if f (z) satisfies the Cauchy-Riemann relations, so does f (z).

– 31 –JHEP07(2024)040

u(z1 ) = e−2πis1A z1

u(z1 ) F (z1 − za , η| − τ ) dz1 ,γjFigure 5. The analogous plot to figure 4 using an asymmetric choice for the basis of homology anddual cohomology: {γ2A , γ23 } and {γ̌23 , γ̌2B }.

– 32 –JHEP07(2024)040

Figure 4. Plots of each component entering (6.4) for n = 3 and (s12 , s1B , z3 , τ ) = (− 13 , − π3 , 13 + 3i , i +111 ) as s1A varies. In each subplot, the real part of the quantity is in blue and its imaginary partis in orange. Here, we choose a symmetric basis for homology and dual homology: {γ2A , γ2B } and{γ̌2A , γ̌2B }. In this case, several quantities plotted above are related by complex conjugation.

for homology and dual homology: {γ2A , γ2B } and {γ̌2A , γ̌2B } and figure 5 corresponds tochoosing an asymmetric basis for homology and dual homology: {γ2A , γ23 } and {γ̌23 , γ̌2B }.
While some of the individual building blocks look very different when comparing the twocases, they combine to form the same complex Riemann-Wirtinger integral! We also compare
the direct evaluation of the complex Riemann-Wirtinger integral (6.2) to the one obtainedby the double copy of (6.4). The difference of these quantities,∆⟨φ3 |φ3 ⟩ = ⟨φ3 |φ3 ⟩− ⟨φ3 |φ3 ⟩(6.2)(6.21)(6.4)

is shown in figure 6 and found to be numerically negligible.
In figure 7, we tabulate the building blocks that enter into the computation of the integral⟨φ3 |φ2 ⟩ via the double copy with n = 4 punctures (i.e. a three-dimensional twisted homology).
In this case, the integral ⟨φ3 |φ2 ⟩ has both real and imaginary parts of similar magnitude.
The integral ⟨φ3 |φ2 ⟩ as defined by (6.2) diverges for s12 > 1. Thus, one defines the integral⟨φ3 |φ2 ⟩ by the double copy (6.4) whenever s12 > 1, see figure 8. The integral ⟨φ3 |φ2 ⟩ as
4
2
defined by the double copy (6.4) for e.g. (s12 , s13 ) = ( 10
, − 10) requires a regularization of theintegral ⟨φ3 |γ 23 ], but also evaluates to a convergent complex integral via its definition (6.2).
Thus, we showcase a harmony between regularization via twisted cycles and convergent
complex integrals.
6.3

The double copy for complex Mandelstam variables

So far we have assumed that the Mandelstam variables {s1A , s1B , s12 , . . . , s1n } are all real.
In this section, we will describe one way to relax this condition.
First and foremost, the action of “complex conjugation” will still be relevant, but now
we will define it (in this section only) to not act on the Mandelstam variables:(s1A , s1j ) → (s1A , s1j ) ,– 33 –(6.22)JHEP07(2024)040

Figure 6. Numerical comparison of the (real part of the) complex Riemann-Wirtinger integral ⟨φ3 |φ3 ⟩by numerical integration (6.2) and by using the double copy formula (6.4). We use (s12 , s1B , z3 , τ ) =1(− 13 , − π3 , 13 + 3i , i + 11
) and plot (a) ⟨φ3 |φ3 ⟩ according to numerical integration of the complex
integral (6.2) and the double copy formula, and (b) the real part of ∆⟨φ3 |φ3 ⟩ for different valuesof s1A . Note that ⟨φ3 |φ3 ⟩ is a real quantity for these values of the Mandelstam variables, so theimaginary parts here are numerically zero.

Figure 8. The quantity ⟨φ3 |φ2 ⟩ according to the complex integral (6.2) (in blue) and the double
copy formula (6.4) (in orange) as a function of s12 . We use the same values of Mandelstam variables,
punctures and moduli as in figure 7. We note that the definition by (6.2) diverges for s12 > 0. Theorange line analytically continues this integral beyond this region.

even when they take complex values. Complex conjugation acts on the puncture positions
zj , η and the modulus τ as usual. Next, we define the complex Riemann-Wirtinger integral
as before (6.2) except that |u(z1 )|2 in the integrand is not the usual absolute value. Instead,|u(z1 )|2 is defined by (6.3) for complex Mandelstam variables.
Then, we need to remember the “reality” condition on the Mandelstam s1B :
s1B = s1B

– 34 –(6.23)JHEP07(2024)040

Figure 7. Numerical samples of every component that enters in the computation of ⟨φ3 |φ2 ⟩ via
the double copy (6.4) for a symmetric choice of homology bases for n = 4 (i.e. 3-dimensional twisted
homology basis). In each subplot, the real part of the quantity is in blue and its imaginary part is
2 1
1
in orange. We use (s13 , s1A , s1B , z3 , z4 , τ ) = (− 10, π , − π3 , 17 + 3i , 25 + 2i , i + 11
) and vary s12 . Notethat because we use the symmetric choice in homology bases, the homology intersection numbers are
related by complex conjugation.

which is non-trivial because s1B is constrained (or defined) by (3.19), and our complex
conjugation, by definition, does not act on Mandelstam variables s1A or s1j . For example,
one can satisfy this s1B = s1B condition by solving for η and η, if we first pick a complex
value of s1B . This is what we do for numerical checks. We will describe another way tosatisfy this condition in (6.25), in a way that works that uses the regular definition of complex
conjugation (section 6.4).
Then, the twisted Riemann bilinear relations are satisfied as in (6.4), except that weneed to explicitly use the conjugated integrals [γ 2j |φa ⟩ defined in (6.15). We note that evenin the most symmetric case, the complex Riemann-Wirtinger integrals evaluate to complex
numbers, due to the complex powers in |u(z1 )|2 . The double copy for complex Mandelstams
is plotted in figures 9 and 10. In figure 9, we compare the l.h.s. and r.h.s. of the double
copy (6.4) and find good numerical agreement. Each component entering this double copy
is presented in figure 10.
6.4

The double copy and modularity

In this section, we relax the reality condition on the Mandelstams (excluding s1B ) in way
that does not change how complex conjugation works. We keep s1B real and solve thecondition (6.6) for s1A instead of η. Here, we can think of fixing s1A as a particular limit of
the loop momentum. Moreover, η is now a free variable and can take its usual interpretation asa formal expansion variable. Interestingly, this approach is naturally associated to the doubly
periodic completions of the Kronecker-Eisenstein functions that one sees after performing
the integral over the loop momentum in string integrals and connects to the so-called elliptic
modular graph forms (eMGFs) [99].
Equation (6.6) ensures that the integrand of the complex Riemann-Wirtinger integral,is doubly-periodic in z1 . However, this integralÑab (z2 , z3 , . . . , zn , τ, s1A , s1j , η, η) := ⟨φ̄a |φb ⟩(6.24)is still a function of the punctures {z2 , z3 , . . . , zn } and the modulus τ but with no clear– 35 –JHEP07(2024)040

Figure 9. Plots for the quantity ⟨φ3 |φ3 ⟩ for n = 3 and (s12 , Im (s1A ), s1B , z3 , τ ) = (− 13 + i, 13 , 13 +2i
1
3 , i + 11 ) as Re(s1A ) varies. We compute the same quantity via the direct integration (6.2) and via
the double copy (6.4). Note that ⟨φ3 |φ3 ⟩ has both real and imaginary parts, because the integrandof (6.2) is not strictly positive for generic complex Mandelstam variables.

modular properties. Now, consider solving for s1A in equation (6.6):19
s1A =Im η −Pn

j=2 s1j Im zj

Im τ

.

(6.25)

Substituting this value of s1A into the complex Riemann-Wirtinger integral Ñab and adding
some (z1 -independent) multiplicative factors, we obtain an “improved” version of the complex
Riemann-Wirtinger integral:
Nkl (z2 , z3 , . . . , zn , τ, s1j , η, η) = Ñkl (z2 , z3 , . . . , zn , s1A , τ, s1j , η, η) (6.25)
× exp −2πiη
Z

=
M

n
X



(Im zj )2 

Im zl
Im zk
+ 2πiη
−
2πs1j
Im τ
Im τ
Im τ
j=2

v(z1 ) Ω(z1 − zl , η|τ ) Ω(z1 − zk , η|τ ) d2 z1 ,(6.26)

where Ω are the doubly periodic completions of the Kronecker-Eisenstein functions F andv(z1 ) = exp

X
n
j=2



s1j

(Im z1j )2
log |ϑ1 (z1j |τ )| − 2π
Im τ
2



.

(6.27)A derivation of equation (6.26) is included in appendix C.
We remark that this integral Nkl (z2 , z3 , . . . , zn , τ, s1j , η, η) is doubly periodic in the
punctures {z2 , z3 , . . . , zn }. Moreover, this version of the complex Riemann-Wirtinger integral
inherits the modular properties of the Kronecker-Eisenstein series Ω(z, η|τ ). In particular,
they are Jacobi forms of weight (1,1) and vanishing index:
Nkl (z2′ , z3′ , . . . , zn′ , τ ′ , s1j , η ′ , η ′ ) = |γτ + δ|2 Nkl (z2 , z3 , . . . , zn , τ, s1j , η, η) ,19

(6.28)

This is strikingly similar to how one integrates out the loop momentum in string theory, see (7.12) in [107].
This generalizes a formula of Ghazouani and Pirio, see equation (63) of [66].

– 36 –JHEP07(2024)040

Figure 10. Plots of every component entering (6.4) for n = 3 and (s12 , Im (s1A ), s1B , z3 , τ ) = (− 13 +1
i, 13 , 13 + 2i
3 , i + 11 ) as Re(s1A ) varies. Note that we use the same basis for homology and dual homology,
but the quantities above, as complex numbers, are not related by complex conjugation. Compare thiscase to the top subplot in figure 4.

forατ + βτ =,γτ + δ
′

zj′ =

zj
,
γτ + δηη =,γτ + δ

αβ
γ δ

′

!

∈ SL(2, Z) .

(6.29)

Additionally, we can consider the Laurent series of Nkl in η and η:
Nkl =

∞ X
∞
X

(a,b) a−1 b−1

Nkl

η

η

,

(6.30)

a=0 b=0
(a,b)

(a,b)

Nkl

(a,b)

= Nkl

(z2 , z3 , . . . , zn , τ, s1j ) is a Jacobi form of weight (b, a) and
a

(a,b)

(z2′ , z3′ , . . . , zn′ , τ ′ , s1j ) = (γτ + δ)b (γτ + δ) Nkl(z2 , z3 , . . . , zn , τ, s1j ) ,(a,b)

for an SL(2, Z) action as in (6.29). In fact, the component functions Nkl
the integrals:
(a,b)

Nkl

Z

(z2 , z3 , . . . , zn , τ, s1j , η, η) =M

(6.31)are given byv(z1 ) f (b) (z1 − zl |τ ) f (a) (z1 − zk |τ ) d2 z1 .

(6.32)

Note that via the twisted period relations (6.4), we can write these as a bilinear combinations
or as the “double copy” of Riemann-Wirtinger integrals up to some z1 -independent factors.
(0,0)
The simplest example of (6.31) is given by N2,2 (z2 , z3 , s12 ) for the (n = 3) RiemannWirtinger integral. This is the residue in η and η. On the r.h.s. of the double copy (6.4), η andη poles can only appear from the Kronecker-Eisenstein series, and no subleading contributionin η or η contribute. Thus, when imposing (6.25) we only require a leading (in η, η) version:s1A →−s12 Im z2 − s13 Im z3.
Im τ(6.33)

Choosing a symmetric choice of homology bases {γ2A , γ23 }, the intersection matrix is∨ ] [γ |γ ∨ ][γ2A |γ2A
2A 23∨ ] [γ |γ ∨ ][γ23 |γ2A
23 23!

2i sin(π(s1A − s12 )) eiπ(s1A −s12 )−e−iπ(s1A −s12 )0sin(πs1A )=sin(πs12 )(6.33)=sin(πs12 u3 )sin(πs12 )!

− 1)) eiπs12 (u3 −1)

2i sin(πs12 (u3−e−iπs12 (u3 −1)(6.33)!

0,(6.34)

where we have fixed z2 = 0, used momentum conservation, and introduced co-moving
coordinates z3 = v3 + u3 τ and v3 , u3 ∈ R (note that only u3 appears at this order). Now,
what used to be a matrix that only depended on the exponents s• depends on the punctures!
We also need the residue (in η) of the Riemann-Wirtinger integrals, under the condition (6.33). Interestingly, this lands precisely on the Riemann-Wirtinger integrals with η = 0:Z[γj |φa ⟩e2πis1A z1 u(z1 ) dz1

=
η −1

γj

(6.33)

Z(6.33)

e2πis12 u3 z1

=
γj



s12
ϑ1 (z1 |τ )dz1 = [γj |ζ1 ⟩ϑ1 (z1 − z3 |τ )(6.33)– 37 –

(6.35)JHEP07(2024)040

then every such Nkl
vanishing index:

where f (z, η) η−1 denotes the coefficient of η −1 of the Laurent expansion of f (z, η). Note,that the twisted cycle [γj | changes with (6.33)20 and|φa ⟩= F (z − za , η|τ ) dz1
η −1= dz1 = |ζ1 ⟩ ,(6.36)

η −1which are twisted 1-forms of different local system (see appendix E.4).
Putting everything together, according to (6.4) and (6.26) the leading term of themodular double copy is given by:(Im z3 )2
2πs12
Im τ

!

sin(πs12 ) isin(πs12 u3 ) 2!

e−iπs12 (u3 −1)[γ2A |ζ1 ⟩× [γ2A |ζ1 ⟩ [γ23 |ζ1 ⟩ ·
·
iπs
(u
−1)
−e 12 32i sin(πs12 (u3 − 1))[γ23 |ζ1 ⟩0= exp(2πs12 u23 Im τ )!

sin(πs12 )Im eiπs12 (u3 −1) [γ23 |ζ1 ⟩[γ2A |ζ1 ⟩sin(πs12 u3 )− sin(πs12 (u3 − 1))[γ23 |ζ1 ⟩[γ23 |ζ1 ⟩ .

(6.37)

z3
Note that in (6.37) we have used momentum conservation, the comoving coordinate u3 = Im
Im τ ,
fixed z2 = 0, and every instance of [γ23 |ζ1 ⟩ should be read as [γ23 |ζ1 ⟩ (6.33) . In the r.h.s.
(0,0)

of (6.37) the reality of the integral Nab is manifest. Also note that the dependence on
the punctures za and zb drops out at leading order in (η, η̄). This equation (without thefirst exponential factor) was first written by Ghazouani and Pirio in the context of Veech
(1,0)
(1,1)
volumes [66]. Formulas for the subleading contributions Nab and Nab are provided inappendix D.
We remark that the improved complex integrals Nkl in (6.26) coincide with generating
series of eMGFs, when one imposes momentum conservation on the latter, and rescales the ηin Ω(z1 − zj , η, |τ ) in the integrand of Nkl ; see e.g. Equation (4.1) of [99]. The α′ -expansionof (6.37) generates double copy representations of eMGFs by sums of products of meromorphic
quantities and complex conjugates. Examples of such representation can be found in [108] at
depth one in terms of eMPLs and [103] in terms of iterated τ -integrals.

7

Towards integrating two punctures

In this section, we speculate on how to generalize the Riemann-Writinger integrals to thecase where two punctures are integrated. Understanding this is an important step towards
treating real one-loop string integrals within the formalism of twisted (co)homology.
To make the discussion concrete consider the following natural generalization of the
Riemann-Wirtinger integral
Z

Z

dz1

dz2 u F (z1a , η1 )F (z2b , η2 ) ,

20

(7.1)

Here, [γj | changes in the sense that the Ľω,η -valued coefficients evaluated with the replacement (6.33),
but the topological cycle remains the same.

– 38 –JHEP07(2024)040(0,0)
Nab = expwhereu = e2πi(s1A z1 +s2A z2 )

n
Y
s1j

n
Y
s2k

j=2

k=3

ϑ1 (z1j )ϑ1 (z2k ) ,(7.2)

is the z1 - and z2 -dependent parts of the n-point Koba-Nielsen. To keep the discussion simple,
we assume that a, b ≥ 3 and a ̸= b. This twist has the following A- and B-cycle monodromies
uzk →zk +1= e2πiskA ,uPn
uzk →zk +τ
2πi(skA τ + j̸=k s1j zj )
=e
u

for

k = 1, 2 .

(7.3)

L2 (s1A , s1i̸=1 , s1B , s2A , s2i̸=2 , s2B ) ∼ Lω1 ,η1 (s1A , s1i̸=1 , s1B ) ⊗ Lω2 ,η2 (s2A , s2i̸=2 , s2B ) , (7.4)
whereη1 = s1A τ + s12 z2 +
η2 = s2A τ + s12 z1 +

n
X
j=3
n
X

s1j zj − s1B ,(7.5)

s1j zj − s2B .

(7.6)

j=3

To see if this is sensible, we compute the monodromy of the z1 A-cycle
b1 F (z2b , η2 )]|z1 →z1 +1
u F (z1a , η1 )F (z2b , η2 )|z1 →z1 +1 = [u1 F (z1a , η1 )]|z1 →z1 +1 × [u(7.7)

b1 is z1 -independent part of the twist. We
where u1 is the z1 -dependent part of the twist and uknow what happens to the first term — it is an overall phase e2πis1A . In the second term,
substituting z1 → z1 + 1 into (7.6) leads to the shift η2 → η2 + s12 indicating a change inthe second local system in the tensor product L2
b1 F (z2b , η2 + s12 )]u F (z1a , η1 )F (z2b , η2 )|z1 →z1 +1 = [u1 F (z1a , η1 )e2πis1A ] × [u= e2πis1A [u F (z1a , η1 )F (z2b , η2 )]s2B →s′

2B

(7.8)

where s′2B = s2B − s12 . So in the end, we get the phase factor e2πis1A and a shift of thelocal system
L2 → L′2 = L2 (s1A , s1i̸=1 , s1B , s2A , s2i̸=2 , s2B − s12 ) .

(7.9)

There is an analogous change of local system for the z1 B-cycle. The only difference isthat we get phase factor e2πis2B and move to a local system with s′2A = s2A + s12 . Clearly,
analogous statements hold for the z2 A- and B-cycles. In fact, one can argue topologically forthe need of more structure (e.g. “shifts of local systems”). Recall that the local system is a
representation of the fundamental group of the underlying manifold. In most situations, thelocal system that enters the twisted (co)homology is of rank 1 and comes from the constantmultiplicative monodromies of the twist. Next, consider the twist u = u(z1 , z2 ; s1A , s2A , s12 )as defined in (7.2) but where we have highlighted the dependence of u on z1 , z2 ; s1A , s2A , s12 .

– 39 –JHEP07(2024)040

A natural guess for the corresponding local system is the tensor product of the local system
considered in the previous sections of this paper

j

i

i

−1
Figure 11. Pictorial proof of the identity Bi A−1
j Bi Aj = Mij on the fundamental group of n ≥ 3punctures on the torus (topologically: a paralellogram with opposite sides identified). The blackarrows denote homotopies. The first picture shows, in red, B-cycle monodromies for the puncture zi
and, in green, A-cycle monodromies for the puncture zj . Our conventions for multiplication are thatthe product AB is first doing A and then doing B, and if they intersect we would write B underneath
A. Note that the other punctures are not pictured.

More abstractly, the monodromies (7.3) arise from the action of the monodromy operators±1
A±1
that perform the displacements of a function along the A-cycle or B-cycle ofj , Bjthe corresponding puncture zj :A±1
j : f (zj ) 7→ f (zj ± 1)

Bj±1 : f (zj ) 7→ f (zj ± τ ) .

and(7.10)

Summarizing, the monodromies of the twist u(z1 , z2 ; s1A , s2A , s12 ) are:±2πisjA
A±1
u(z1 , z2 ; s1A , s2A , s12 ) j = 1, 2
j : u(z1 , z2 ; s1A , s2A , s12 ) 7→ e

(7.11)

B1±1 : u(z1 , z2 ; s1A , s2A , s12 ) 7→ e±2πis̃1B u(z1 , z2 ; s1A , s2A ± s12 , s12 ) ,(7.12)

B2±1 : u(z1 , z2 ; s1A , s2A , s12 ) 7→ e±2πis̃2B u(z1 , z2 ; s1A ± s12 , s2A , s12 )(7.13)

where the factors e±2πis̃jB are independent of z1 and z2 . Explicitly,s̃1B = s1A τ +s̃2B = s2A τ +nXj̸=1,2nX

s1j zj(7.14)

s2j zj ,(7.15)

j̸=1,2and are not to be confused with the skB . Since these phases are independent of the integration
variables, they can be pulled outside of the integral (7.1). We comment that the shift in
s1A in s12 caused by the monodromy B2 above physically corresponds to the shift in loopmomentum of ℓ → ℓ + k2 , in massless kinematics.

– 40 –JHEP07(2024)040jNow, in choosing the factors of the twist u(z1 , z2 ; s1A , s2A , s12 ) above, the action of themonodromy that takes the puncture z1 clockwise around z2 , M12 on the twist is e−2πis12 . It
turns out that the generator M12 is not independent from the generators Aj , Bj . Instead,these generators actually satisfy the identity:21−1
M12 = A−1
1 B2 A1 B2 .

(7.16)We showcase an identity of this type in figure 11. This can be explicitly verified by actingwith the r.h.s. of the above identity on the twist(7.17)

which aligns with the topological expectation.
The crucial point above is the following: if it is possible to construct a 1-dimensional
representation (or rank-1 local system) of the fundamental group of the configuration spaceof the punctured torus, any such representation ρ would be commutative and satisfy−1
ρ(A−1
1 B 2 A1 B 1 ) = 1 .

(7.18)

This, in turn, means that the e2πis12 = 1. That is, s12 ∈ Z instead of being generic s12 . Thisis potentially an interesting starting point for studying Riemann-Wirtinger integrals with
two integrations. Otherwise, perhaps one should work with higher-rank local systems.
While the shifting of local systems is uncomfortable at first, it is actually known how
to map a basis in one local system to a basis in the other using the monodromy relationsof [1]. This perspective seems promising and could potentially be an important step towards
treating actual one-loop string integrals! We leave the investigation of such integrals andtheir potential double copy to future work.
A similar difficulty arises when considering the following, even simpler integral:Zγϑs112 (z1 |τ ) e2πis1A z1 F (z1 , η|τ ) dz1 ,(7.19)

which looks like a Riemann-Wirtinger integral with n = 2 punctures. However, we needto relax the momentum conservation condition so that s12 is non-zero in order to get a
non-trivial integral. Then the following equation becomes relevant:−1s12
A1 B1 A−1
7→ e−2πis12 [ϑ1 (z12 |τ )]s12 .
1 B1 : [ϑ1 (z12 |τ )](7.20)

Thus, any rank-1 local system that describes the integral (7.20) must have s12 ∈ Z − {0}.
Interestingly, Stieberger [14] recently found two instances of a double copy formula that work
for integrals of the form (7.19) for s12 = 1 at generic complex structure (τ ). It would beinteresting to see if the constructions therein extend for other s12 ∈ Z≥1 .
21

See corollary 5.1 and figure 2a of [109]. We identify the Ai here with ρi there, and Bj here with τithere. To be precise, different combinations give different “’kinds” of Mij : going over or under the other fixedpunctures.

– 41 –JHEP07(2024)040−1−2πis12A−1u(z1 , z2 ; s1A , s1B , s12 ) ,1 ◦ B2 ◦ A1 ◦ B2 : u(z1 , z2 ; s1A , s1B , s12 ) 7→ e

8Future directions and discussion– 42 –JHEP07(2024)040

Riemann-Wirtinger integrals serve as a simple toy model for one-loop string integrands (seesection 3). In particular, we have full mathematical control over their twisted (co)homology.
This facilitates the computation and study of their differential equations, monodromy relations
and, most importantly, leads to a double copy formula for a complex variant of the RiemannWirtinger integrals.
In this work, we reviewed the construction of the twisted cohomology and homology of
the Riemann-Wirtinger family of integrals in sections 4 and 5. In particular, we provide
several convenient bases of cohomology and homology. We then introduce the intersection
number and intersection index — inner products on cohomology and homology — since
they underpin our double copy construction. As a simple example for the utility of theintersection number and intersection index we verify distinguished linear relations between
elements of (co)homology in the main text.
Then, in section 6, we combine the ingredients of sections 4 and 5 to construct the double
copy of complex Riemann-Wirtinger integrals. We provide numerical checks of this double
copy for real Mandelstams (section 6.2) and complex Mandelstams (section 6.3). Then,in section 6.4, we introduce a single-valued and modular Riemann-Wirtinger integral forwhich we can verify the double copy order by order in the now free parameters η and η̄.
In particular, this provides a double copy representation for some generating functions ofeMGFs [99]. Finally, in section 7, we speculate on how the analysis of this work could be
extended to Riemann-Wirtinger integrals where more than one puncture is integrated.
While Riemann-Wirtinger integrals capture many of the gross features of one-loop string
integrands, the constraint (3.19) entangling the η-variable with the moduli is difficult tointerpret. In alternative approaches in mathematics and string theory [110, 111], η is anindependent parameter organizing different integrands of one-loop string amplitudes intoa generating series. However, the formalism of twisted (co)homology seems to require theconstraint (3.19) hinting that perhaps a more robust theory of (co)homology is needed todescribe the usual generating series of one-loop string integrands. Within the (co)homology
framework at η = 0, the double copy of the constant function, seen in four-point open- andclosed- superstring amplitudes [112] and n-point tachyonic amplitudes [113], was tested both
numerically and analytically in sections 6.2, 6.3 and 6.4. Moreover, we have also numericallytested the double copy of some single-valued combination of g (k) ’s for η = 0. Given that the
gap between Riemann-Wirtinger integrals and string amplitudes is easier to bridge at η = 0,
this case is a particularly promising starting point to deduce double copy formulae of string
amplitudes from those of the Riemann-Wirtinger integral studied in this work.
Perhaps the most direct connection to the usual string integral picture is the modular
double copy of section 6.4 and its relation to generating functions of eMGFs. Here, RiemannWirtinger integrals are augmented with additional factors independent of the integration
variable, converting the Kronecker-Eisenstein functions in the integrand into their doubly
periodic completion. The resulting double copy corresponds to certain generating functions ofeMGFs. It would be interesting to find an expansion scheme of Riemann-Wirtinger integralsthat generates expansions for eMGFs in terms of eMPLs and their complex conjugates.
Additionally, the form of the modular double copy in 6.4 is interesting because we havesolved (3.19) for the variable s1A corresponding to the loop momentum in chiral splitting.
Thus, to give a string-integral interpretation for this, the closed-string integral after loopmomentum integration can be written as a double copy of open-string integrals where theloop momentum, s1A , has a specific value.

Generalizations of Riemann Wirtinger integrals to higher genus have been pioneeredin [118, 119] with a discussion of cohomology bases. It would be interesting to extend thehomology analysis of [1, 2, 66] and the double-copy formulae of this work to the highergenus setting of [118, 119]. Twisted (co)homology methods and double-copy approaches
at higher genus will have a wealth of implications of string amplitudes, Feynman integrals
and mathematics.
Finally, it will be rewarding to formulate the genus-one KLT relations of Stieberger [13, 14]in the language of twisted (co)homology investigate alternative derivations from intersection
indices. A possible starting point for this would be to give a twisted-(co)homology understanding of the relations for mixed open- and closed-string amplitudes at genus zero [120, 121]
and at genus one [11], which play a key role in the KLT relations of [13, 14]. Since thegenus-one KLT relations therein work for multiple integrated punctures, these works must
process key insights for Riemann-Wirtinger integral with multiple integrated punctures.
It would also be interesting to obtain analytic expressions for RW integrals in termsof eMPLs. This would facilitate an analytic double copy and further our understanding of
single-valued eMPLs. While a complete story is missing, progress towards this goal has beenpresented in appendix E. Here, we derive the differential equations satisfied by RiemannWirtinger integrals for η ̸= 0 (section E.1) and η = 0 (appendix E.4). In particular, we verify
the Gauss-Manin connection of [2] (η ̸= 0) and [66] (η = 0 and n = 3) using independent
methods. In section E.2, we solve the η ̸= 0 differential equations to O(α′ ) explicitly interms of elliptic polylogarithms. Boundary values for the bases of twisted contours presented
in section 5.1 are also provided in section E.3. In particular, the form and solution of theη ̸= 0 DEQ is strikingly similar to that of actual string integrals [116]. The η = 0 differential
equations and basis are in some sense more complicated due to the presence of unavoidable
doubles poles. However, while unfamiliar to most iterated integral practitioners, it is possible
to make sense of iterated integrals whose kernels have higher order poles [122, 123] and isin some sense expected at higher genus.

– 43 –JHEP07(2024)040

Of course, it would also be interesting to extend the analysis here to the case wheremore than one puncture is integrated as discussed in section 7. In particular, the (n − 1)dimensional Riemann-Wirtinger integral should be a straightforward generalization of thetwo-dimensional Riemann-Wirtinger integral. This generalization would be a huge step
towards understanding string integrands in the formalism of twisted (co)homology: placing
homology — i.e. monodromy relations and counting numbers of independent open-string
amplitudes [5, 7–9] — and cohomology — i.e. finding bases of integrands for differential
equations [110, 111, 114–116] — on the same ground. This is particularly interesting because
in the string-theory setup, the homology counting is most readily done when the integrands
are meromorphic (i.e. in the chiral splitting formalism), while the systematic counting of
cohomology representatives is more readily done in the doubly-periodic setting [117].

Acknowledgments

A

Local systems

In this appendix, we give a perhaps more geometric description of a local system. This isincluded for completeness only and the interested reader can find more details in [124].
First, we define the monodromy of ω on γ where ω is a closed 1-form on M and[γ] ∈ π1 (M ) is a loop in the first homotopy group. Fixing ω provides a map π1 → C
Rby [γ] 7→ γ ω. In fact, this is a homomorphism between groups. Our goal is to use thishomomorphism to define a representation of the fundamental group. To get a representationof the fundamental group, we compose the above homomorphism with the exponential map
R
exp : C → C∗ = GL(1, C) by [γ] 7→ exp γ ω. Next, we use the above to define a complex
line bundle known as a local system.
Next, let M̃ be the universal cover of M . By the definition of a universal cover, thefundamental group π1 (M ) has a free action on M̃ (all stabalizers of π1 are trivial) such thatthe quotient of the action is M . This means that M̃ is a principal π1 (M )-bundle over M .
Since we have a principal π1 (M )-bundle and a representation π1 → C∗ , we can construct
the associated vector bundle (M̃ × C)/π1 (M ) → M . In this example, C∗ = GL(1; C) is thegroup of 1 × 1 matrices so the associated vector bundle is a line bundle. This line bundleLω = (M̃ × C)/π1 (M ) → M is called the local system. Note that subscript emphasises thatthe construction depends on the choice of ω.
While this line bundle is actually trivial, there are new natural smooth structures, suchas flat connections, that can be defined. In particular, there exists a unique flat connection∇ω = d + d log(u) on Lω that pulls back to d on the trivial line bundle M̃ × C. The factthat Lω is trivial means that there exists a global trivialization and implies that we can workwith complex valued forms rather than Lω -valued forms. This only works for cohomology
and we must use Lω -valued cycles for homology.

B

Computing intersection indices

We will now demonstrate with the aid of a few simple examples, the calculation of theintersection indices mentioned in section 5.2, for a more detailed derivation of (5.19) thereader is advised to revisit the original works on this topic [1, 66, 104]. Note that for the– 44 –JHEP07(2024)040

The authors would like to thank Oliver Schlotterer for many interesting discussions andparticipation in the early stages of this work. The authors would also like to thank Francis
Brown, Eduardo Casali, Lorenz Eberhardt, Yoshiaki Goto, Saiei-Jaeyeong Matsubara-Heo,
Sebastian Mizera, Franziska Porkert, Giulio Salvatori, Marcus Spradlin, Piotr Tourkine,
Anastasia Volovich, and Federico Zerbini for useful discussions. AP would like to thank theinstitute for advanced study for its hospitality and the organizers of the String Amplitudes
at Finite α′ workshop where this work was initiated. This work was supported in part by
the US Department of Energy under contract DESC0010010 Task F (RB, AP, LR) andby Galkin Foundation Fellowship (LR). The research of CR is supported by the European
Research Council under ERC-STG-804286 UNISCAMP.

rest of this appendix, in the figures, the twisted cycles will be denoted by dark blue andthe dual will be in green, consistent with the convention stated in section 5.2 for topologicalintersection indices. Moreover, the intersection points will be colored red and the branch
cut B on the torus will be denoted by a red dashed line.
As a simple example of calculating an intersection index, we start by evaluating theintersection index of γ2j with itself, that is [γ2j |γ̌2j ]. As seen from figure 12, the twisted
cycle intersects its dual exactly at three points (x1 , x2 , x3 ). Further note that, the way one
defines the intersection of the two cycles, is such that in doing so the dual twisted cycle does
not cross the branch cut B. Using the conventions defined in section 5 for the topologicalintersection indices and referring to figure 12 we can easily read these off at the three points as
top
top[m0 |ℓ2j ]top
x1 = −1, [ℓ2j |ℓ2j ]x2 = 1 and [Sj |ℓ2j ]x3 = 1. Then using the master formula for theintersection index of twisted cycles and their duals (5.8), and the expressions for regularised
cycle Reg[γ2j ], we derive the desired intersection index as follows:11
top[γ2j |γ̌2j ] = [m0 |ℓ2j ]top
+ [ℓ2j |ℓ̌2j ]topx1 2πis12
x2 + [Sj |ℓ̌2j ]x3 2πis1j
e
−1
e
−1
1 − e2πi(s12 +s1j )=,(1 − e2πis12 )(1 − e2πis1j )(B.1)

which agrees with (5.19). Notice in the last example, we did not pick up any phase dueto the discontinuity around the branch cut B, this is because we chose to intersect thetwisted cycles in such a way that we avoid crossing the branch cut. However, we could
have very well considered self-intersecting the twisted cycle such that one ends up withfewer intersection points, but the last term in (B.1) picks up an overall phase due to themonodromy for the Koba-Nielsen twist (3.2) around B. The next example that we are going
to consider will demonstrate how to account for the monodromy properties of the twist
while performing such calculations.
Let us now evaluate the [γ2B |γ̌2A ] intersection index. Figure 13 describes the topologicalintersection of the two twisted cycles, in particular we see that the twisted cycles intersect(2)(2)
exactly at two points (P0 , P2 ). Any point on the line segment l2A that lies just outside
the fundamental domain, is related to a point on the circular arc m2 that lies just inside
up to a phase e2πis1A , corresponding to the A-cycle monodromy. As a result this time one– 45 –JHEP07(2024)040

Figure 12. Self intersection of the γ2j cycles. The twisted-cycle intersection is relevant for the[γ2j |γ̌2j ] intersection index. The intersection occurs at three points (x1 , x2 , x3 ) as depicted above in
red. The topological intersection indices at these points are given on the right for reference. The
branch cut B is denoted as a red dashed curve starting at z2 and ending at zj .

(2)picks up a phase at P2 , as for the twists in equation (5.8) we have
(2)

(2)

2πis1A z −2πis1A (z+1)
um1 (P2 )u−1e
= e−2πis1A
γ̌2A (P2 ) = e(B.2)

note that in the above equation the Kronecker theta functions cancel each other out due
to the SL(2, Z) modular transformation property ϑ1 (z + 1) = −ϑ1 (z) and the momentumP
conservation for the Mandelstam variables nk=1 s1k = 0. Then referring to figure 12, wehave that the topological intersection indices at these points are [m0 |ℓ̌2A ]top(2) = −1 andP0[m1 |ℓ̌2A ]top(2) = 1, thus again using equations (5.12) and (5.13) and plugging them into theP2formula (5.8) one gets
2πi(−s1B +s12 )1 − e−2πis1B
top −2πis1A 1 − e
+
[m
|
ℓ̌
]
e
1 2A (2)P0P2e2πis12 − 1e2πis12 − 1
1 − e−2πis1B − e−2πis1A + e2πi(s12 −s1B −s1A )=1 − e2πis12[γ2B |γ̌2A ] = [m0 |ℓ̌2A ]top
(2)

(B.3)

in agreement with (5.19).
We further explore this notion of picking up non-trivial phase factors due to different
branch choices at a given simplex and its dual, with a highly non-trivial example of intersection
index of the twisted B-cycle with itself i.e. [γ2B |γ̌2B ] (see figure 14 for reference). Inferringfrom the figure 14, we see that the dual cycle intersects twisted cycle exactly at four points(x1 , x2 , y1 , y2 ). At each of these points, using the convention introduced earlier in this section,
the topological intersection indices aref
top[mi1 |ℓ̌2B ]top
x1 = −1 = [m1 |ℓ̌2B ]y1

and

f
top
[mi2 |ℓ̌2B ]top
x2 = 1 = [m2 |ℓ̌2B ]y2(B.4)

where we have that mi refers to the initial arc circling z2 and mf refers to the final arc circlingz2 + τ . Unlike before, there is now a need to distinguish between the two arcs {mi2 , mf2 } onC/Λτ , the branch choice u · F for each of these simplices is different, as they lie on different
domains of the torus. However, much like the previous example, the effect that this branchchoice has is that it leads to non-trivial monodromy phase factors. The first non-trivial– 46 –JHEP07(2024)040

Figure 13. Intersection of the A-cycle and B-cycle. The twisted-cycle intersection is relevant for the(2)(2)[γ2B |γ̌2A ] intersection index. The intersection occurs at two points (P0 , P1 ) as depicted above in red.

contribution comes from the intersection between the arc mi2 on the twisted cycle Reg[γ2B ]and the dual cycle γ̌2B at x2 . This leads toumi (x2 )F (x2 )(uγ̌2B (x2 )F (x2 ))−12

= u(x2 ; s1A , {s1k })F (x2 ; s1A , {s1k }) (u(x2 − τ ; −s1A , {−s1k })F (x2 − τ ; −s1A , {−s1k }))−1= e2πis1B ,(B.5)

where in the last line we used the B-cycle monodromy relation for the twist given by (3.6) andthe same for the Kronecker-Eisenstein generating function given by (3.12). Another non-trivial
contribution comes from the branch choice at the intersection between mf2 ⊂ Reg[γ2B ] andγ̌2B at y2 . Similar arguments hold as before, and one again picks up the same phase factor
umf (y2 )F (y2 )(uγ̌2B (y2 )F (y2 ))−12

= u(y2 + τ ; s1A , {s1k })F (y2 + τ ; s1A , {s1k }) (u(y2 ; −s1A , {−s1k })F (y2 ; −s1A , {−s1k }))−1= e2πis1B ,(B.6)

however, this time it is due to the monodromy around the twisted cycle as opposed to the caseat x2 , where one picks up a phase around the dual-twisted cycle. Finally, using the expressions
for the regularised cycles (5.12) and (5.13), the topological intersection indices (B.4), andthe phases (B.5) and (B.6) one concludes after a few lines of algebra
2πis1B1i
top e[γ2B |γ̌2B ] = [mi1 |ℓ̌2B ]top
+
[m
|
ℓ̌
]
x1 2πis12
2 2B x2 2πis12
e
−1
e
−1
2πis12ee2πis12− [mf1 |ℓ̌2B ]tope−2πis1B − [mf2 |ℓ̌2B ]top
y1 2πis12
y2 2πis12
e
−1
e
−1
2πis
2πis
2πis
12
1B
1B
(e
− 1)(e−e)=−,e2πis1B (e2πis12 − 1)

as required by (5.19).

– 47 –(B.7)

JHEP07(2024)040

Figure 14. Self intersection of the B-cycles. The twisted-cycle intersection is relevant for the [γ2A |γ̌2B ]intersection index. The intersection occurs at four points (x1 , x2 , y1 , y2 ) as depicted above in red.

C

Solving for s1A

In this section, we provide a derivation of (6.26).
We start by inserting (6.25) into es1A (z1 −z1 ) :e

2πis1A (z1 −z1 )

=e

Im η
2πi Im(z1 −z1 )τ=e

2πi Im τ1 (η−η)


n

2πi X
exp −
s1j Im zj (z1 − z1 )Im τ j=2




Im z

exp 4πIm z1nXs1j Im zj .

(C.1)

j=2

Im z1

F (z1 − zl , η|τ )F (z1 − zk , η|τ )e2πi Im τ (η−η)= F (z1 − zl , η|τ )e2πiηIm (z1 −zl +zl )Im τF (z1 − zk , η|τ )e−2πiη

Im (z1 −zk +zk )
Im τ

Im zl
Im zk
− 2πiη
,
Im τ
Im τ


= Ω(z1 − zl , η|τ )Ω(z1 − zk , η|τ ) exp 2πiη

(C.2)

where we have obtained the definition of the doubly-periodic non-holomorphic Eisenstein series.
Next, look at the second factor in (C.1) and the theta functions inside the complex
Riemann-Wirtinger integral (6.2):
Y
n





|ϑ1 (z1 − zj |τ )|2s1j exp 4π

j=2

= exp

X
n

n
Im z1 X
s1j Im zj
Im τ j=2



s1j log |ϑ1 (z1 − zj |τ )|2 exp 2π

j=2


n
X
s1j
j=2

+ 2π
= exp

X
n





s1j log |ϑ1 (z1 − zj |τ )|2 exp − 2π

j=2

= exp

X
n
j=2

Im τ

s1j

− (Im z1 )2 + 2Im z1 Im zj − (Im zj )2

n
X
s1j

j=2
n
X

Im τ

s1j

j=2





(Im z1 )2 + 2π

j=2

Im τ

(Im zj )2nX(Im (z1 − zj ))2(Im zj )2
+ 2π
s1j
Im τ
Im τ
j=2

(Im (z1 − zj ))2
log |ϑ1 (z1 − zj |τ )| − 2π
Im τ
2

n
X
s1j



n
X





(Im zj )2
exp 2πs1j, (C.3)
Im τ
j=2




where we note that we completed the square to form (Im (z1 − zj ))2 and that the termPns1j2
j=2 Im τ (Im z1 ) vanishes due to momentum conservation.
Gathering the results of (C.2) and (C.3), we obtain the following formula for the complex
Riemann-Wirtinger integral:⟨φ̄k |φl ⟩ = exp 2πiη
×

Z

exp
M

n
X
Im zl
Im zk
(Im zj )2 
− 2πiη
+ 2π
s1j
Im τ
Im τ
Im τ
j=2

X
n
j=2



s1j

(Im (z1 − zj ))2
log |ϑ1 (z1 − zj |τ )| − 2πIm τ
2× Ω(z1 − zl , η|τ ) Ω(z1 − zk , η|τ ) d2 z1 .
Equation (6.26) follows immediately from (C.4).

– 48 –(C.4)


JHEP07(2024)040

Now, let’s look at the first factor in (C.1) with the Kronecker-Eisenstein series in the
integrand of (6.2):

DThe modular double copy: subleading terms
(0,1)

(1,1)

We will now write the modular forms N2,2 and N2,2 that an be obtained from RiemannWirtinger integrals when (n = 3). For concreteness, we will focus on the case of real
Mandelstam variable s12 and fixing z2 = 0. We will also use the comoving coordinatez3
u3 = ImIm τ .
We will need to introduce some notation for the expansion of the Riemann-Wirtinger
integral [γ|φj ⟩ and its complex-conjugate as follows:[γ|φj ⟩ (6.33) =∞ X∞
X
n=−1 m=0∞ X∞
X

η n η m [γ|φj ⟩(m,n) ,η m η n [γ|φj ⟩

(m,n)

(D.1)

,

(D.2)

n=0 m=−1where we need to include powers of both η and η because of (6.33). Because of the realityof s12 we have[γ|φj ⟩(m,n) = [γ|φj ⟩
(0,1)

(n,m)

.

(D.3)

(1,1)

Thus, we will be able to write N2,2 and N2,2 in terms of only the Riemann-Wirtinger
integrals [γ|φ2 ⟩ . From this point onward, we omit any explicit reference to (6.33), whichis understood to hold for the rests of this appendix.
(0,1)
The component of the modular double copy, N22 , is given by:
(0,1)

N22

i
2
csc(πs12 u3 ) sin(πs12 )e2πs12 u3 Im τ
(D.4)
2

 iπs12 (u3 −1) 1πe2πis12(0,−1)(0,−1) 
×
e
[γ
,
φ
⟩
−
[γ
,
φ
⟩
[γ23 , φ2 ⟩(0,−1)
23
2
2
2A
Im τ
e2πis12 u3 − 1=+ [γ23 , φ2 ⟩0+ (Im τ ) e(0,−1)−iπs12 (u3 −1)[γ2A , φ2 ⟩(0,−1) − [γ23 , φ2 ⟩(0,−1)− [γ23 , φ2 ⟩(0,0) [γ23 , φ2 ⟩(0,−1) + [γ2A , φ2 ⟩(0,0) [γ23 , φ2 ⟩(0,−1)− [γ23 , φ2 ⟩(0,−1) [γ23 , φ2 ⟩(1,−1) + [γ2A , φ2 ⟩(0,−1) [γ23 , φ2 ⟩(1,−1)+ e2πis12 (u3 −1) [γ23 , φ2 ⟩(0,0) [γ23 , φ2 ⟩(−1,0) − [γ2A , φ2 ⟩(−1,0)+e

2πis12 (u3 −1)[γ23 , φ2 ⟩(0,−1)

[γ23 , φ2 ⟩(1,−1) − [γ2A , φ2 ⟩(1,−1)





.

(0,1)

Note that in the formula for N22 above we have collected the powers of Im τ that appear(0,1)
from the double copy, other than the power in the exponential in the first line. Like N22(1,1)
above, we write N22 as a sum of three terms, by isolating negative powers of Im τ :
(1,1)

N22

i
1
1
2
csc(πs12 u3 ) sin(πs12 )e2πs12 u3 Im τ
N (2) +
N (1) + (Im τ )0 N (0) , (D.5)
2(Im τ )2Im τ=

– 49 –

JHEP07(2024)040[γ|φj ⟩ (6.33) =and where N (2) , N (1) , N (0) are given by:
N (2) = −iπ 2 eiπs12 (u3 −1)
2πis12(0,−1)(0,−1) cot(πs
)
e
[γ
,
φ
⟩
−
[γ
,
φ
⟩
[γ23 , φ2 ⟩(0,−1)
12u
23
2
2
2A
3
e2πis12 u3 − 1
× [γ23 , φ2 ⟩(0,−1) − [γ23 , φ2 ⟩(0,−1) + [γ2A , φ2 ⟩(0,−1)
N (1) =πeπis12 (u3 −1) 2πis12ee2πis12 u3 − 1








,

(D.6)

− [γ2A , φ2 ⟩(1,−1) + [γ2A , φ2 ⟩(0,0) [γ23 , φ2 ⟩(0,−1)(0,−1)− [γ2A , φ2 ⟩(0,−1) (D.7)[γ23 , φ2 ⟩(1,−1) − −[γ23 , φ2 ⟩(0,0)+ [γ23 , φ2 ⟩(1,−1) (e2πis12 − 1)[γ23 , φ2 ⟩(0,−1) + [γ2A , φ2 ⟩(0,−1)− [γ23 , φ2 ⟩(0,0) (e2πis12 − 1)[γ23 , φ2 ⟩(0,−1) + [γ2A , φ2 ⟩(0,−1)









+ [γ23 , φ2 ⟩(0,−1) [γ23 , φ2 ⟩(1,−1) − [γ23 , φ2 ⟩(0,0)− [γ2A , φ2 ⟩(1,−1) + [γ2A , φ2 ⟩(0,0)
N

(0)

=e

−πis12 (u3 −1)







,

− [γ23 , φ2 ⟩(1,0) [γ23 , φ2 ⟩(0,−1) + [γ2A , φ2 ⟩(1,0) [γ23 , φ2 ⟩(0,−1)(D.8)− [γ23 , φ2 ⟩(1,−1) [γ23 , φ2 ⟩(1,−1) + [γ2A , φ2 ⟩(1,−1) [γ23 , φ2 ⟩(1,−1)− [γ23 , φ2 ⟩(0,0) [γ23 , φ2 ⟩(0,0) + [γ2A , φ2 ⟩(0,0) [γ23 , φ2 ⟩(0,−1)− [γ23 , φ2 ⟩(0,−1) [γ23 , φ2 ⟩(1,0) + [γ2A , φ2 ⟩(0,−1) [γ23 , φ2 ⟩(1,0)+e

2πis12 (u3 −1)[γ23 , φ2 ⟩(1,0) [γ23 , φ2 ⟩(0,−1) − [γ2A , φ2 ⟩(0,−1)+ [γ23 , φ2 ⟩(1,−1) [γ23 , φ2 ⟩(1,−1) − [γ2A , φ2 ⟩(1,−1)+ [γ23 , φ2 ⟩(0,0) [γ23 , φ2 ⟩(0,0) − [γ2A , φ2 ⟩(0,0)

+ [γ23 , φ2 ⟩(0,−1) [γ23 , φ2 ⟩(1,0) − [γ2A , φ2 ⟩(1,0)





.

As a supplementary material, we attach a Mathematica notebook with the formulas for
(1,0)
(1,1)
N22 and N22 .

E

Towards the analytic Riemann-Wirtinger integral in terms of eMPLs

In this appendix, we compute the differential equations satisfied by the Riemann-Wirtinger
integrals in section E.1. In section E.2, we solve the differential equations to order O(α′ ) interms of eMPLs for generic boundary values. Boundary values for the contours introduced in
section 5.1 are derived in E.3. Lastly, in appendix E.4, we describe the cohomology in theη → 0 limit and discuss some of new features of the corresponding differential equations.

– 50 –JHEP07(2024)040− [γ23 , φ2 ⟩



E.1

Differential equations for the φ-basis

Using intersection numbers, we compute the Gauss-Manin differential equation satisfied
by the Riemann-Wirtinger integrals where we differentiate with respect to the external
un-integrated punctures zi≥2 .
The differential operators in the external punctures must preserve the monodromy
structure of the local system in order for the differential equations to close. For f dz1 ∈ Ω1η ,the Gauss-Manin differential operators that preserve monodromies are [2]:∇ext,a f dz1 := (∂a f ) dz1 + (ωa f ) dz1 + s1a (∂η f ) dz1

(E.1)
(E.2)

Here, f is a meromorphic function on M with B-cycle monodromy e−2πiη . Understanding
the precise form of the above differential operators requires understanding how varying theπpunctures and moduli on the total space M → M → B = {(z2 , . . . , zn , τ ) ∈ Cn−1 ×H|zi ̸= zj }descends to M . For Riemann-Wirtinger integrals, the differential operator ∇M/B is simply∇ (see [66] appendix B for the derivation in the η = 0 case).22 It is also important to notethat we cannot throw away ∇M/B (∂η f ) even though it is a total covariant derivative because∂η f does not have the allowed multiplicative monodromy.
To motivate the form of the Gauss-Manin differentials above, we examine how theconnection on M transforms under the B-cycle. We let d̃ be the exterior derivative on
M and set ω̃ = d̃ log(u). This is the obvious uplift of the connection on M to M. The
B-cycle monodromy of ω̃ isω̃ = d̃ log(u)−→ ω̃ + d̃ log e2πis1A τ +2πi

Pn



n
X

!

s z
i=2 1i iz1 →z1 +τ= ω̃ + 2πi d̃ s1A τ +s1i zi= ω̃ + 2πi d̃η,

(E.3)

i=2

When η = 0, the above connection is doubly periodic. Next, we compute the B-cycle
˜ = d̃ + ω̃∧. Explicitly,
monodromy of the image of the covariant derivative on M: ∇˜ F (z1a , η) d̃z1
∇
=e

−2πiη



−→

z1 →z1 +τ

˜ + 2πi d̃η∧
∇





(∇τ − ∇1 )F (z1a , η) d̃τ ∧ d̃z1 +e−2πiη F (z1a , η) (d̃z1 + d̃τ ) ,nX∇i F (z1a , η) d̃zi ∧ d̃z1 + · · · ,

(E.4)

i=2

where the dots above indicate differential forms that are independent of d̃z1 . By adding1
˜ one obtains a differential form whose− 2πi∇1 ∂η to the dτ -part of the covariant derivative ∇d̃z1 part has the allowed B-cycle monodromy. In fact, the components of this differential
are precisely (E.1) and (E.2). Note that we have made the action of the partial derivatives
on η explicit in (E.1) and (E.2) but left this implicit in (E.4).
The differential equation for the family of Riemann-Wirtinger integrals or Gauss-Manin
connection on the base space B isdext [γ|φa ⟩ = Aab [γ|φb ⟩ ⇐⇒ ∇ext |φa ⟩ = Aab ∧ |φb ⟩22

See also [125] or [126] for the general theory.

– 51 –(E.5)

JHEP07(2024)040

for
a = 2, 3, . . . , n ,1∇ext,τ f dz1 := (∂τ f ) dz1 + (ωτ f ) dz1 + s1A (∂η f ) dz1 −∇(∂η f ) .
2πi M/Bwhere−1
Aab |dzi = Cbc⟨φ̌c |∇ext,i φa ⟩ = −1⟨φ̌b |∇ext,i φa ⟩.
s1b

(E.6)

Breaking this up into the off-diagonal and diagonal parts, the general formula for the
dzc -components of the differential equation are

Aoff-diag
c


ab

=

X

δbd δac s1b F (zbc , η) −

d̸=c

X

δad δbc s1c F (zca , η) ,

"


ab

(E.7)

d̸=c



= δab − s1c (1 − δca ) g (1) (zac ) + δac 2πis1A +X

s1d g (1) (zcd )

#

,

(E.8)

d̸=c

where a, b = 2, . . . n. Similarly, the general formula for the dτ -component is

Aoff-diag
τ







Adiag
τ

(1 − δab )=−s1b ∂η F (zba , η) dτ ,
ab
2πi

(E.9)

 n

δab X
(2)
=
s1c g (zac ) − s1a ∂η F (0, η) dτ ,ab
2πi c=2(E.10)



where
ϑ′′ (η)∂η F (0, η) = 1−
ϑ1 (η) ′
ϑ1 (η) 2

ϑ1 (η)= − (℘(η) + G2 (τ )) ,(E.11)

ϑ′′′ (0)and G2 (τ ) = − 3ϑ1′ (0) is the second Eisenstein series. For example, the dz2 - and dτ -components
1of A for n = 4 are


(1)



(1)

2πis1A +s13 g23 +s14 g24 s13 F (z32 , η) s14 F (z42 , η)


(1)

,
A z2 = −s12 F (z23 , η)s12 g230(1)−s12 F (z24 , η)0s12 g24
P4Aτ =s1c (2)c=2 2 g (z2c )
P

1 

2πi

0
0

0
4
s1c (2)c=2 2 g (z3c )0(E.12)
00P4

s1c (2)c=2 2 g (z4c )s12 ∂η F (0, η) s13 ∂η F (z32 , η) s14 ∂η F (z42 , η)1 −s12 ∂η F (z23 , η) s13 ∂η F (0, η) s14 ∂η F (z43 , η) .
2πi
s12 ∂η F (z24 , η) s13 ∂η F (z34 , η) s14 ∂η F (0, η)(E.13)We have also checked that equations (E.7)–(E.10) match the differential equations obtainedby different methods in [2]. Moreover, our results agree with those obtained using standard
integration-by-parts and the Fay identity [116] once the condition (3.19) is imposed andexpand on this below.
It is interesting that these equations differ from those previously obtained in the mathand physics literature, especially in the context of KZB equations [110, 111, 115, 116]. Since
the authors of [2] have already commented on how their differential equations relate to thosein [110], we focus on the differences in the zj -differential equation of this work and section(3.3.1) of [116]. Imposing momentum conservation on the differential equations of [116]– 52 –JHEP07(2024)040

Adiag
c



yields (E.12) and (E.13) up to one key difference: what we mean by the derivative ∂zi in ourwork is actually a partial derivative at constant s1B , (∂zi )s1B , while the partial derivative ∂zi
of [116] makes no such distinction. Thus, these two derivatives are related by:(∂zi )s1B = ∂zi [117] + s1i ∂η ,(E.14)∂a Ab − ∂b Aa + [Aa , Ab ] = 0 ,(E.15)∂τ Aa − ∂a Aτ + [Aτ , Aa ] = 0 ,(E.16)

for all a, b = 2, . . . , n and a ̸= b. To show integrability, we will also need the help fromthe following identity
(1)

(1)

∂η Fjk + (gij − gik )Fjk + Fji Fik = 0 ,(E.17)

which follows from the O(ξ 0 ) term of (3.13) with (i, j, k) = (2, 1, 3), ζi = ξ and ζj = η − ξ.
Concentrating on the integrability of the dza -components, one finds that



0 = ∂c Ae − ∂e Ac + [Ac , Ae ]ab"

= (1 − δab )

X



(1)

(1)

− (δac δbe − δae δbc ) s1b s1d ∂η Fba + (gdb − gda )Fba + Fda Fbdd̸=c,e(E.18)+ (δac δbd − δbc δad ) s1b s1e

− (δae δbd − δbe δad ) s1b s1c



(1)
(1)
∂η Fba + (geb − gea)Fba + Fbe Fea(1)(1)∂η Fba + (gcb − gca)Fba + Fbc Fca




#

,

where we have introduced the shorthand Fab = F (zab , η). The τ -integrability follows similarly,
but requires more complicated Fay identities.
E.2

The O(α′ ) solution to the differential equations

In this section, we study the α′ expansions of the differential equations (E.7)–(E.10) with n = 4punctures and the resulting solutions. The purpose of this section is to obtain some zero-thorder information about the α′ -expansion of Riemann-Wirtinger integrals; to identify the kindof objects that appear and how they relate to known functions in the elliptic Feynman integral
and loop string amplitude literature. At leading order in the α′ -expansion, the RiemannWirtinger integrals have an analytic structure very similar to the leading α′ -expansion of theanalogous string integrals [116]. However, at higher orders, the two solutions start to diverge.
We start by making the α′ dependence apparent in the Mandelstams by setting sij = α′ s̃ij ,where s̃ij = ki · kj are the new α′ independent Mandelstams. Next, we expand the KroneckerEisenstein functions F (zij , η) in the cohomology basis elements in orders of α′ remembering– 53 –JHEP07(2024)040

where we one has to exchange the punctures z1 and z2 in [116] to convert into the conventions used here. Once we take equation (E.14) into account, we can readily recover thezi −differential equation in our work from [116].
As an important cross-check, we show that the above differential equation is integrable.
P
Writing the connection on kinematic space as A = ni=2 Ai dzi + Aτ dτ , we want to showthat η = −s1B + α′ (s̃1A τ + ni=2 s̃1i zi ). While all of the Mandelstam variables have a true
factor of α′ , it is unclear if s1B should also be assigned a power of α′ . In the following wetake the conservative approach and assume that s1B does not scale with α′ to ensure thatη ̸= 0 since the structure of the cohomology changes in the η → 0 limit.
To sub-leading order in α′ ,
P

"
′



F (zij , η) ≈ F (zij , −s1B ) 1+α s̃1A τ +nXs̃1i zi

h

g

(1)

(zij −s1B )−g(1)(−s1B )

i#.

(E.19)

i=2

Similarly, the kinematic connections (E.7)–(E.10) have an α′ expansion
∞
X

(ℓ)

(α′ )ℓ Ãa ,(E.20)ℓ=1

that does not truncate due to the non-uniform dependence of η on α′ . This means thatthe differential equations are not in α′ -form complicating the expansion of the path ordered
exponential. Still, one can obtain the α′ expanded solution order-by-order where the orderk DEQ has contributions from all previous orders. Explicitly, we havedext I(0)γ = 0,
dext I(1)γ =dext I(k≥1)=γ(E.21)

n
X
i=2
n
X

(1)

(1)

(0)
dzi Ãi · I(0)γ + dτ Ãτ · Iγ ,

dzi

i=2

X

(l)

Ãi · I(m)
+ dτ
γ

l+m=k

(E.22)(l)XÃτ · I(m),γ

(E.23)l+m=kwhere we have set Iγ := α′ [γ|φ⟩. The factor of α′ in the definition of Iγ is there to ensure that
there are no α′ -poles in the series expansion of Iγ for the contours introduced in section 5.
To explore the properties of these functions, we solve the DEQs to the first non-trivial
order in α′ . Here, the leading order approximation of the connection matrices is


Ã(1)
τ


ab

=

 n

δab X
s1c g (2) (zac ) + s1a ∂s1B F (0, −s1B )2πi c=2(1 − δab )+s1b ∂s1B F (zba , −s1B ) ,
2πi






(1)
Ãc=2,...,4=ab

(E.24)#"
Xδbd δac s̃1b F (zbc , −s1B ) −Xδad δbc s̃1c F (zca , −s1B )

d̸=c

d̸=c

"

+ δab − s̃1c (1 − δca ) g(1)(zac ) + δac 2πis̃1A +X

s̃1d g(1)

#(zcd ).

(E.25)d̸=c′ ℓ (ℓ)ℓ=0 (α ) cγ as the boundary value. While the boundary(ℓ)
values for a basis of contours is given in section E.3, we will keep cγ generic in this section∗
We also set Iγ (zi=2,...,n, τ ∗) =P∞

to avoid subtleties with the limit τ → i∞ where the torus degenerates into a nodal sphere.
Obviously, the first term in the solution is simply the leading term of the boundary value:
R
(1) ∗
(0)
(0)
(1) ∗
(0) ∗
Iγ = cγ . Next, we integrate in τ to find Iγ (zi≥2, τ ) = ττ∗ dτ ′ Ãτ (zi≥2, τ ′ ) · Iγ (zi≥2, τ ′ ).

– 54 –JHEP07(2024)040

Aa =Explicitly,nXnXϑ′ (z ∗ , τ ) ηDe (τ ∗ )
(0)
(1) ∗
∗
Iγ,a
(zi≥2 , τ ) = c(0)s1b log ′1 ∗ab ∗+cγ,b s1b F (zba, τ ) + c(1)
γ,a
γ,a ,
ϑ
(z
,
τ
)
η
De (τ )
1
ab
b=2
b=2

"

#

(E.26)

where ηDe (τ ) is the Dedekind eta-function,23
F (z, τ ) =

τ∗

′

′

dτ ∂s1B F (z, −s1B |τ ) =

"

Z τ
τ∗

dτ

1

′

s21B

−

∞
X

#

(k − 1)g(k)′(z, τ )(−s1B )

k−2

,

k=2

∞
h
i
τ − τ∗ X
k−2
k−1(k−1) ∗(k−1) ∗
∗
,
τ
)
−
Ω
(z
,
τ
)
,
−
(−s
)
(2πi)
Ω
(z
1B
ba
ba
s21B
k=2

(E.27)and Ω(k) (z, τ ) are the known primitives of the g (k) (z, τ ) [127]g (k) (z, τ ) = (2πi)k−1 ∂z Ω(k) (z, τ ) =(2πi)k−1∂τ Ω(k−1) (z, τ ) .
k−1(E.28)The primitives Ω(k) are the symbol letters of elliptic multiple polylogarithms (eMPLs). In
fact, the Ω(k) are actually depth-1 eMPLs themselves. It is also important to note that
F (0, τ ) is well-defined since ∂s1B F (0, −s1B ) appears in the differential equations. To see this,observe that the primitives are finite at the origin since the g (k≥2) (0, τ ) < ∞ are finite at theorigin. Moreover, note that the above expression for F only makes sense if the differenceτ − τ ∗ is finite otherwise one has to apply the tangential base point regularization of [128]Rτand set i∞dτ ′ = τ . However, we find boundary values in section E.3, which ensure that theτ -integral is finite without applying tangential base point regularization.
Once a boundary value is known at finite τ or when the τ integration converges, thezi -integrals are readily evaluated in terms of familiar eMPLs




(1)
Iγ,a
(zi≥2 , τ ) = 2πis̃1A Γ̃ ( 00 ; za −za∗ |τ ) +

X

h

i

∗
s̃1b Γ̃ ( 10 ; zab |τ ) − Γ̃ ( 10 ; zab|τ )  c(0)
γ,a

b̸=a

+

X

(0)
s̃1b cγ,b∗(1) ∗[Φ (zab ; s1B |τ ) − Φ (zab; s1B |τ )] + Iγ,a(zi≥2 , τ ) ,(E.29)b̸=awhere we define the z-integral of the Kronecker-Eisenstein function to beΦ (zij ; s1B |τ ) :=

X

sk−1
1B Γ̃

k ; z |τ
0 ij



,

(E.30)

k≥0

and Γ̃ ( 00 ; za −za∗ |τ ) = za −za∗ . Note that this expression may only be well defined after shuffle
regularization [108].
(1)
Unsurprisingly, the first α′ correction, Iγ , has a functional form very similar to thatfound in [116] where there was no constraint on η. However, at order O(α′2 ) we start toP
see integrals with powers of s̃1A τ + ni=2 s̃1i zi in the integrand and terms with different(1)transcendentality start to mix. Thus, we expect that only the leading term Iγ is actually
representative of the analogous string integral.
23

The subscript on the Dedekind eta-function is to differentiate from the variable η = s1A τ +– 55 –

Pni=2

s1i zi .

JHEP07(2024)040=

Z τ

Unfortunately, twisted (co)homology seems to force the condition (3.19) on us. To getcloser to real string integrals, we must find a way to understand the (co)homology for unconstrained η (3.19). However, at the level of the differential equation, we can always shift the ηderivatives similarly to (E.14) to define differential operators that are unconstrained by (3.19).
E.3

Boundary valuesϑ1 (z1a + η0 + s1A τ )ϑ1 (η0 + s1A τ )ϑ′1 (0)ϑ1 (z1a )−→ eπiz1a

0 < Re(s1A ) < 1 ,

for

τ →i∞

(E.31)

1,
τ →i∞ sin(πz1a )−→(E.32)

we find that the τ → i∞ limit of the Riemann-Wirtinger integrand becomes
dw1
,
w1a

u F (z1a , η) dz1 −→ utree
τ →i∞(E.33)

where wi = e2πizi are the new coordinates on the sphere andnYutree = w1s1As− 1i s1i
wi 2 w1i!

,(E.34)

i=2

is the analogous twist for tree-level string integrals. Above, η0 = ni=2 s1i zi − s1B is the τ independent part of η and drops out of all quantities in the limit. Equations (E.33) and (E.34)
P

s

− 1A

are the n-point generalization of the analogous formulas in [65] where the factors of wi 2 are
missing. Like [65], we assume that 0 < Re(s1A ) < 1 since this leads to convergent integrals.
Just from the integrand, it is easy to see that the τ → i∞ limit results in a linear combinationof generalized Lauricella-D hypergeometric functions (see figure 15).
The contours γ2j are the easiest to describe since the path from z2 to zj on the torus
maps to an analogous path from w2 to wj with no phase factors. Gauge fixing z2 = 0, we find[γ2j |φk ⟩i∞ := lim [γ2j |φa ⟩ =
τ →i∞

Z wj

utree
w2 =1

dw1
.
w1a

(E.35)On the other hand, the A- and B-cycle integrals become[γ2A |φk ⟩i∞ := lim [γ2j |φa ⟩ = (1 − e

2πis1Aτ →i∞[γ2B |φk ⟩i∞ := lim [γ2j |φa ⟩ =
τ →i∞

utree

)
w2 =1

Z 0

utree
w2 =1

Z ∞

dw1
,
w1a

dw1
− e2πis1B
w1a– 56 –(E.36)

Z ∞

utree
w2 =1

dw1
.
w1a

(E.37)JHEP07(2024)040

In this section, we show that the τ → i∞ limit of the Riemann-Wirtinger family corresponds
to generalized Lauricella-D hypergeometric functions. This degeneration to a nodal sphere
yields one method for computing the boundary values for the differential equations derived
in section E.2. For n = 3, this degeneration can be found in [65]. However, we fix a typo
there and provide an n-point generalization that has been numerically verified. Finally, using
a neat trick, we also provide boundary conditions at finite τ for all but the A- and B-cycles.
To understand how to compute this degenerate limit, we need to know the form of
the integrand in the τ → i∞ limit and how the cycles on the torus map to cycles on the
nodal sphere. Using

wn
··
·

w4
w3
∞

1

Figure 15. The branch choice for the tree-level twist (Koba-Nielsen) factor given by (E.34). The
branch choice in the fundamental domain of the torus maps to the red dotted line connecting the marked
points on the rational nodal curve parameterized by w = e2πiz . The A- and B-cycle discontinuities
are captured by the branch cut from 0 to ∞ denoted by the purple dashed line.

Note that the A- and B-cycle degenerations come with explicit factors of the A- and B-cycle
monodromies even though the twist utree knows nothing about s1B .
The phase factor and contour on the nodal sphere for the A-cycle integral (E.36) isfairly straightforward to derive. The path from z1 = 0 to z1 = 1 on the torus maps to thecontour from arg(w1 ) = 0 to arg(w1 ) = 2π with |w1 | = 1 on the nodal sphere. Moreover, thecorresponding punctures are all inside the unit circle since Im zi ≥ 0
τ

wn
··
·

zn

w4

zn−1

[γ2A | =

z5

·

w3

−→

··

0

τ →i∞

∞.

1

(E.38)z4z3
z2 = 01

We can then deform the contour to be: ∞ + i0− → 1 → ∞ + i0+

wn

w3
0

1

·

·

w4

τ →i∞

wn
··

··

[γ2A | −→
2πis1A = (1 − e
)


∞



w4
w3
0

1




,


∞

(E.39)

where the phase comes from crossing the purple branch cut. Equation (E.39) is precisely
the contour in the boundary value (E.36)!
On the other hand, the degeneration of the B-cycle is more subtle since the τ → i∞limit of a cycle only makes sense for paths whose imaginary part is much smaller than |τ |.

– 57 –JHEP07(2024)0400

Thus, we cannot directly map the B-cycle to the nodal sphere. Instead, we define a new paththat includes the B-cycle and can be deformed to one with imaginary part much smaller
than |τ |. To this end, we consider the following linear combination:[γ2B | + [γ2A |z1 →z1 +τ − [γ2B |z1 →z1 +1 = (1 − e2πis1A )[γ2B | + e2πis1B [γ2A | ,
τ

τ

=
z5

·

(E.40)

zn

zn
zn−1

zn−1

=

··

·

z5

.

··

z3

z2 = 0z2 = 0

1

1Then, the above path maps to the nodal sphere as follows
wn
··
·

w4

[γ2B | + [γ2A |z1 →z1 +τ − [γ2B |z1 →z1 +1 −→

.

w3

τ →i∞
0

(E.41)∞

1

Accounting for the phase from crossing the branch cut, we find



)





 (E.42)



·

τ →i∞

··

[γ2B | + [γ2A |z1 →z1 +τ − [γ2B |z1 →z1 +1 −→ = (1 − e

2πis1A



wn

w4
w3
0

∞

1

Combining (E.40) and (E.42) and inserting (E.39), we obtain the contour in (E.37).
As a sanity check, we show that these degenerate integrals satisfy the over-completeness
relation (5.6). Taking the τ → i∞ limit of (5.6) yields
lim (5.6) = (1 − e2πis1A )

τ →i∞

Z ∞

utree
0

Z

=
C∞

utree

n
dw1 X
−
e−2πi(s12 +···+s1j ) (1 − e2πis1j )[γ2j |φa ⟩i∞ ,w1a j=3ndw1 X−
e−2πi(s12 +···+s1j ) (1 − e2πis1j )[γ2j |φa ⟩i∞ ,w1a j=3(E.43)

where all of the dependence on the B-cycle monodromy has dropped out and we have identified
the first term with the contour C∞ in figure 16. Next, we note thatnXe−2πi(s12 +···+s1j )(1 − e2πis1j)[γ2j |φa ⟩i∞ =

j=3

=

n 
X
j=3
n 
X

1−e

Pj−1

 Z wjPn

 Z wj−2πis
k=2 1k

wj−1

1−e

2πi

k=j

Z
Cn

– 58 –

s1k

utree
wj−1

j=3

=−

utree

utree

dw1
.
w1a

dw1
w1a

dw1
w1a
(E.44)JHEP07(2024)040

z4

z4
z3

0

0

C∞
C0
Cn

wn
1

w3

w4

1

w3

w4

∞

Figure 16. Contour deformation argument for the over-completness of cycles on the sphere. Via
Cauchy’s theorem, the integral over the contour C0 vanishes. Moreover, C0 can be deformed into thecontours C∞ + Cn , which can further be decomposed into the contours of figure 15. Thus, one finds a
linear relation between the contours of figure 15.

Putting the last two equations together, yields
Zlim (5.6) =τ →i∞

C∞ +Cn

utree

dw1
=
w1a

Z
C0

utree

dw1
=0
w1a

(E.45)

since the contour C0 = C∞ + Cn in figure 16 is clearly contractible with no singularities inside
and thus the corresponding integral vanishes due to Cauchy.
The drawback to equations (E.35)–(E.37) is that one has to be careful when propagating
the boundary value at τ = i∞ to finite τ since equation (E.26) is naively divergent at τ = i∞.
Fortunately, we can avoid the complication of regulating the τ -integration for the γ2j cycles
because we can find a boundary value at finite τ by taking the zi to a special configuration!
By taking all of the punctures to the origin, the τ -dependence drops out. However, thislimit must be taken carefully using regularized objects to get a finite answer. In the end, we
find the following prescription for the boundary value where all punctures are sent to the origin
[γ2i |φa ⟩0 := Yk̸=2,ilim (−2πizk )zk →0−s1k 

lim (−2πizi )zi →0−s12 −s1iZdz1 u F (z1a , η|τ ) . (E.46)
γ2i

In the above, we set z1 = xzi and the integration region is x ∈ [0, 1] in x-space runs from 0
to 1. It is also important to note that the limit zi → 0 is taken before all others.
One can think of the prefactors in (E.46) as defining a regularized limit consistent withthe shuffle regularisation of eMPLs [108, 115]. The eMPL of weight and length one makes anappearance in the twist because it is simply related to the log of the Jacobi theta function
and its regularization is well understood. The unregulated eMPL isΓ̃ ( 10 ; z|τ ) = lim
ϵ→0

Z z

dw g (1) (w; τ ) = Ω(1) (z, τ ) − lim Ω(1) (ϵ, τ ) ,ϵ→0ϵ(E.47)

where Ω(1) (z, τ ) = log (ϑ1 (z|τ )/ηDe (τ )) (recall from section E.2 that ηDe is the Dedekind eta
function and Ω(1) is an elliptic symbol letter [127]). As one can clearly see, this is divergent
for all z1 except z1 = 0 where it vanishes. For any z1 ̸= 0, one needs to regularize Γ̃ ( 10 ; z|τ )– 59 –JHEP07(2024)040∞

wnin order to make sense of the lower integration boundary. The shuffle regularized eMPL is
obtained by subtracting off the logarithmic singularity at the lower boundary [108]Γ̃reg ( 10 ; z|τ ) := lim
ϵ→0

Z z

dw g

(1)

(w; τ ) + log(1 − e

2πiϵ



)

ϵ

= log(1 − e2πiz ) − πiz + 4π

1
(1 − cos(2πkz))eπiklτ .
2πk
k,l>0
X

(E.48)[γ2i |φa ⟩ =Z

dz1 e2πis1A z1 exp
γ2i

X
n



s1j Γ̃reg ( 10 ; z1 − zj |τ ) F (z1 − za , η|τ ) ,(E.49)

j=2

where momentum conservation ensures that additional “constants” introduced from theregularization drop out. The prefactors in (E.46) can now be understood as canceling the nonanalytic behavior of Γ̃reg ( 10 ; z|τ ) for z ∼ 0. Since Γ̃reg ( 10 ; z1 |τ ) and Γ̃reg ( 10 ; z1 − zi |τ ) diverge
at both the boundaries of γ2i (z1 = 0 and z1 = zi ), we need a factor of (−2πizi )−s12 −s1i
to render the zi → 0 limit finite. Similarly, Γ̃reg ( 10 ; z1 − zk |τ ) for k ̸= i diverges at the
integration boundary z1 = 0. Therefore, we need a factor of (−2πizk )−s1k for all k ̸= 2, i.
Another way to take the limit is to use (E.47) when z ∼ 0 and (E.48) otherwise. To see
how this works in practice, consider the boundary value [γ24 |φa ⟩0 when n = 4. Taking thelimit z4 → 0 is subtle because the integration contour is being “squished” to a neighborhoodof 0 and the Kronecker-Eisenstein functions diverges when a = 2 and a = 4. Making thesubstitution z1 = xz4 , yields
1
F (z1 , η|τ )
x




φ = F (z1 − z3 , η|τ ) dz1 =  0  dx + O(z4 ) .
1− 1−x
F (z1 − z4 , η|τ )









(E.50)

Next, we expand the twist around z4 = 0. Since the regulated eMPLs Γ̃reg ( 10 ; z1 |τ ) andΓ̃reg ( 10 ; z1 − z4 |τ ) diverge at both the boundaries of γ24 , we use (E.47) to define the limitϑ1 (xz4 )= log x ,
ϑ1 (ϵ)
ϑ1 ((x − 1)z4 )lim Γ̃ ( 10 ; z1 − z4 |τ ) = lim lim log= log(x − 1) ,z4 →0
z4 →0 ϵ→0
ϑ1 (ϵ)lim Γ̃ ( 10 ; z1 |τ ) = lim lim log

z4 →0

z4 →0 ϵ→0(E.51)(E.52)

where we have set ϵ = z4 andlim Γ̃reg ( 10 ; z1 − z3 |τ ) = Γ̃reg ( 10 ; −z3 |τ ) .

z4 →0(E.53)

Using (E.47) again, the z3 → 0 limit of the only remaining eMPL becomeslim Γ̃ ( 10 ; −z3 |τ ) = lim lim log

z3 →0

z3 →0 ϵ→0

– 60 –

ϑ1 (−z3 )= log(−1) = iπ ,
ϑ1 (ϵ)(E.54)JHEP07(2024)040

Now, the regularized eMPL has a logarithmic singularity at z = 0, Γ̃reg ( 10 ; z ∼ 0|τ ) ∼log(−2πiz), but is finite everywhere else. Writing the Riemann-Wirtinger integral in termsof the regulated eMPL, we find

where we have set ϵ = z3 . Putting everything together, we find the finite τ boundary value[γ24 |φ⟩0 =

Z 1

1
x



dx (−1)−s12 xs12 (1 − x)s14  0 
0
1
− 1−x




 1 
Γ(1 + s12 )Γ(1 + s14 )  s12 = (−1)−s12 0 .

Γ(1 + s12 + s14 )(E.55)− s114

 1



s12



 0 


 .. 
 . 


 0 
Γ(1
+
s
)Γ(1
+
s
)
12
1i


[γ2i |φ⟩0 = (−1)−s12 1 ,Γ(1 + s12 + s1i ) − s1i 

 0 


 . 
 . 
 . 

(E.56)

0where the second non-trivial entry shows up in the ith row.
Notice that these boundary values are independent of τ ! In fact, one can verify thatthe (E.56) matches (E.35): [γ2j |φa ⟩0 = [γ2j |φa ⟩i∞ . One has to be careful when taking thezi → 0 limit of the hypergeometric functions produced by (E.35) and this equivalence waschecked for n = 3, 4 using the HPL and HypExp Mathematica packages [129, 130]. Moreover,this boundary value conspires to cancel the naive divergence in (E.26) from setting τ ∗ = i∞when integrating the τ -part of differential equation. Substituting this boundary value
into (E.26), we find
∗
(1) ∗
∗
(1)
I(1)
γ2j (zi≥2 , τ ) = Iγ2j (zi≥2 , τ ) = c(E.57)

where c(1) is the order α′ term in the boundary value. When all zi → 0, the log termin (E.26) drops out due to momentum conservation and the boundary value (E.56) ensuresthat the sum over the F ’s vanish.
E.4

The η → 0 limit

The η-dependent differential forms considered in section 4 are useful because they can bethought of as generating integrals where one performs a formal series expansion in η. On
the other hand, true string integrals do not depend on the parameter η. For this reason,
it is interesting to examine the η → 0 limit of the Riemann-Wirtinger integrals. However,unlike true string integrands, the Riemann-Wirtinger integrand must not have B-cycle
monodromies.24 Anything that breaks this condition is not part of our cohomology.
24

The integral over the loop-momentum guarantees the double periodicity of string integrals. Therefore, it
is possible to have string integrands that are not doubly periodic.

– 61 –JHEP07(2024)040

Notice that this is the same boundary value for the analogous integral defined with η
independent of the punctures [116] (up to signs coming from a different convention for the
Mandelstams). Generalisation to arbitrary n is straight forward,(p)It turns out that the spanning set {ξa } of H 1 (M, Lω,η ) (equation (4.7)) has a smoothη → 0 limit that also spans the cohomology when η = 0: H 1 (M, Lω,η=0 ) [1]. Explicitly,
we let ζ (p) = limη→0 ξ (p) and find(p)

ζ1

= dz1 ,

ζp(p) = ∂z1 g (1) (z1p ) dz1 ,(p)ζa≥2 =

h(E.58)ig (1) (z1a ) − g (1) (z1p ) dz1

fora ̸= p.

(p)2πis1A |ζ1 ⟩ +

n
X

(p)

s1j |ζj ⟩ ≃ 0(E.59)j=2
j̸=pthat follows from the η → 0 limit of (4.8). Moreover, in this limit, the defining equationfor η places restrictions on the punctures in the problem
0 = s1A τ +nX

s1i zi − s1B .

(E.60)

i=2

Importantly, for the differential equation, this means that the differentials satisfy the relation
0 = s1A dτ +nX

s1i dzi .

(E.61)

i=2

We can compute the intersection numbers directly in the η ̸= 0 limit case using (4.13) as
before. Of course, this yields the same result as taking the η → 0 limit of (4.18)–(4.20). For
example, at n = 4 with p = 4, the explicit form of the intersection matrix is

(4)

(4)

0




0
⟨ζa |ζb ⟩ 
=

2πi
 0


0

0

s13
s12 s14− s114− s114s12
s13 s142iπs1 −s12 [g (1) (z24 )−g (1) (z34 )]s14 (s14 +1)

2iπs1 +s13 [g (1) (z24 )−g (1) (z34 )]1
s14 +1s14 (s14 +1)1
s14 −12iπs1 +s13 [g (1) (z24 )−g (1) (z34 )] (s14 −1)s14

(1)
(1)
2iπs1 −s12 [g (z24 )−g (z34 )]  ,(s14 −1)s14
(4) (4)⟨ζ4 |ζ4 ⟩2πi(E.62)
where
(4)

(4)

⟨ζ4 |ζ4 ⟩1=− 4π 2 s21 − 2s14 (s12 − s14 ) G2 (τ )2πi(s14 − 1)s14 (s14 + 1)








+ 4iπs1 s12 g (1) (z24 ) + s13 g (1) (z34 ) + s12 s13 g (1) (z24 ) − g (1) (z34 )+ 2s14 s12 g (2) (z24 ) + s13 g (2) (z34 )
(p)

(p)



.

2

(E.63)

Choosing ζp and any set of n − 2 other ζa̸=p forms a basis of the cohomology
1
H (M, Lω,η=0 ). After we choosing a basis, the DEQs can be computed in the usual way. For– 62 –JHEP07(2024)040

Each element of this set has no B-cycle monodromy. Also note that, to have a basis, theelement ζ (p) with a double pole must be included. This spanning set is also subject tothe relation

(p)example, choosing ζa>1 for our basis and choosing to eliminate dz4 using the η = 0 condition,
we find the following dz2 component of the differential equationAζ |dz2 =9X

Ξ i Xi(E.64)

i=1

where the Xi are monomials in the g-functions (including the constant function)Xi ∈ 1, g (1) (z23 ) , g (1) (z34 ) , g (1) (z24 ) , g (2) (z34 ) , g (2) (z24 ) ,


2



, g (1) (z24 ) g (1) (z34 ) , g (1) (z24 )

2 

,

(E.65)and the coefficient matrices are





1 s12
− 2iπs1 ss13 (ss212 −s14 ) 2iπs
s2
12 14

14

2iπs1 s12s214

Ξ1 = 

2iπs1 s12
s214

0

0

s313
s12 s214
s213
s214





Ξ4 =  −

00

Ξ6 = 
0

−

0
−

0

0

0

Ξ8 =  0

0









Ξ3 = 

s213
s214

s2

2s213
s214
2s12 s13s214

12
0 (s14 +1)s14

− s12s2s13
14

s212
s214

0

s212
1 s12 s13
− 4iπss214(1−s14 )s214
s312
2iπs1 s12 (s12 −s13 ) ,− s s2(1−s14 )s214
13 14



0

0



s12 s13s214
2iπs1 s12− (−s14 −1)s14

12
0 s2s
14 +1



0









− ss131 0
12Ξ2 =  1 − ss12
0,
13
0
0 0


4π 2 s2 s
s13 G2 (τ )− (1−s 1 )s122
s14
14 14 
4π 2 s2 s,− s12sG142 (τ ) − (1−s 1 )s122 
14 14 4iπs1 s12 s13(1−s14 )s214
2iπs1 s12 (s12 −s13 ) ,− (1−s )s2

14 14

0

0

Ξ5 = 
0

0



s12
0 s14
+1

0






,




Ξ7 =  0 00(s12 +1)s13
0 0 − s12(1−s
)s2
14

00

2s s213− (1−s12 )s2
14 14 

2s212 s13
,(1−s14 )s214 14s212 (s12 +1)(1−s14 )s214

0


0

0

Ξ9 =  0

0





0

s12 s130 (s14+1)s142s12 s13 
s214
2s2 
− s212 ,
14(E.66)

0




,

s2 (s +1)

13 13
− (1−s
)s2
14

14




s12 s13 (s13 +1) .
(1−s14 )s214 0

It is interesting to note that the η = 0 differential equations are much more complicatedthan their η ̸= 0 cousins. Most of the complexity comes from having to use a form witha double pole in our basis. This translates into double poles in the differential equation.
That is, the double poles coming from the monomials g (1) (z34 )


2

2

, g (1) (z24 ) g (1) (z34 ) andg (1) (z24 ) are not spurious. Connections with higher order poles appear at higher genus
and it is possible to make sense of iterated integrals with such a connection [122, 123].
The η = 0 twisted (co)homology was previously studied in [66]. Here, the authors study
the moduli space of flat tori and elliptic hypergeometric functions using algebro-geometrictechniques similar to those used in this paper. In particular, they compute the n = 2
differential equations (see their appendix B) and construct the double copy in order to obtainan explicit expression for the so-called Veech map. In both cases, we find agreement.

– 63 –JHEP07(2024)040

g (1) (z34 )Open Access. This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY4.0), which permits any use, distribution and reproduction inany medium, provided the original author(s) and source are credited.

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